Abstract:
Embodiments of this invention include computer-implemented mathematical methods to develop software and/or hardware implementations that use wavelet transforms (WT) to pre-process video frames that can then be compressed using a variety of codecs to produce compressed video frames. Such compressed video frames can then be transmitted, decompressed, post-processed using the post-processing methods disclosed in the invention and displayed in their original size and quality using software and/or hardware implementations of embodiments of the invention, thereby producing real-time high-quality reproduction of video sequences. Embodiments of devices that can implement the methods of this invention include mainframe computers, desktop computers, personal computers, laptop computers, tablet computers, wireless computers, television sets, set top boxes, cellular telephones, and computer readable media.

Description:
CLAIM OF PRIORITY 
       [0001]    This application is a Continuation of U.S. application Ser. No. 12/806,153 filed Aug. 6, 2010 (now U.S. Pat. No. 8,031,782 issued Oct. 4, 2011), which is a Continuation-In-Part of U.S. application Ser. No. 12/657,100 now abandoned), which is a Continuation under 35 U.S.C. 111 of PCT International Application PCT/US2009/004879 filed Aug. 26 2009, titled “Systems and Methods for Compression, Transmission and Decompression of Video Codecs,” Angel DeCegama, inventor, which claims priority under 35 U.S.C. 119(e) to U.S. Provisional Patent Application No.: 61/190,585, filed Aug. 29, 2008, entitled: “Improved Methods for Compression Transmission and Decompression of Video Codecs,” Angel DeCegama inventor. The above patent and each of the above applications is herein incorporated fully by reference. 
     
    
     COLOR DRAWINGS 
       [0002]    The patent or application file contains at least one drawing executed in color. Copies of this patent or patent application publication with color drawing(s) will be provided by the Office upon request and payment of the necessary fee. 
       COPYRIGHT NOTICE 
       [0003]    This application contains material that is subject to protection under copyright laws of the United States (18 U.S.C.) and other countries. 
       FIELD OF THE INVENTION 
       [0004]    This invention relates to the field of video compression when it is desired to minimize the amount of bytes needed to reproduce the different video frames with a desired visual quality while reducing the corresponding costs of storage and transmission. Particularly, this invention relates to wavelet transformation methods to improve the performance of any video codec to compress and decompress video frames. 
       BACKGROUND 
       [0005]    Video communications systems are being swamped by increasing amounts of data and current video compression techniques have remained almost stagnant in recent years. New techniques are needed to significantly cut the video data storage and transmission requirements. 
       SUMMARY 
       [0006]    Although video compression, transmission, decompression and display technologies have been under rapid development, many of the methods used are applicable to only a few types of video technologies. Thus, a new problem in the field is how to develop compression and decompression methods that are widely applicable to a variety of video codecs so that computer storage and transmission requirements are minimized. 
         [0007]    To solve this and other problems, I have discovered and developed new methods for rapid compression, decompression and display that are applicable to a wide range of video codecs that significantly reduce computer storage and transmission requirements which maintaining high-quality video output, so that movies, entertainment, live television, and the like can be viewed in real time. 
         [0008]    In some embodiments of this invention, it is possible to reduce data storage and transmission requirements by 50% for a given information content and user quality of service. In other embodiments, reductions of over 70% can be easily achieved, compared to the codec-alone level of compression. Such advantageous effects can be obtained with little or not appreciable loss of visual quality. 
         [0009]    These embodiments of this invention solve important problems in video transmission and storage without any disruption of existing video compression codecs and can be applied without the need for special handling of a codec. Certain embodiments of this invention simply enhance the function of existing codecs by making them perform better with computer-based methods to process each frame before being compressed and after being decompressed. It is not a disruptive technology. It is simply an enabling technology. Aspects of this invention are based on the mathematics of the Wavelet Transform (WT). 
         [0010]    Embodiments of the invention involve pre-processing a video image, decimating the image using WT, taking the decimated WT of a given video frame down several levels and keeping only the low-frequency part at each level. An example of such a “reduction method” is described herein below. Then, a video codec operates in its usual fashion on such a “pre-processed frame” or “reduced frame” to compress the image and further decrease the amount of information that is needed to be stored and transmitted. Then, the data representing the pre-processed frame compressed using a codec can be transmitted with high efficiency because the number of bits of information has been reduced substantially compared to either the original frame, or the frame that had been processed by the codec alone. 
         [0011]      FIG. 1  depicts a schematic drawing  100   a  of a method of this invention. Input Video File  10  is received by a computer input device, and is then pre-processed by Size Reduction method  20  of this invention to produce a Reduced Video File  30 . Reduced Video File  30  is then processed by Codec  40   a,  thereby producing Compressed Video File  50 . Then, Codec  40   b  decompresses Compressed Video File  50  to produce Decompressed Video File  60  being ¼, 1/16, 1/64 etc. of the original size of Input Video File  10 . Application of a post-processing method of Frame Expansion  70  produces Output Video File  80  having a full size image which is then displayed on video monitor  90  or is stored in a memory device  100 . Optionally, Transmission Process  55  can transmit Compressed Video File  50  to a remote location, where it can be decompressed by the codec and post-processed according to methods of this invention, and displayed on a video monitor or stored for future use. 
         [0012]      FIG. 2  depicts a general schematic drawing  200  showing how an original frame is reduced using frame pre-processing methods of this invention. Original Frame  210  is first treated to decimate low frequency (LF) components of the WT  220 , which are kept. High frequency (HF) components  230  are discarded. Then,  220  is then further pre-processed in step  235 , again decimating low frequency (LF) components of the WT  240 , which are kept. High frequency (HF) components  250  are discarded. The process can be repeated as desired through 3, 4, 5, 6, or even more levels. 
         [0013]      FIG. 3  depicts a general schematic drawing  300  showing frame size expansion according to an embodiment of this invention. Reduced Size  310 , is vertically expanded in step  315 , thereby producing Vertically Expanded Frame  320 , which is then horizontally expanded in step  325  producing Horizontally Expanded Frame  330 . Horizontally Expanded Frame  330  is vertically expanded in step  335  thereby producing Vertically Expanded  340 , which is horizontally expanded in step  345 , thereby producing Horizontally Expanded Frame  350 . The process can be repeated as many times as desired through 3, 4, 5, 6 or more levels. 
         [0014]    Pre-processing methods of this invention can result in a size reduction for the frame of ¼ for one level of transformation or 1/16 for two levels of transformation, and so on. This can be done for all frames in a given video sequence. Then the reduced frames are used to create a new video file, for example, in .avi format. It can be appreciated that other file formats can also be used, including for example, OG3, Asf, Quick Time, Real Media, Matroska, DIVX and MP4. This file is then input to any available codec of choice and it is compressed by the codec following its standard procedure to a size which typically ranges from 40% (one level of WT) to less than 20% (two levels or more of WT) of the compressed size obtained without the step of frame size reduction. Such files can be stored and/or transmitted with very significant cost savings. By appropriately interfacing with the codec, such procedure can be carried out frame by frame instead of having to create an intermediate file. 
         [0015]    For decompression of each frame, the codec is used in its normal way and a file (for example, in .avi format) of a size approximately equal to ¼ (one level of WT) or 1/16, 1/64, etc. (two levels or more of WT) of the original uncompressed file size is obtained. The final step is to generate a file where all the frames are full size with the file size being the same as that of the original uncompressed file. The methods and systems to accomplish that without loss of quality with respect to the decompressed frames produced by the codec without the initial frame size reduction are described herein. This step can be accomplished frame by frame without producing the intermediate file, further improving the efficiency of the process. 
         [0016]    It can be appreciated that a series of frames can be pre-processed, compressed, transmitted, decompressed, post-processed and displayed in real time, thereby producing a high quality video, such as a movie or live broadcast. Because the steps in pre-processing, compression, decompression, post-processing can be carried out very rapidly, a reproduced video (e.g., a movie or live broadcast), can be viewed in real time. 
         [0017]    Thus, in certain aspects, this invention provides a system for video image compression and decompression, comprising: 
         [0018]    a first computer module for image frame pre-processing using direct wavelet transformation (WT); 
         [0019]    a video codec; 
         [0020]    a second computer module for image frame post-processing using low-frequency parts of the WT and the last pixel of every row and column of the original image before the WT and 
         [0021]    an output device. 
         [0022]    In other aspects, this invention provides a system wherein said first computer module comprises: 
         [0023]    an input buffer for storing a video image frame; 
         [0024]    a memory device storing instructions for frame pre-processing, wherein said instructions are based upon direct wavelet transformation (WT); 
         [0025]    a processor for implementing said instructions for frame pre-processing, and 
         [0026]    an output. 
         [0027]    In further aspects, this invention includes systems wherein said second computer module comprises: 
         [0028]    an input buffer; 
         [0029]    a memory device storing instructions for frame post-processing, wherein said instructions are based upon using low-frequency parts of the WT plus the last pixel of every row and column of the original image before the WT; 
         [0030]    a processor for implementing said instructions for frame post-processing; and 
         [0031]    an output. 
         [0032]    In still further aspects, this invention provides systems further comprising another storage device for storing a post-processed frame of said video image. 
         [0033]    In other aspects, a system of this invention includes instructions for frame pre-processing using decimated WT and retaining low frequency part of said decimated WT and discarding high-frequency part of the decimated WT. 
         [0034]    In still other aspects, a system of this invention includes instructions for frame post-processing to recreate a post-processed frame by using low-frequency parts of the WT and the last pixel of every row and column of the original image before the WT. 
         [0035]    In other aspects a system of this invention includes instructions for frame post-processing to recreate a full sized post-processed frame by using low-frequency parts of the WT and the last pixel of every row and column of the original image before the WT. 
         [0036]    In further aspects, this invention provides an integrated computer device for pre-processing a video image frame, comprising: 
         [0037]    a computer storage module containing instructions for frame pre-processing according to decimated WT; and 
         [0038]    a processor for processing said decimated WT by retaining low-frequency parts and discarding high-frequency parts. 
         [0039]    In additional aspects, this invention provides an integrated computer device for post-processing a video image frame, comprising: 
         [0040]    a computer storage module containing instructions for frame post-processing using low-frequency parts of the WT and the last pixel of every row and column of the original image before the WT; and 
         [0041]    a processor for processing said computations to re-create a full-size video image. 
         [0042]    In still further aspects, this invention provides a computer readable medium, comprising: 
         [0043]    a medium; and 
         [0044]    instructions thereon to pre-process a video frame using WT. 
         [0045]    In still additional aspects, this invention provides a computer readable medium, comprising: 
         [0046]    a medium; and 
         [0047]    instructions thereon to post-process a reduced video frame to re-create a video frame of original size using low-frequency parts of the WT plus the last pixel of every row and column of the original image before the WT. 
         [0048]    In certain of these above aspects, a computer readable medium is a diskette, compact disk (CD), magnetic tape, paper or punch card. 
         [0049]    In aspects of this invention, the Haar WT is used. 
         [0050]    In other aspects of this invention Daubechies-4, Daubechies-6, Daubechies-8, biorthogonal or asymmetrical wavelets can be used. 
         [0051]    Systems of this invention can provide high-quality reproduction of video images in real time. In some aspects, systems can provide over 50% reduction in storage space. In other aspects, systems can provide over 50% reduction in transmission cost, with little perceptible loss of visual quality. In other aspects, systems of this invention can provide 70% to 80% reduction in storage costs. In additional aspects, systems of this invention can provide 70% to 80% decrease in transmission costs, with little or no perceptible loss of visual quality compared to codec alone compression, transmission and decompression. 
         [0052]    In other aspects, this invention provides a method for producing a video image of an object, comprising the steps: 
         [0053]    a. providing a digitized image frame of said object; 
         [0054]    b. providing a decimated WT of said digitized image frame; 
         [0055]    c. discarding high-frequency components of said decimated WT thereby producing a pre-processed frame; 
         [0056]    d. compressing said pre-processed frame using a video codec producing a compressed video frame; 
         [0057]    e. decompressing said compressed video frame using said codec; and 
         [0058]    f. recreating a full sized image of said frame using post-processing using low-frequency parts of the WT and the last pixel of every row and column of the original image before the WT. 
         [0059]    In other aspects, a method of this invention provides after step d above, a step of transmitting said compressed image to a remote location. 
         [0060]    In still other aspects, this invention provides a method, further comprising displaying said full sized image on a video monitor. 
         [0061]    In further aspects, this invention provides a method, wherein said step of pre-processing includes a single level frame size reduction according to the following steps:
       For every row in input frame pFrameIn       
 
         [0000]    
       
         
               
             
           
               
                   
               
             
             
               
                 { 
               
               
                  ○ Compute the decimated low-frequency Haar WT for consecutive 
               
               
                   pixels 
               
               
                  ○ Store in half the width of pFrameIn the resulting values 
               
               
                  ○ At end of row, store the last pixel unchanged 
               
               
                 } 
               
               
                   
               
             
          
         
       
       
         
           
             For every 2 consecutive rows in modified pFrameIn 
           
         
       
     
         [0000]                                {        ○ Compute the decimated low frequency Haar WT of corresponding         pixels in the 2 rows column by column        ○ Store the resulting values in output frame pFrameOut        ○ Advance row position by 2       }                    
Store the last row of the modified pFrameIn in pFrameOut last row.
 
         [0064]    In certain of these aspects, this invention provides a method, further comprising a second level pre-processing step. 
         [0065]    In other of these aspects, this invention provides a method, further comprising a second level step and a third level step of pre-processing. 
         [0066]    In additional aspects, this invention provides a method, wherein said step of post-processing includes a first level frame size expansion according to the following steps:
       Copy last row of input frame pFrameIn into intermediate output frame Img with the same pixels per row as pFrameIn and double the number of rows.   For each pixel position of pFrameIn and Img       
 
         [0000]    
       
         
               
             
           
               
                   
               
             
             
               
                 { 
               
               
                  ○ Calculate the new pixel values of 2 rows of Img starting at the bottom 
               
               
                   and moving up column by column according to the formulas 
               
               
                   y 2n  = (2x n  + y 2n+1 )/3 and 
               
               
                   y 2n−1  = (4x n  − y 2n+1 )/3 
               
               
                   (the x&#39;s represent pixels of pFrameIn and the y&#39;s represent pixels of 
               
               
                   Img) 
               
               
                  ○ Store the calculated pixels in Img 
               
               
                 } 
               
               
                   
               
             
          
         
       
       
         
           
             For every row of Img 
           
         
       
     
         [0000]    
       
         
               
             
           
               
                   
               
             
             
               
                 { 
               
               
                  ○ Start with the last pixel and store it in the last pixel of the 
               
               
                   corresponding row of the output frame pFrameOut 
               
               
                  ○ Compute the pixels of the rest of the row from right to left according 
               
               
                   to the above formulas where now the x&#39;s represent the pixels of Img 
               
               
                   and the y&#39;s represent the pixels of pFrameOut 
               
               
                  ○ Store the calculated pixels in pFrameOut 
               
               
                 } 
               
               
                   
               
             
          
         
       
     
         [0070]    In certain of these aspects, this invention includes a method, further comprising the step of a second level frame size expansion. 
         [0071]    In other of these aspects, this invention includes a method, further comprising a second level step and a third level step of frame size expansion. 
         [0072]    In certain embodiments, this invention includes a method, wherein a codec is selected from the group consisting of MPEG-4, H264, VC-1, and DivX. 
         [0073]    In other embodiments, this invention includes a method, wherein said codec is a wavelet-based codec or any other kind of codec. 
         [0074]    In certain aspects, methods of this invention can provide high-quality video reproduction of video images in real time. In some aspects, methods can provide over 50% reduction in storage space. In other aspects, methods can provide over 50% reduction in transmission cost, with little perceptible loss of visual quality. In other aspects, the reduction in storage space may be over 70% to 80%, with little reduction in video quality compared to codec-alone compression, transmission and decompression. 
         [0075]    There are a multitude of applications for video compression in areas such as security, distant learning, videoconferencing, entertainment and telemedicine. 
     
    
     
       BRIEF DESCRIPTION OF THE FIGURES 
         [0076]    This invention is described with reference to specific embodiments thereof. Other features of this invention can be appreciated in view of the Figure, in which: 
           [0077]      FIG. 1  shows a video compression system using the invention in conjunction with a video codec. 
           [0078]      FIG. 2  shows a frame size reduction step of an embodiment of the invention that reduces the length of the columns and rows of an original frame by applying a decimated WT repeatedly level after level of reduction. 
           [0079]      FIG. 3  shows an expansion of an embodiment of the invention of columns and rows to recover their original lengths level after level. 
           [0080]      FIG. 4  shows a process of frame size reduction of an embodiment of the invention by the application of a low-frequency Haar Wavelet Filter to the rows and columns of a video frame. 
           [0081]      FIG. 5  shows a process of an embodiment of the invention for recovering the original size of a video frame by the application of a recovery algorithm of this invention to a previously reduced frame. 
           [0082]      FIG. 6  shows a process of an embodiment of the invention for reducing the frame size one more level compared to the process shown in  FIG. 4 . 
           [0083]      FIG. 7  shows expansion by one-level of a two-level sized reduction of an embodiment of the invention. The original full size can then be recovered by the process of  FIG. 5 . 
           [0084]      FIG. 8  shows a process of an embodiment of the invention for going from 2-Level frame size reduction to 3-Level frame size reduction. 
           [0085]      FIG. 9  shows a process of an embodiment of this invention for frame size expansion from 3-Level size reduction to 2-Level size reduction. Additional levels of expansion can be handled similarly. 
           [0086]      FIG. 10  shows a photograph of a video frame, of 1080i video compressed by H264 to 6 Mbps and then decompressed and displayed. 
           [0087]      FIG. 11  shows a photograph of the codec-processed image of the photograph shown in  FIG. 10  compressed by H264 with pre-processing according to an embodiment of this invention to 3 Mbps and then decompressed, post-processed and displayed. 
           [0088]      FIG. 12  shows a photograph of the frame shown in  FIG. 10  compressed by H264 with pre-processing according to an embodiment of this invention to 1.5 Mbps and then decompressed, post-processed and displayed. 
           [0089]      FIG. 13  shows a photograph of a frame of 1080i video compressed by VC-1 to 6 Mbps and then decompressed and displayed. 
           [0090]      FIG. 14  shows a photograph of the same frame as in  FIG. 13  compressed by VC-1 after pre-processing according to an embodiment of this invention to 3 Mbps and then decompressed, post-processed and displayed. 
           [0091]      FIG. 15  shows a photograph of the same frame as in  FIG. 12  compressed by VC-1 after pre-processing according to an embodiment of this invention to 1.5 Mbps and then decompressed, post-processed and displayed. 
           [0092]      FIG. 16  shows a photograph of a frame of 1080i video compressed by H264 to 6 Mbps and then decompressed and displayed. 
           [0093]      FIG. 17  shows a photograph of the frame shown in  FIG. 16  compressed by H264 with pre-processing according to an embodiment of this invention to 3 Mbps and then decompressed, post-processed and displayed. 
           [0094]      FIG. 18  shows a photograph of the same frame as shown in  FIG. 16  compressed by H264 and pre-processed according to an embodiment of this invention to 1.5 Mbps and then decompressed, post-processed and displayed. 
           [0095]      FIG. 19  shows a photograph of a frame of 1080i video compressed by H264 to 6 Mbps and then decompressed and displayed. 
           [0096]      FIG. 20  shows a photograph of the same frame as in  FIG. 19  compressed by H264 and pre-processed according to an embodiment of this invention to 3 Mbps and then decompressed, post-processed and displayed. 
           [0097]      FIG. 21  shows a photograph of the same frame as in  FIG. 19  compressed by H264 and pre-processed according to an embodiment of this invention to 1.5 Mbps and then decompressed, post-processed and displayed. 
           [0098]      FIG. 22  shows a schematic diagram of a system of this invention to implement frame pre-processing and post-processing methods according to embodiments of this invention. 
           [0099]      FIGS. 23A and 23B  depict schematic drawings of pre-processing ( FIG. 23A ) and posts-processing ( FIG. 23B ) devices of this invention. 
           [0100]      FIGS. 24A and 24B  depict schematic drawings of computer readable devices containing instructions for pre-processing ( FIG. 24A ) and post-processing ( FIG. 24B ) of this invention. 
           [0101]      FIG. 25  is a graphical depiction of a method for calculating the inverse wavelet transform (IWT) using Haar wavelets. 
           [0102]      FIG. 26  is a graphical depiction of a method for calculating wavelet transform (WT) low frequency (LF) and high frequency (HF) values process using Haar wavelets. 
       
    
    
     DETAILED DESCRIPTION 
       [0103]    Aspects of this invention are based on the mathematics of the Wavelet Transform (WT). Embodiments of the invention involve taking the decimated WT of a given video frame down several levels and keeping only the low-frequency part at each level. Embodiments of this invention include new systems and methods for decreasing the amount of space needed to store electronic files containing video images. 
         [0104]    In certain embodiments, a frame of a video file is pre-processed by methods and systems of this invention to reduce its size by factors of 4, 16, 64 or even further. Then a video codec is applied to compress the frame of significantly reduced size to produce a compressed file which is significantly smaller than the frame would be without the use of the frame pre-processing. In some embodiments, all frames of a video file can be processed in a similar fashion. Such compressed file can then be stored and/or transmitted before decompression. The final step is to recover one or more individual video frames in their original size with comparable quality. This is accomplished by the second part of the invention which is used after a codec decompression step. 
         [0105]    As used herein the term “video image” has the same meaning as “video frame,” and the term “image” has the same meaning as “frame” when used in the context of video information. 
         [0106]    As used herein, the terms “frame pre-processing,” “frame size preprocessing” and “frame size reduction” mean processes where a video image or video frame is reduced in accordance with aspects of this invention prior to encoding (compression) by a codec. 
         [0107]    As used herein, the terms “frame post-processing,” “frame size post-processing” and “frame expansion” mean processes whereby an image decoded by a codec is further expanded according to methods of this invention to produce a high-quality image. 
         [0108]    The term “codec” refers to a computerized method for coding and decoding information, and as applied to this invention, refers to a large number of different technologies, including MPEG-4, 11-264, VC-1 as well as wavelet-based methods for video compression/decompression disclosed in U.S. Pat. No. 7,317,840, herein incorporated fully by reference. 
         [0109]    The term “computer readable medium” or “medium” as applied to a storage device includes diskettes; compact disks (CDs) magnetic tape, paper, flash drive, punch cards or other physical embodiments containing instructions thereon that can be retrieved by a computer device and implemented using a special purpose computer programmed to operate according to methods of this invention. A “non-physical medium” includes signals which can be received by a computer system and stored and implemented by a computer processor. 
         [0110]    Embodiments of the present invention are described with reference to flowchart illustrations or pseudocode. These methods and systems can also be implemented as computer program products. In this regard, each block or step of a flowchart, pseudocode or computer code, and combinations of blocks (and/or steps) in a flowchart, pseudocode or computer code can be implemented by various means, such as hardware, firmware, and/or software including one or more computer program instructions embodied in computer-readable program code logic. As will be appreciated, any such computer program instructions may be loaded onto a computer, including a general purpose computer or a special purpose computer, or other programmable processing apparatus to produce a machine, such that the computer program instructions which execute on the computer or other programmable processing apparatus implement the functions specified in the block(s) of the flowchart(s), pseudocode or computer code. 
         [0111]    Accordingly, blocks of the flowcharts, pseudocode or computer code support combinations of methods for performing the specified functions, combinations of steps for performing the specified functions, and computer program instructions, such as embodied in computer-readable program code logic for performing the specified functions. It will also be understood that each block of the flowchart illustrations, and combinations of blocks in the flowchart illustrations, can be implemented by special purpose hardware-based computer systems which perform the specified functions or steps, or combinations of special purpose hardware and computer-readable program code logic means. 
         [0112]    Furthermore, these computer program instructions, such as embodied in computer-readable program code logic, may also be stored in a computer-readable memory that can direct a computer or other programmable processing apparatus to function in a particular manner, such that the instructions stored in the computer-readable memory produce an article of manufacture including instructions which implement the function specified in the block(s) of the flowchart(s), pseudocode or computer code. The computer program instructions may also be loaded onto a computer or other programmable processing apparatus to cause a series of operational steps to be performed on the computer or other programmable processing apparatus to produce a computer-implemented process such that the instructions which execute on the computer or other programmable processing apparatus provide steps for implementing the functions specified in the block(s) of the flowchart(s). 
       Embodiments of the Invention 
       [0113]    In embodiments of this invention, a feature is the ability to recreate a given image or video frame from the low-frequency component of its WT which can be ¼, 1/16, 1/64 etc. the size of the original image or video frame. This can be done precisely by applying the mathematics of direct wavelet transformation (WT) and the computations described be low. 
         [0114]    Take, for example, the Haar wavelet. The direct Haar WT low-frequency coefficients are a 2 =0.5 and a 1 =0.5 and the high-frequency coefficients are b 2 =+0.5 and b 1 =−0.5. The IWT low-frequency coefficients are aa 2 =1.0 and aa 1 =1.0 and the IWT high-frequency coefficients are bb 1 =−1.0 and bb 1 =+1.0. The WT is applied to the individual pixel rows and columns of a given image or video frame. This is done separately for the luminance (Y) and chrominance (U, V) components of the different pixels of each row and column. It can also be done for the R, G and B planes. 
         [0115]    Let&#39;s define a set of y i s to constitute the different values of one such component of a given row or column of an image or video frame. Let&#39;s also define a set of x i s to be the corresponding WT low-frequency values and a set pf z i s to be the corresponding WT high-frequency values. 
         [0116]    We can then write for the Haar WT (with decimation and no wraparound): 
         [0000]    
       
      
       X 
       0 
       =a 
       2 
       y 
       0 
       +a 
       1 
       y 
       1  
       Z 
       0 
       =b 
       2 
       y 
       0 
       +b 
       1 
       y 
       1  
      
     
         [0000]    
       
      
       X 
       1 
       =a 
       2 
       y 
       2 
       +a 
       1 
       y 
       3  
       Z 
       1 
       =b 
       2 
       y 
       2 
       +b 
       1 
       y 
       3  
      
     
         [0000]        X   n   =a   2   y   2n   +a   1   y   2n+1    Z   n   =b 2 y   2n   +b   1   y   2n+1    
         [0117]    The procedure for these calculations is shown in  FIG. 26 . Knowing both the x i s and the z i s we can reconstruct exactly the y i s by calculating the corresponding IWT. 
         [0000]    
       
         
           
             
               
                 
                   
                     y 
                     0 
                   
                   = 
                     
                    
                   
                     
                       
                         aa 
                         2 
                       
                        
                       
                         x 
                         0 
                       
                     
                     + 
                     
                       
                         bb 
                         2 
                       
                        
                       
                         z 
                         0 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       x 
                       0 
                     
                     - 
                     
                       z 
                       0 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     y 
                     1 
                   
                   = 
                     
                    
                   
                     
                       
                         aa 
                         1 
                       
                        
                       
                         x 
                         1 
                       
                     
                     + 
                     
                       
                         bb 
                         1 
                       
                        
                       
                         z 
                         1 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       x 
                       1 
                     
                     + 
                     
                       z 
                       1 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     y 
                     2 
                   
                   = 
                     
                    
                   
                     
                       
                         aa 
                         2 
                       
                        
                       
                         x 
                         1 
                       
                     
                     + 
                     
                       
                         bb 
                         2 
                       
                        
                       
                         z 
                         1 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       x 
                       1 
                     
                     - 
                     
                       z 
                       1 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     y 
                     3 
                   
                   = 
                     
                    
                   
                     
                       
                         aa 
                         1 
                       
                        
                       
                         x 
                         2 
                       
                     
                     + 
                     
                       
                         bb 
                         1 
                       
                        
                       
                         z 
                         2 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       x 
                       2 
                     
                     + 
                     
                       z 
                       2 
                     
                   
                 
               
             
           
         
       
       
         
           ⋯ 
         
       
       
         
           
             
               y 
               
                 
                   2 
                    
                   
                       
                   
                    
                   n 
                 
                 - 
                 1 
               
             
             = 
             
               
                 x 
                 n 
               
               + 
               
                 z 
                 n 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     y 
                     zn 
                   
                   = 
                     
                    
                   
                     
                       
                         aa 
                         2 
                       
                        
                       
                         x 
                         n 
                       
                     
                     + 
                     
                       
                         bb 
                         2 
                       
                        
                       
                         z 
                         n 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       x 
                       n 
                     
                     - 
                     
                       
                         z 
                         n 
                       
                       . 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               y 
               
                 
                   2 
                    
                   
                       
                   
                    
                   n 
                 
                 + 
                 1 
               
             
              
             
                 
             
              
             is 
              
             
                 
             
              
             given 
              
             
                 
             
              
             and 
              
             
                 
             
              
             need 
              
             
                 
             
              
             not 
              
             
                 
             
              
             be 
              
             
                 
             
              
             
               calculated 
               . 
             
           
         
       
     
         [0118]    The above equations represent the IWT process shown in  FIG. 25 . Assuming that y 2n+1  is known, we can then write: 
         [0000]    
       
         
           
             
               y 
               
                 2 
                  
                 n 
               
             
             = 
             
               
                 
                   x 
                   n 
                 
                 - 
                 
                   
                     b 
                     2 
                   
                    
                   
                     y 
                     
                       2 
                        
                       n 
                     
                   
                 
                 - 
                 
                   
                     b 
                     1 
                   
                    
                   
                     y 
                     
                       
                         2 
                          
                         n 
                       
                       + 
                       1 
                     
                   
                 
               
               = 
               
                 
                   x 
                   n 
                 
                 - 
                 
                   0.5 
                    
                   
                     y 
                     
                       2 
                        
                       n 
                     
                   
                 
                 + 
                 
                   0.5 
                    
                   
                     y 
                     
                       
                         2 
                          
                         n 
                       
                       + 
                       1 
                     
                   
                 
               
             
           
         
       
       
         
           and 
         
       
       
         
           
             
               y 
               
                 2 
                  
                 n 
               
             
             = 
             
               
                 
                   
                     x 
                     n 
                   
                   1.5 
                 
                 + 
                 
                   
                     0.5 
                     * 
                     
                       y 
                       
                         
                           2 
                            
                           n 
                         
                         + 
                         1 
                       
                     
                   
                   1.5 
                 
               
               = 
               
                 
                   
                     2 
                      
                     
                       x 
                       n 
                     
                   
                   + 
                   
                     y 
                     
                       
                         2 
                          
                         n 
                       
                       + 
                       1 
                     
                   
                 
                 3 
               
             
           
         
       
     
         [0119]    Similarly, we can keep moving back towards y 0 . 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           y 
                           
                             
                               2 
                                
                               n 
                             
                             - 
                             1 
                           
                         
                         = 
                           
                          
                         
                           
                             x 
                             n 
                           
                           + 
                           
                             z 
                             n 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             x 
                             n 
                           
                           + 
                           
                             
                               b 
                               2 
                             
                              
                             
                               y 
                               
                                 2 
                                  
                                 n 
                               
                             
                           
                           + 
                           
                             
                               b 
                               1 
                             
                              
                             
                               y 
                               
                                 
                                   2 
                                    
                                   n 
                                 
                                 + 
                                 1 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             x 
                             n 
                           
                           + 
                           
                             0.5 
                              
                             
                               y 
                               
                                 2 
                                  
                                 n 
                               
                             
                           
                           - 
                           
                             0.5 
                              
                             
                               y 
                               
                                 
                                   2 
                                    
                                   n 
                                 
                                 + 
                                 1 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               4 
                                
                               
                                 x 
                                 n 
                               
                             
                             - 
                             
                               y 
                               
                                 
                                   2 
                                    
                                   n 
                                 
                                 + 
                                 1 
                               
                             
                           
                           3 
                         
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   ⋮ 
                    
                   
                     
 
                   
                    
                   
                     
                       y 
                       2 
                     
                     = 
                     
                       
                         
                           2 
                            
                           
                             x 
                             1 
                           
                         
                         + 
                         
                           y 
                           3 
                         
                       
                       3 
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       y 
                       1 
                     
                     = 
                     
                       
                         
                           4 
                            
                           
                             x 
                             1 
                           
                         
                         - 
                         
                           y 
                           3 
                         
                       
                       3 
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       y 
                       0 
                     
                     = 
                     
                       
                         
                           2 
                            
                           
                             x 
                             0 
                           
                         
                         + 
                         
                           y 
                           1 
                         
                       
                       3 
                     
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
         [0120]    Similar equations can be obtained by moving from top to bottom and from left to right. 
         [0121]    In this case, y 0  is known and we can calculate the remaining pixels (y i s) of any row and column, i.e., 
         [0000]    
       
         
           
             
               y 
               0 
             
             = 
             
               
                 
                   x 
                   0 
                 
                 - 
                 
                   z 
                   0 
                 
               
               = 
               
                 
                   
                     x 
                     0 
                   
                   - 
                   
                     
                       
                         
                           y 
                           0 
                         
                         - 
                         
                           y 
                           1 
                         
                       
                       2 
                     
                      
                     
                         
                     
                      
                     that 
                      
                     
                         
                     
                      
                     gives 
                      
                     
                         
                     
                      
                     
                       y 
                       1 
                     
                   
                 
                 = 
                 
                   
                     3 
                      
                     
                       y 
                       0 
                     
                   
                   - 
                   
                     2 
                      
                     
                       x 
                       0 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               y 
               1 
             
             = 
             
               
                 x 
                 1 
               
               + 
               
                 z 
                 1 
               
             
           
         
       
       
         
           
             
               y 
               2 
             
             = 
             
               
                 
                   x 
                   1 
                 
                 - 
                 
                   
                     z 
                     1 
                   
                    
                   
                       
                   
                    
                   that 
                    
                   
                       
                   
                    
                   gives 
                    
                   
                     : 
                   
                    
                   
                       
                   
                    
                   
                     y 
                     2 
                   
                 
               
               = 
               
                 
                   2 
                    
                   
                     x 
                     1 
                   
                 
                 - 
                 
                   
                     y 
                     1 
                   
                    
                   
                       
                   
                    
                   and 
                 
               
             
           
         
       
       
         
           
             
               y 
               3 
             
             = 
             
               
                 x 
                 2 
               
               + 
               
                 z 
                 2 
               
             
           
         
       
       
         
           
             
               y 
               4 
             
             = 
             
               
                 
                   x 
                   2 
                 
                 - 
                 
                   
                     z 
                     2 
                   
                    
                   
                       
                   
                    
                   that 
                    
                   
                       
                   
                    
                   gives 
                    
                   
                     : 
                   
                    
                   
                       
                   
                    
                   
                     y 
                     4 
                   
                 
               
               = 
               
                 
                   2 
                    
                   
                     x 
                     2 
                   
                 
                 - 
                 
                   
                     y 
                     3 
                   
                    
                   
                       
                   
                    
                   and 
                 
               
             
           
         
       
       
         
           
             
               y 
               5 
             
             = 
             
               
                 3 
                  
                 
                   y 
                   4 
                 
               
               - 
               
                 2 
                  
                 
                   x 
                   2 
                 
               
             
           
         
       
       
         
           ⋮ 
         
       
       
         
           
             
               y 
               
                 
                   2 
                    
                   n 
                 
                 - 
                 1 
               
             
             = 
             
               
                 x 
                 n 
               
               + 
               
                 z 
                 n 
               
             
           
         
       
       
         
           
             
               y 
               
                 2 
                  
                 n 
               
             
             = 
             
               
                 
                   x 
                   n 
                 
                 - 
                 
                   
                     z 
                     n 
                   
                    
                   
                       
                   
                    
                   that 
                    
                   
                       
                   
                    
                   gives 
                    
                   
                     : 
                   
                    
                   
                       
                   
                    
                   
                     y 
                     
                       2 
                        
                       n 
                     
                   
                 
               
               = 
               
                 
                   2 
                    
                   
                     x 
                     n 
                   
                 
                 - 
                 
                   
                     y 
                     
                       
                         2 
                          
                         n 
                       
                       - 
                       1 
                     
                   
                    
                   
                       
                   
                    
                   and 
                 
               
             
           
         
       
       
         
           
             
               y 
               
                 
                   2 
                    
                   n 
                 
                 + 
                 1 
               
             
             = 
             
               
                 3 
                  
                 
                   y 
                   
                     2 
                      
                     n 
                   
                 
               
               - 
               
                 2 
                  
                 
                   
                     x 
                     n 
                   
                   . 
                 
               
             
           
         
       
     
         [0122]    Another possibility is to start at any given pixel of a row or column and move backward or forward computing the remaining pixels using the above two sets of equations. The advantage of doing this is that this approach may be considered a form of encryption. If the starting point pixel is not known for every row and column, using the wrong starting pixels will probably result in unrecognizable frames. The actual starting pixel can be generated randomly using a pseudo-random number generator with a specific random seed value. All the receiver of the video compressed with the techniques of the invention needs to do is to use such random seed value at the decompression end. Such value is like an encryption key for legitimate users. It can be easily entered into the decompression algorithm for high quality video reconstruction. 
         [0123]    Note that the x i  values are approximate because the actual original values have been processed by the codec being used to compress/decompress. Therefore, the y i s calculated with the preceding algorithm are also approximations. The question is how good the approximations are. This can be determined by the perceived visual quality of the output video and by calculating the PSNR values for every output video frame for different bit rates and comparing such values for the codec alone and for the values obtained using methods of this invention. 
         [0124]    In terms of the PSNR values, the following Table 1 shows the comparison in PSNR values (in decibels (db)) between the codec alone and the codec with the invention for different video examples. 
         [0000]    
       
         
               
             
               
               
               
               
               
               
             
               
               
               
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 PSNR Values (in db) 
               
             
          
           
               
                 HD Videos 
                   
                   
                   
                   
                   
               
               
                 (1920×1080) 
                 Casino Royale 
                 Baseball 
                 King Kong 
                 Terminator 
                 Yellowstone 
               
               
                   
               
             
          
           
               
                 H264 - 3 Mbps 
                 127.05 
                 125.32 
                 126.91 
                 126.82 
                 110.66 
               
               
                 ADC2 - 3 Mbps 
                 128.14 
                 128.677394 
                 128.132767 
                 128.118862 
                 114.999 
               
               
                 H264 - 2 Mbps 
                 102.69 
                 107.22 
                 105.49 
                 106.18 
                 108.76 
               
               
                 ADC2 - 2 Mbps 
                 128.1179233 
                 128.6735 
                 128.127 
                 128.1181 
                 114.987 
               
               
                 H264 - 1 Mbps 
                 81.29 
                 86.13 
                 81.23 
                 82.15 
                 76.82 
               
               
                 ADC2 - 1 Mbps 
                 128.117233 
                 128.6731 
                 128.121 
                 128.11879 
                 114.966 
               
               
                 H264 - 0.5 Mbps 
                 69.19 
                 72.67 
                 70.83 
                 72.21 
                 65.31 
               
               
                 ADC2 - 0.5 Mbps 
                 128.117224 
                 128.6728 
                 128.119 
                 128.11876 
                 114.953 
               
               
                 H264 - 0.25 Mbps 
                 54.33 
                 57.92 
                 56.11 
                 56.62 
                 48.12 
               
               
                 ADC2 - 0.25 Mbps 
                 128.117223 
                 128.6724 
                 128.117432 
                 128.118744 
                 114.952515 
               
               
                   
               
             
          
         
       
     
         [0125]    Table 1 clearly shows the high quality of the video reconstructed by the invention even at very high levels of compression, whereas the quality of the video decompressed by any standard codec (H264 in this example) quickly decays with lower bit-rates which makes such standard codecs unsuitable for practical applications requiring good visual quality at very low bit rates. These mathematical results confirm the visual perception evaluations of the results of the invention. 
         [0126]    Therefore, besides the x i s values, one more value can also be stored to be able to recreate precisely the entire original row or column. The additional memory required is very small overhead when one considers that we are dealing with hundreds or even thousands of pixels for each row and column of images or video frames of typical applications. 
         [0127]    By applying such a procedure to every row and column of an image or video frame, the size can be reduced to approximately ¼ of the original that can be reproduced exactly from its reduced version. This process can be repeated on the reduced images or video frames for further size reductions of 1/16, 1/64, etc., of the original. 
         [0128]    Of course, this cannot be done indefinitely because the precision of the calculations must be limited in order to avoid increasing the required number of bits instead of reducing it and some information is being lost at each ¼ reduction. However, extensive tests showed that the quality of image reproduction is maintained up to 2 or 3 reduction levels with size reduction of up to 16 or 64 times before compression by the codec. Such levels of compression are very significant in terms of video storage and transmission costs. 
         [0129]    Such tests consisted of using a number of diverse uncompressed video clips, i.e., sports, action movies, educational videos, etc, of about 10 minute duration and requiring tens of Gigabytes of storage space for each. Such video clips were then compressed by a number of different codecs as well as by the methods of the invention. The resulting compressed files were compared for size and then decompressed and played back side by side to compare the perceived quality. Such tests clearly demonstrated that methods of the invention can be suitable for providing additional substantial compression of files, and decompression of the files without significant loss of quality. Photographic examples of some of these tests are provided herein as  FIGS. 10-21 . 
         [0130]    The reproduction by any codec of the reduced size frame is precise enough to be able to apply the above calculations for recovery of the original full size frames with similar quality to that of the frames recovered by the codec without the initial frame size reduction step. 
       Frame Post-Processing and Frame Expansion 
       [0131]    This step of the invention can produce high quality full-screen frames for display on a TV set or PC Monitor. Because of the amount of data involved, standard approaches can be very time-consuming and cannot produce high quality enlargements in any case. 
         [0132]    The techniques developed to complete the frame expansion methods of the invention can be simple computationally, i.e., fast, and can generate enlarged images of high quality with no pixelization and showing none of the blocking artifacts that plague state-of-the-art techniques. The methods of this invention can be applied repeatedly with similar results and enlargement factors of 4 every time it is applied. 
         [0133]    In addition, the process can be further extended by using, after more than 2 or 3 reduction levels, any of the expansion filters disclosed in U.S. Pat. No. 7,317,840 to enlarge very small images with high quality. U.S. Pat. No. 7,317,840 is expressly incorporated fully by reference herein. 
         [0134]    Overall enlargement factors of more than 1000 have been demonstrated with such an extension. 
         [0135]    The image expansion technique disclosed in such a patent is based on the fact that the given image can be considered to be the level 1 low frequency component of the WT of a higher resolution image which is four times larger. One way to accomplish this is to estimate the missing high frequency WT coefficients of level 1 from the given low frequency coefficients. 
         [0136]    A discussion of wavelet theory is provided in “Ten Lectures on Wavelets”, I. Daubechies, Society for Industrial and Applied Mathematics, Philadelphia, 1992, incorporated herein fully by reference. However, in brief, wavelets are functions generated from a single function Ψ by dilations and translation. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       Ψ 
                       n 
                       j 
                     
                      
                     
                       ( 
                       x 
                       ) 
                     
                   
                   = 
                   
                     
                       Ψ 
                        
                       
                         ( 
                         
                           
                             
                               2 
                               j 
                             
                              
                             x 
                           
                           - 
                           n 
                         
                         ) 
                       
                     
                     
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Where j corresponds to the level of the transform, and hence governs the dilation, and n governs the translation. 
         [0137]    The basic idea of the wavelet transform is to represent an arbitrary function f as a superposition of wavelets. 
         [0000]    
       
         
           
             
               
                 
                   f 
                   = 
                   
                     
                       ∑ 
                       
                         j 
                         , 
                         n 
                       
                     
                      
                     
                       
                         a 
                         n 
                         j 
                       
                        
                       
                         ( 
                         f 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
         [0138]    Since the Ψ n   j  constitute an orthonormal basis, the wavelet transform coefficients are given by the inner product of the arbitrary function and the wavelet basis functions: 
         [0000]        a   n   j ( f )=&lt;Ψ n   j   , f&gt;   (3)
 
         [0139]    In a multiresolution analysis, one really has two functions: a mother wavelet Ψ and a scaling function φ. Like the mother wavelet, the scaling function φ generates a family of dilated and translated versions of itself: 
         [0000]      φ n   j ( x )=2 −j/2 φ(2 −j   x−n )   (4)
 
         [0140]    When compressing data files representative of images, it can be desirable to preserve symmetry. As a result, the requirement of an orthogonal basis may be relaxed (although it is not necessary) and biorthogonal wavelet sets can be used. In this case, the Ψ n   j  no longer constitute an orthonormal basis, hence the computation of the coefficients a n   j  is carried out via the dual basis, 
         [0000]        a   n   j ( f )=&lt;  Ψ   n   j   , f&gt;   (5)
 
         [0000]    where  Ψ  is a function associated with the corresponding synthesis filter coefficients defined below. 
         [0141]    When f is given in sampled form, one can take these samples as the coefficients x n   j  for sub-band j=0. The coefficients for sub-band j+1 are then given by the convolution sums: 
         [0000]    
       
         
           
             
               
                 
                   
                     X 
                     n 
                     
                       j 
                       + 
                       1 
                     
                   
                   = 
                   
                     
                       ∑ 
                       k 
                     
                      
                     
                       
                         h 
                         
                           
                             2 
                              
                             n 
                           
                           - 
                           k 
                         
                       
                        
                       
                         X 
                         k 
                         j 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     6 
                      
                     a 
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    for high frequency coefficients; and 
         [0000]    
       
         
           
             
               
                 
                   
                     C 
                     n 
                     
                       j 
                       + 
                       1 
                     
                   
                   = 
                   
                     
                       ∑ 
                       k 
                     
                      
                     
                       
                         g 
                         
                           
                             2 
                              
                             n 
                           
                           - 
                           k 
                         
                       
                        
                       
                         X 
                         k 
                         j 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     6 
                      
                     b 
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    for high frequency coefficients.
 
This describes a sub-band algorithm with:
 
         [0000]    
       
         
           
             
               
                 
                   
                     h 
                     n 
                   
                   = 
                   
                     
                       ∫ 
                       
                         
                           ϕ 
                            
                           
                             ( 
                             
                               x 
                               - 
                               n 
                             
                             ) 
                           
                         
                          
                         
                           ϕ 
                            
                           
                             ( 
                             x 
                             ) 
                           
                         
                          
                         
                            
                           x 
                         
                       
                     
                     
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   
                     7 
                      
                     a 
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    representing a low-pass filter and
   (7b)  g   1 =(−1)h —1+1 , representing a high-pass filter. Consequently, the exact reconstruction is given by:   
 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       X 
                       i 
                       j 
                     
                     = 
                     
                       · 
                       
                         
                           
                             ∑ 
                             _ 
                           
                           n 
                         
                          
                         
                           ( 
                           
                             
                               
                                 h 
                                 
                                   
                                     2 
                                      
                                     n 
                                   
                                   - 
                                   1 
                                 
                               
                                
                               
                                 X 
                                 n 
                                 
                                   j 
                                   + 
                                   1 
                                 
                               
                             
                              
                             
                               + 
                               _ 
                             
                              
                             
                               
                                 g 
                                 
                                   
                                     2 
                                      
                                     n 
                                   
                                   - 
                                   1 
                                 
                               
                                
                               
                                 C 
                                 n 
                                 
                                   j 
                                   + 
                                   1 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where    h 2n−1  and    g 2n−1  represent the reconstruction filters. 
         [0143]    The relation between the different filters is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     g 
                     n 
                   
                   = 
                   
                     
                       
                         
                           ( 
                           
                             - 
                             1 
                           
                           ) 
                         
                         n 
                       
                        
                       
                         h 
                         
                           
                             - 
                             n 
                           
                           + 
                           1 
                         
                       
                        
                       
                           
                       
                        
                       or 
                        
                       
                           
                       
                        
                       
                         g 
                         n 
                       
                     
                     = 
                     
                       
                         
                           ( 
                           
                             - 
                             1 
                           
                           ) 
                         
                         
                           n 
                           + 
                           1 
                         
                       
                        
                       
                         
                           h 
                           _ 
                         
                         
                           
                             - 
                             n 
                           
                           + 
                           1 
                         
                       
                        
                       
                           
                       
                        
                       
                         ( 
                         biorthogonal 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     9 
                      
                     a 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       g 
                       _ 
                     
                     n 
                   
                   = 
                   
                     
                       
                         
                           ( 
                           
                             - 
                             1 
                           
                           ) 
                         
                         n 
                       
                        
                       
                         
                           h 
                           _ 
                         
                         
                           
                             - 
                             n 
                           
                           + 
                           1 
                         
                       
                        
                       
                           
                       
                        
                       or 
                        
                       
                           
                       
                        
                       
                         
                           g 
                           _ 
                         
                         n 
                       
                     
                     = 
                     
                       
                         
                           ( 
                           
                             - 
                             1 
                           
                           ) 
                         
                         
                           n 
                           + 
                           1 
                         
                       
                        
                       
                         h 
                         
                           
                             - 
                             n 
                           
                           + 
                           1 
                         
                       
                        
                       
                           
                       
                        
                       
                         ( 
                         biorthogonal 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     9 
                      
                     b 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ∑ 
                       n 
                     
                      
                     
                       
                         · 
                         
                           h 
                           n 
                           - 
                         
                       
                        
                       
                         h 
                         
                           n 
                           + 
                           
                             2 
                              
                             k 
                           
                         
                       
                     
                   
                   = 
                   
                     
                       δ 
                       
                         k 
                         , 
                         o 
                       
                     
                      
                     
                         
                     
                      
                     
                       ( 
                       
                         delta 
                          
                         
                             
                         
                          
                         function 
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   
                     9 
                      
                     c 
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    where h n  and g n  represent the low-pass analysis filter and the high-pass analysis filter respectively, and h n  and g n  represent the corresponding synthesis filters. 
         [0144]    We now turn to a matrix modified formulation of the one-dimensional wavelet transform. Using the above impulse responses h n  and g n , we can define the circular convolution operators at resolution 2 j : H j , G j , H j , G j . These four matrices are circulant and symmetric. The H j  matrices are built from the h n  filter coefficients and similarly for G j  (from g n ), H j  (from h n ) and G j  (from g n ). 
         [0145]    The fundamental matrix relation for exactly reconstructing the data at resolution 2 −j  is 
         [0000]        H   j      H     j   +G   j      G     j   =I   j    (10)
 
         [0000]    where I j  is the identity matrix. 
         [0146]    Let  X   j+1  be a vector of low frequency wavelet transform coefficients at scale 2 −(j+1)  and let  C   x   j+1  be the vector of associated high frequency wavelet coefficients. 
         [0147]    We have, in augmented vector form: 
         [0000]    
       
         
           
             
               
                 
                   
                      
                     
                       
                         
                           
                             
                               x 
                               _ 
                             
                             
                               j 
                               + 
                               1 
                             
                           
                         
                       
                       
                         
                           
                             
                               C 
                               _ 
                             
                             x 
                             
                               j 
                               + 
                               1 
                             
                           
                         
                       
                     
                      
                   
                   = 
                   
                     
                        
                       
                         
                           
                             
                               H 
                               j 
                             
                           
                           
                             O 
                           
                         
                         
                           
                             O 
                           
                           
                             
                               G 
                               j 
                             
                           
                         
                       
                        
                     
                     × 
                     
                        
                       
                         
                           
                             
                               
                                 x 
                                 _ 
                               
                               j 
                             
                           
                         
                         
                           
                             
                               
                                 x 
                                 _ 
                               
                               j 
                             
                           
                         
                       
                        
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where  X   j+1  is the smoothed vector obtained from  X   j . The wavelet coefficients  C   x   j+1  contain information lost in the transition between the low frequency bands of scales 2 −j  and 2 −(j+1) . 
         [0148]    The reconstruction equation is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       X 
                       _ 
                     
                     j 
                   
                   = 
                   
                     
                        
                       
                         
                           
                             _j 
                           
                           
                             _j 
                           
                         
                         
                           
                             H 
                           
                           
                             G 
                           
                         
                       
                        
                     
                      
                     X 
                      
                     
                        
                       
                         
                           
                             
                               
                                 x 
                                 _ 
                               
                               
                                 j 
                                 + 
                                 1 
                               
                             
                           
                         
                         
                           
                             
                               
                                 C 
                                 _ 
                               
                               x 
                               
                                 j 
                                 + 
                                 1 
                               
                             
                           
                         
                       
                        
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
         [0149]    Since, from equation (11),  X   j+1 =H j   X   j , we can, in principle, recover  X   j  from  X   j+1  merely by inverting H j . However, this is generally not practical both because of the presence of inaccuracies in  X   j+1  and because H j  is generally an ill-conditioned matrix. As a result, the above problem is ill-posed and there is, in general, no unique solution. 
         [0150]    If we discard the high frequency coefficients,  C   x   j+1 , then equation (12) reduces to  y     j   =H j   X   j+1    
         [0000]    which is a blurred approximation of  X   j . 
         [0151]    From equation (11),  X   j+1 =H j X j , which gives 
         [0000]        H   j      X     j+1   =H   j    H   j      X     j  or   (13a)
 
         [0000]        X   j+1 −H j    X   j .   (14)
 
         [0000]    In our problem, the  X   j+1  (transformed rows or columns of level j+1) are known and the problem is to determine the  X   j  of the next higher level. 
         [0152]    This can be thought of as an image restoration problem in which the image defined by the vector  X   j  has been blurred by the operator H j , which due to its low-pass nature, is an ill-conditioned matrix. 
         [0153]    Regularization, as in “Methodes de resolution des problems mal poses”, A. N. Tikhonov and V. Y. Arsenin, Moscow, Edition MIR, incorporated herein fully by reference, is a method used to solve ill-posed problems of this type. This method is similar to a constrained least squares minimization technique. 
         [0154]    A solution for this type of problem is found by minimizing the following Lagrangian function: 
         [0000]        J (   X     j ,α)=|   X     j+1   −H   j      X     j |1 2   +α|G   j      X     j | 2    (15)
 
         [0000]    where G j  is the regularization operator and α is a positive scalar such that α→0 as the accuracy of  X   j+1  increases. 
         [0155]    It is also known from regularization theory that if H j  acts as a low-pass filter, G j  must be a high-pass filter. In other words, since H j  is the low-pass filter matrix of the wavelet transform, G j , must be the corresponding high-pass filter matrix. 
         [0156]    Equation (15) may be also written with respect to the estimated wavelet transform coefficients 
         [0000]      C   x   j+1  and  {circumflex over (X)}   j+1  (from equation (11)). 
         [0000]        J (   X     j ,α)=|   X     j+1     X     j+1 | 2   +α| C     x   j+1 | 2 .   (16)
 
         [0157]    Using the exact reconstruction matrix relation shown in Equation 10, we get: 
         [0000]          X     j+1   =H   j      H     j      X     j+1   +G   j      G     j      X     j+1 .   (16a)
 
         [0158]    Also, we can write 
         [0000]          {circumflex over (X)}     (j+1)   =H   j      X     j   =H   j (   H     j      X     (j+1)   +  G     j      C     x   (j+1)    (16b)
 
         [0000]    (keep in mind that  X   j  is estimated.)
 
Then subtracting (16b) from (16a) gives:
 
         [0000]          X     j+1   − {circumflex over (X)}     j+1   =G   j      G     j      X     j+1   −H   j    G   j      C     x   (j+1)    (16c)
 
         [0000]    Substituting (16c) into (16) results in: 
         [0000]        J (   C     x   (j+1) ,α)=| G   j      G     j      X     j+1   −H   j      G     j      C     x   (j+1) | 2   +α| C     x   j+1 | 2 .   (17)
 
         [0000]    By setting the derivative of J with respect to  C   x   j+1 , equal to zero, we can obtain the following estimate for the high frequency coefficients  C   x   j+1 : 
         [0000]          C     x   j+1   =M  X     j+1    (18)
 
         [0000]    where the estimation matrix M is given by 
         [0000]        M=|α I   j   +  G     t   j    H   t   j    H   j      G     j | −1      G     t   j    H   t   j    G   j      G     j    (19)
 
         [0000]    In which the subscript “t” refers to the matrix transpose. 
         [0159]    Since the goal is to calculate an estimate of  X   j  from  X   j+1 , using equation (12), we can write 
         [0000]          X     j   =T  X     j+1    (20)
 
         [0000]    where T is the matrix 
         [0000]        T=  H     j   +  G     j    M    (21)
 
         [0000]    In other words, it is not necessary to calculate the high frequency coefficients  C   x   j+1 , although their determination is implicit in the derivation of the matrix T. 
         [0160]    One can appreciate that, since we are dealing with a decimated Wavelet Transform, the matrix T is not square, but rather, it is rectangular. Its dimensions are n·n/2 where n is the size of the data before any given level of transformation. This can be verified from the following sizes for the Wavelet Transform matrices: H and G are n/2·n matrices and H and G are n·n/2. Notice that αI+G t  H t  H G is a square matrix of size n/2·n/2 and is invertible if α&gt;o for all wavelet filters. 
         [0161]    Another aspect of this invention is the structure of the matrix T. The rows of T are made up of just two short filters that repeat themselves every two rows with a shift to the right of one location. All other elements of the matrix T are zero. This means that every level of the Wavelet Transform can be recreated from the previous level (of half the size) by convolving both filters centered at a specific location of the available data with such data. This results in two new values from every given value thus doubling the size of the data at every level of signal decompression or expansion. There is no need to multiply the matrix T with the given vector. The two filters depend on the coefficients of the wavelet filters used to transform the original data in the case of compression while any wavelet filter coefficients can be used to determine the two expansion filters. The most significant criteria being quality and speed. 
         [0162]    For example, for a Daubechies-6 wavelet, the two filters that make up the matrix T are: 
         [0000]      x 1 =0.04981749973687 
         [0000]        x   2 =−0.19093441556833
 
         [0000]      x 3 =1.141116915831444 and 
         [0000]        y   1 =−0.1208322083104
 
         [0000]      y 2 −0.65036500052623
 
         [0000]      y 3 =0.47046720778416 
         [0163]    and the T matrix is: 
         [0000]    
       
         
           
             T 
             = 
             
               
                 
                   
                     x 
                     1 
                   
                 
                 
                   
                     x 
                     2 
                   
                 
                 
                   
                     x 
                     3 
                   
                 
                 
                   
                       
                   
                 
                 
                   
                       
                   
                 
                 
                   
                       
                   
                 
                 
                   
                     
                       0 
                       ′ 
                     
                      
                     s 
                   
                 
               
               
                 
                   0 
                 
                 
                   
                     y 
                     1 
                   
                 
                 
                   
                     y 
                     2 
                   
                 
                 
                   
                     y 
                     3 
                   
                 
                 
                   
                       
                   
                 
                 
                   
                       
                   
                 
                 
                   
                     
                       0 
                       ′ 
                     
                      
                     s 
                   
                 
               
               
                 
                   0 
                 
                 
                   
                     x 
                     1 
                   
                 
                 
                   
                     x 
                     2 
                   
                 
                 
                   
                     x 
                     3 
                   
                 
                 
                   
                       
                   
                 
                 
                   
                       
                   
                 
                 
                   
                     
                       0 
                       ′ 
                     
                      
                     s 
                   
                 
               
               
                 
                   0 
                 
                 
                   0 
                 
                 
                   
                     y 
                     1 
                   
                 
                 
                   
                     y 
                     2 
                   
                 
                 
                   
                     y 
                     3 
                   
                 
                 
                   
                       
                   
                 
                 
                   
                     
                       0 
                       ′ 
                     
                      
                     s 
                   
                 
               
               
                 
                   0 
                 
                 
                   0 
                 
                 
                   
                     x 
                     1 
                   
                 
                 
                   
                     x 
                     2 
                   
                 
                 
                   
                     x 
                     3 
                   
                 
                 
                   
                       
                   
                 
                 
                   
                     
                       0 
                       ′ 
                     
                      
                     s 
                   
                 
               
               
                 
                   
                       
                   
                 
                 
                   
                       
                   
                 
                 
                   
                       
                   
                 
                 
                   
                       
                   
                 
                 
                   
                     etc 
                     . 
                   
                 
                 
                   
                       
                   
                 
                 
                   
                       
                   
                 
               
             
           
         
       
     
         [0164]    Using other wavelet bases, similar expansion filters can be obtained. The following Table 2 provides the wavelets and lengths of filters obtained with a Matlab program of for some typical wavelets. 
         [0000]                                TABLE 2                       Wavelet   Expansion Filters Lengths                           Daubechies-4   2           Daubechies-6   3           Daubechies-8   4           Biorthogonal   3-4           Asymmetrical   2                        
It can be appreciated that better expansion quality can be obtained using longer filters, whereas naturally shorter filters can provide faster expansion.
 
         [0165]    It is important to notice that these expansion filters do not depend on the size of the data. By contrast, the undecimated Wavelet Transform results in full matrices with no zeros and whose elements change with the size of the data. 
         [0166]    Thus, the practical advantages of the method disclosed in this patent are obvious in terms of computational complexity and capability to recreate signals with high quality from low frequency information alone. 
         [0167]    With respect to images and video frames, the method is applied first to columns and then to rows. Also, for color images, the method is applied separately to the luminance (Y) and the chrominance (UV) components. 
         [0168]    The procedures of the invention can be extended to other wavelets in addition to the Haar Wavelet, although the calculations are more complicated and time consuming. In some embodiments, the corresponding equations for the WT and IWT lead to a sparse system of linear equations in which only a small number of its matrix elements are non-zero, resulting in a band diagonal matrix in which the width of the band depends on the number of Wavelet coefficients. There are software packages applicable to such systems, e.g., Yale Sparse Matrix Package, but the Haar method above provides the quality and speed that make such more complicated approaches unnecessary for situations in which real-time processing is an important requirement. 
         [0169]    Take for example the Daubechies-4 Wavelet with low-frequency coefficients: 
         [0000]        a 4=0.4829629131445341/√2
 
         [0000]        a 3=0.8365163037378077/√2
 
         [0000]        a 2=0.2241438680420134/√2
 
         [0000]        a 1=−0.1294095225512603/√2
 
         [0170]    and high frequency coefficients 
         [0000]      b4=a1 
         [0000]        b 3=− a 2
 
         [0000]      b2=a3 
         [0000]        b 1 =−a 4 
         [0171]    The coefficients for the inverse wavelet transform are:
       low frequency:       
 
         [0000]        aa 4=−0.1294095225512603*√2
 
         [0000]        aa 3=0.2241438680420134*√2
 
         [0000]        aa 2=0.8365163037378077*√2
 
         [0000]        aa 1=0.4829629131445341*√2
       high frequency       
 
         [0000]        bb 4=− aa 1
 
         [0000]      bb3=aa2 
         [0000]        bb 1=− aa 3
 
         [0000]      bb1=aa4 
         [0174]    Similarly to the case of the Haar Wavelet, we can express the values of the pixels, y i &#39;s, of a row or a column of an image or video frame given the values of their low frequency WT, x i &#39;s, and the values of their high frequency WT, z i &#39;s, as 
         [0000]    
       
      
       y 
       0 
       =aa 
       4 
       x 
       0 
       +aa 
       2 
       x 
       1 
       +bb 
       4 
       z 
       0 
       +bb 
       2 
       z 
       1  
      
     
         [0000]    
       
      
       y 
       1 
       =aa 
       3 
       x 
       1 
       +aa 
       1 
       x 
       2 
       +bb 
       3 
       z 
       1 
       +bb 
       1 
       z 
       2  
      
     
         [0000]    
       
      
       y 
       2 
       =aa 
       4 
       x 
       1 
       +aa 
       2 
       x 
       2 
       +bb 
       4 
       z 
       1 
       +bb 
       1 
       z 
       2  
      
     
         [0000]    
       
      
       y 
       3 
       =aa 
       3 
       x 
       2 
       +aa 
       1 
       x 
       3 
       +bb 
       3 
       z 
       2 
       +bb 
       1 
       z 
       3  
      
     
         [0175]    etc. 
         [0000]    
       
      
       y 
       2n−2 
       =aa 
       4 
       x 
       n−1 
       +aa 
       2 
       x 
       0 
       +bb 
       4 
       z 
       n−1 
       +bb 
       2 
       z 
       0  
      
     
         [0000]        y   2n−1   =aa   3   x   0   +aa   1   x   1   +bb   3   z   0   +bb   1   z   1  (wraparound) 
         [0000]    but we have that 
         [0000]        z   0   =b   4   y   2n−3   +b   3   y   2n−2   +b   2   y   2n−1   +b   1   y   0  (wraparound) 
         [0000]    
       
      
       z 
       1 
       =b 
       4 
       y 
       2n−1 
       +b 
       3 
       y 
       0 
       +b 
       2 
       y 
       1 
       +b 
       1 
       y 
       2  
      
     
         [0000]    
       
      
       z 
       2 
       =b 
       4 
       y 
       1 
       +b 
       3 
       y 
       2 
       +b 
       2 
       y 
       3 
       +b 
       1 
       y 
       4  
      
     
         [0176]    etc. 
         [0000]    
       
      
       z 
       n−2 
       =b 
       4 
       y 
       2n−6 
       +b 
       3 
       y 
       2n−5 
       +b 
       2 
       y 
       2n−4 
       +b 
       1 
       y 
       2n−3  
      
     
         [0000]    
       
      
       z 
       n−1 
       =b 
       4 
       y 
       2n−4 
       +b 
       3 
       y 
       2n−3 
       +b 
       2 
       y 
       2n−2 
       +b 
       1 
       y 
       2n−1  
      
     
         [0177]    Therefore we have a system of equations in hundreds or thousands of variables in which most of the coefficients are zero, i.e., a sparse diagonal linear system. As indicated, there are packages to solve such systems but the Haar Wavelet provides the fastest and most efficient way to reconstruct an image from its low frequency WT. Therefore, using other wavelets is unnecessary and not cost-effective. 
         [0178]    These and other embodiments can be used to produce high-quality replay of video images arising from a number of different sources. For example, movies, and live telecasts (e.g., news broadcasts). 
       EXAMPLES 
       [0179]    The following examples are being presented to illustrate specific embodiments of this invention. It can be appreciated that persons of ordinary skill in the art can develop alternative specific embodiments, based upon the disclosures and teachings contained herein. Each of these embodiments is considered to be part of this invention. 
       Example 1 
     Frame Reduction and Expansion Techniques 
       [0180]    As above indicated, the application of the decimated Haar WT to a given video frame results in a frame that is ¼ the original size because only the low-frequency Haar wavelet filter is applied. It has been proven above that the high-frequency Haar wavelet filter need not be applied if just the last original value before wavelet transformation of a row or column pixel is saved. With this information, all the preceding original pixels of a row or column can be calculated exactly. 
         [0181]    This process can be repeated again on the resulting reduced frame for additional sized reduction to 1/16 of the original and so on. This process is described in detail below. 
       Example 2 
     One-Level Frame Size Reduction and Expansion Back to the Original Size 
       [0182]      FIG. 4  depicts drawing  400  showing a one-level frame size reduction to ¼ of its original size according to an embodiment of this invention. Frame A  410  has horizontal dimension x and vertical dimension y. Column LC 415  and row LR 420  are identified, and pixel (X)  425  is shown. First step  426  is horizontal frame reduction, producing Frame B  430 , having horizontal dimension x/2 and vertical dimension y. Column (LC)  435  and row (LR/2)  440  are identified, as is pixel X  445 . Second step  446  is vertical frame reduction, producing Frame C  450 , having horizontal size x/2 and vertical size y/2. Column (LC/2)  455 , row (LR/2)  460  and pixel (X)  465  are identified. 
         [0183]    In  FIG. 4 , “A” is the original frame with dimensions x and y. The decimated low-pass Haar WT is applied to A horizontally, resulting in a frame “B” of dimensions (x/2) and y. The last column of A, i.e., (“LC”), is copied to the last column of B, x/2+1. 
         [0184]    Next, the decimated low-pass Haar WT is applied to the (x/2)+1 columns of B resulting in frame “C” of dimensions (x/2)+1 and (y/2). The last row of B, i.e., (“LR/2”), is copied to the last row of C, i.e., y/2+1. Notice that pixel X (R, G, B, or Y, U, V component) is kept through this process. LR/2 is the decimated WT of LR and LC/2 is the decimated WT of LC. 
         [0185]    The process of recovering the original frame from C of  FIG. 4  is shown in  FIG. 5 . First the last row of C is used to precisely recover the columns of B using the frame post-processing reconstruction calculations disclosed above. Finally, the reconstruction method is applied to B horizontally starting with the values of LC reconstructed from the value of X using the reconstruction calculations from right to left to recover A exactly. 
         [0186]    This procedure can be interfaced to any existing codec (as part of the research of this invention, it has been applied with similar results to the most popular codecs, (i.e., MPEG-4, H.264, VC-1, DivX) to improve its compression performance significantly (60% to 80% reduction in storage and transmission costs for all extensively tested video files) with minimal loss of video quality compared to that produced by the original codec after decompression. First, a frame pre-processing size reduction process of an embodiment of the invention is applied to the original frames of a given video file. Then, a codec is applied to each frame in the given video file. Application of methods of this invention produces a much smaller file than without the frame pre-processing size reduction step. The resulting compressed video file of this invention can then be stored and/or transmitted at a greatly reduced cost. 
         [0187]    An implementation of a one level frame pre-processing step of an embodiment of the invention corresponds to the pseudocode below:
       For every row in input frame pFrameIn       
 
         [0000]    
       
         
               
             
           
               
                   
               
             
             
               
                 { 
               
               
                  ○ Compute the decimated low-frequency Haar WT for consecutive 
               
               
                   pixels 
               
               
                  ○ Store in half the width of pFrameIn the resulting values 
               
               
                  ○ At end of row, store the last pixel unchanged 
               
               
                 } 
               
               
                   
               
             
          
         
       
       
         
           
             For every 2 consecutive rows in modified pFrameIn 
           
         
       
     
         [0000]                                {        ○ Compute the decimated low frequency Haar WT of corresponding         pixels in the 2 rows column by column        ○ Store the resulting values in output frame pFrameOut        ○ Advance row position by 2       }                    
Store the last row of the modified pFrameIn in pFrameOut last row.
 
         [0190]    This process produces a reduced frame of ¼ the size of the original frame. 
         [0191]    Optionally, an implementation of a second level step includes the additional instruction:
       Repeat with pFrameOut as the input for this step.
 
Using a second level step reduces the size of the frame to 1/16 of the original size.
       
 
         [0193]    Optionally, an implementation of a third level step includes the additional instruction:
       Repeat with pFrameOut of the previous step as the input for this step.
 
Using a third level step reduces the size of the frame to 1/64 of the original size.
       
 
         [0195]    In this description, the computation of the decimated low-frequency Haar WT simply involves taking the average of two consecutive pixels, i.e., the average of each of their components (Y, U, V or R, G, B), and making such average the value of the WT in the corresponding position as we move along the rows of the image first and then down the columns of the WT of the rows. This WT of the original image results in a new image that looks very much like the original but at ¼ its size. 
         [0196]    An implementation of a one level frame post-processing step of this invention corresponds to the pseudocode below:
       Copy last row of input frame pFrameIn into intermediate output frame Img with the same pixels per row as pFrameIn and double the number of rows.   For each pixel position of pFrameIn and Img       
 
         [0000]    
       
         
               
             
           
               
                   
               
             
             
               
                 { 
               
               
                  ○ Calculate the new pixel values of 2 rows of Img starting at the bottom 
               
               
                   and moving up column by column according to the formulas 
               
               
                   y 2n  = (2x n  + y 2n+1 )/3 and 
               
               
                   y 2n−1  = (4x n  − y 2n+1 )/3 
               
               
                   (the x&#39;s represent pixels of pFrameIn and the y&#39;s represent pixels of 
               
               
                   Img) 
               
               
                  ○ Store the calculated pixels in Img 
               
               
                 } 
               
               
                   
               
             
          
         
       
       
         
           
             For every row of Img 
           
         
       
     
         [0000]                                {        ○ Start with the last pixel and store it in the last pixel of the         corresponding row of the output frame pFrameOut        ○ Compute the pixels of the rest of the row from right to left according         to the above formulas where now the x&#39;s represent the pixels of Img         and the y&#39;s represent the pixels of pFrameOut        ○ Store the calculated pixels in pFrameOut       }                    
This one level post-processing step increases the size of the frame by 4-fold compared to the input frame size.
 
         [0200]    Optionally, to, provide a second level post-processing, an embodiment of this invention includes the following step:
       Repeat with pFrameOut of the previous level being the input in this case. This produces a frame that is 16-fold larger than the original input frame.       
 
         [0202]    Optionally, to provide a third level post-processing, an embodiment of this invention further includes the step:
       Repeat with pFrameOut of the previous level being the input of this level. This produces a frame that is 64-fold larger than the original input frame.       
 
         [0204]    In this description, the resulting pFrameOut is an almost identical reproduction of the original image before the frame pre-processing step of size reduction because of the formulas of the invention used in the computation of the new pixels. 
         [0205]    For decompression and display, the codec is applied for decompression and then the above frame post-processing or size expansion procedure of an embodiment of the invention is used prior to displaying high-quality video in its original full-size. 
         [0206]      FIG. 5  depicts a drawing  500  showing a one-level frame size expansion according to an embodiment of this invention. Reduced Frame C  510  has horizontal size x/2+1 and vertical size y/2+1. Frame C  510  has Column  515  and Row  520 , with pixel (X)  525  shown. Vertical expansion step  526  produces Frame B  530 , having a horizontal dimension x/2+1 and vertical dimension y. Column (LC)  535  is identified. Horizontal expansion step  536  produces Frame A  540  having horizontal dimension x and vertical dimension y as in the original frame. 
         [0207]    Because of the lossy compression of existing standard video codecs, there is some minor loss of video quality compared to the original before compression by the codec but the methods of the invention described here do not result in any perceived degradation of quality when compared to that produced by the codec on its own from a much larger file. 
       Example 3 
     Multiple-Level Frame Size Reduction and Expansion 
       [0208]    The process of embodiments of the invention described in  FIGS. 4 and 5  can be continued by one or more levels starting with the C frame instead of the A frame. There are additional right columns and bottom rows to be saved but they are one half the sizes of the previous level and, consequently, they don&#39;t appreciably detract from the saving in storage and transmission bandwidth. 
         [0209]      FIG. 6  depicts a drawing  600  of a two-level size reduction according to an embodiment of this invention including another level of frame reduction compared to  FIG. 4 . In  FIG. 6 , frame C  610  has horizontal dimension x/2+1 and vertical dimension y/2+1. Columns (LC2)  620  and (LC1/2)  615  are identified. Rows (LR2)  630  and (LR1/2)  625  are also identified. Pixels (X)  635  and (Y)  640  are identified. By application of size reduction step  636  in the horizontal dimension, frame D  650  is produced, having horizontal dimension x/4 and vertical dimension y/2. Columns (LC2)  620  and (LC1/2)  615  are identified, as are rows (LR2/2)  645  and (LR1/4)  650 . Pixels (X)  637 , (Y)  642  and (Z)  655  are shown. With size reduction step  638  in the vertical direction, frame E  660  is produced, having horizontal dimension x/4 and vertical dimension y/4. Columns (LC2/2)  665  and (LC1/4)  670 , rows (LR2/2)  645  and (LR1/4)  650  are shown. Pixels (X)  675 , (Y)  680 , (Z)  685  and (W)  690  are also shown. 
         [0210]      FIG. 7  depicts drawing  700  of a one-level expansion of a two-level reduction according to an embodiment of this invention. Frame E  710  has horizontal dimension x/4 and vertical dimension y/4. Columns  720  and  725 , Rows  730  and  735 , and pixels (X)  740 , (Y)  745 , (Z)  750  and (W)  755  are shown. Application of frame expansion step  756  in the vertical dimension (vertical frame post-processing), produces frame F  760  having horizontal dimension x/4 and vertical dimension y/2. Columns  765  and  770 , Row  775 , and Pixels (X)  780  and (Z)  785  are shown. With horizontal expansion step  786  (horizontal frame post-processing), frame G  790  is produced having horizontal dimension x/2 and vertical dimension y/2. Column  792 , Row  794  and Pixel (X)  796  are shown. 
         [0211]      FIG. 8  depicts drawing  800  of a three-level size reduction according to an embodiment of this invention. Frame E  810  has horizontal dimension x/4 and vertical dimension y/4. Columns  812  and  814 , Rows  816  and  818 , and Pixels (X)  820 , (Y)  822 , (Z)  824  and (W)  826  are shown. With horizontal size reduction step  828 , frame H  830  is produced, having horizontal dimension x/8 and vertical dimension y/4. Columns  832 ,  834  and  836 , Rows  838  and  840 , and Pixels (X)  842 , (Y)  844 , (Z)  846 , (W)  848 , (R)  850  and (S)  852  are shown. With additional vertical size reduction step  849 , frame I  860  is produced, having horizontal dimension x/8 and vertical dimension y/8. Columns  861 ,  862  and  863 , Rows  864 ,  865  and  866 , and Pixels (T)  867 , (R)  868 , (S)  869 , (U)  870 , (Y)  871 , (Z)  872 , (V)  873 , (W)  874  and (X)  875  are shown. 
         [0212]      FIG. 9  depicts drawing  900  of a one-level frame expansion of a three-level frame reduction according to an embodiment of this invention. Frame I  910  has horizontal dimension x/8 and vertical dimension y/8. Columns  912 ,  913  and  914 , and Rows  915 ,  916  and  917  are shown. Pixels (X)  918 , (W)  919 , (V)  926 , (U)  922 , (Y)  921 , (Z)  920 , (T)  923 , (R)  924  and (S)  925  are shown. Application of frame expansion step  927  in the vertical dimension produces frame J  930 . Frame J  930  has horizontal dimension x/8 and vertical dimension y/4. Columns  931 ,  932  and  933 , and Rows  934  and  935 , and Pixels (X)  936 , (W)  937 , (Y)  939 , (Z)  938 , (R)  940  and (S)  941  are shown. Application of frame expansion step  945  in the horizontal dimension produces frame K  950 . Frame  950  has horizontal dimension x/4 and vertical dimension y/4. Columns  951  and  952 , Rows  953  and  954 , and Pixels (X)  955 , (W)  956 , (Y)  957  and (Z)  958  are shown. 
         [0213]    It can be appreciated that similar approaches can be applied to provide additional levels of reduction and expansion without departing from the scope of this invention. 
       Modes of Operation 
       [0214]    The above ideas and methods can be implemented in a number of different ways. For example, (1) frame size reduction of only one level but codec compression at different levels of bit assignment per compressed frame; and (2) frame size reduction of multiple levels and codec bit assignment as a function of the reduced frame size of the different levels. 
       Example 4 
     Frame Pre-Processing Code 
       [0215]    This Example provides one specific way in which the principles of this invention can be implemented to pre-process frames in the horizontal and vertical dimensions to reduce their size prior to compression and decompression using a codec. 
         [0000]    
       
         
               
             
           
               
                   
               
             
             
               
                 Void CAviApi : : RCBFrameReduce( int x. int y. int nFrames. Unsigned char* pFrameIn 
               
               
                 unsigned char * pFrameOut ) 
               
               
                 { 
               
               
                 Unsigned char *pFin0. *pFin1. *pFin2 : 
               
               
                 int szin.i, j, k01.k02.k1.k2ix: 
               
               
                 szin=3*(x/2) *y: 
               
               
                 unsigned char *Fin2=(unsigned char *) malloc(sizeof (unsigned char) *szin): 
               
               
                 //unsigned char *Fin3(unsigned char *) malloc(sizeof (unsigned char) *szin): 
               
               
                 pFin0=pFrameIn: 
               
               
                 pFin1=pFin2: 
               
               
                 //pFin2=pFin3: 
               
               
                 pFrameIn=pFin0+6: 
               
               
                 ///////////////////////////////////////HORIZONTAL//////////////////////////////////////// 
               
               
                 I=0: 
               
               
                 while (i&lt;y) { 
               
               
                  j=1: 
               
               
                  while (j&lt;x/2) { 
               
               
                   Fin2[0] = (pFrameIn[0]&gt;&gt;1) + (pFrameIn[3]&gt;&gt;1): 
               
               
                   Fin2[1] = (pFrameIn[1]&gt;&gt;1) + (pFrameIn[4]&gt;&gt;1): 
               
               
                   Fin2[2] = (pFrameIn[2]&gt;&gt;1) + (pFrameIn[5]&gt;&gt;1): 
               
               
                   pFrameIn += 6: 
               
               
                   Fin2 +=3: 
               
               
                  J++: } 
               
               
                 pFrameIn−=3: 
               
               
                 Fin2[0]=pFrameIn[0]: 
               
               
                 Fin2[1]=pFrameIn[1]: 
               
               
                 Fin2[2]=pFrameIn[2]: 
               
               
                 pFrameIn+=9: 
               
               
                 Fin2+=3: 
               
               
                 j++: } 
               
               
                 //CopyMemory(Fin3.Fin2.szin): 
               
               
                 ///////////////////////////////////////VERTICAL///////////////////////////////////////// 
               
               
                 Fin2=pFin1: 
               
               
                 i=1: 
               
               
                 while (i&lt;y/2) } 
               
               
                  j=0: 
               
               
                  while (j&lt;x/2) { 
               
               
                   pFrameOut[0]=(Fin2[0]&gt;&gt;1)+(Fin2[3*x/2]&gt;&gt;1): 
               
               
                   pFrameOut[1]=(Fin2[1]&gt;&gt;1)+(Fin2[1+3*x/2]&gt;&gt;1): 
               
               
                   pFrameOut[2]=(Fin2[2]&gt;&gt;1)+(Fin2[2+3*x/2]&gt;&gt;1): 
               
               
                 //   pFrameOut[0]=(*pFin1)&gt;&gt;1+(*pFin2)&gt;&gt;1: 
               
               
                 //   pFrameOut[1]=(*pFin1+1))&gt;&gt;1+(*(pFin2+1))&gt;&gt;1: 
               
               
                 //   pFrameOut[2]=(*pFin1+2))&gt;&gt;1+(*(pFin2+2))&gt;&gt;1: 
               
               
                 pFrameOut+=3: 
               
               
                 Fin2+=3: 
               
               
                 //pFin1+=3: 
               
               
                 //pFin2+=3: 
               
               
                 j++: } 
               
               
                 Fin2+=3*x/2: 
               
               
                 //pFin1+=3*x/2: 
               
               
                 //pFin2+=3*x/2: 
               
               
                 i++: } 
               
               
                 Fin2−=3*x/2: 
               
               
                 //pFin1−=3*x/2: 
               
               
                 memcpy(pFrameOut.Fin2.3*x/2): 
               
               
                 //memcpy(pFrameOut.pFin1.3*x/2): 
               
               
                 ///////////////////////////////////////////////// 
               
               
                 } 
               
               
                 /////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// 
               
               
                 //////////////////////////////////////////////////////// 
               
               
                   
               
             
          
         
       
     
       Example 5 
     Frame Post-Processing Code 
       [0216]    This Example provides one specific way in which the principles of this invention can be implemented to post-process frames in the horizontal and vertical dimensions to expand their size after decompression of a video file using a codec. 
         [0000]    
       
         
               
             
           
               
                   
               
             
             
               
                 void CAviApi : :RGBFrameExpand(int x. int y. int nFrames. Unsigned char* pFrameIn. 
               
               
                 unsigned char * pFrameOut) 
               
               
                 ( 
               
               
                 //////////////////////////////////////////////////// 
               
               
                 int sz2=y: 
               
               
                 int sz01=x*4: // bytes/row RGB 
               
               
                 int sz: 
               
               
                 unsigned char *Img=(unsigned char *) malloc(sizof (unsigned char)*sz2*sz01*2): 
               
               
                 //int swnhnce=6: 
               
               
                 //int n=swnhnce: 
               
               
                 //int swADC4: 
               
               
                 float temp: 
               
               
                 //float cnhncfctr: 
               
               
                 //float Y.U.V.Frmx: 
               
               
                 unsigned char Zero[8]: 
               
               
                 int i.j.jx.ix.k01.k02.k1.k2.k3.lim.lim1: 
               
               
                 unsigned char *es0.*es1.*es2.*es3.*es4.*es5.*es6.*es7.*es8: 
               
               
                 unsigned char *es5j.es6j.es7j.es8j: 
               
               
                 char *ces3.*ces4.*ces9: 
               
               
                 unsigned char *esi1.*esi2.*esi3 
               
               
                 char *cesi3: 
               
               
                 ///////////////////VERTICAL////////////////// 
               
               
                 #pragma omp parallel shared(sz.sz2.sz01.Img.pFrameIn) private(k1.k2.es0.es1.j.ix.i.temp.es3) 
               
               
                 { 
               
               
                 ////////////////////////////////////////////////////// 
               
               
                 sz=2*sz2: 
               
               
                 ////////////////////////////////////////////////////// 
               
               
                 k1=(sz2-1)*sz01: 
               
               
                 k2=(sz-1)*sz01: 
               
               
                 es1-&amp;Img[k2]: 
               
               
                 es0=&amp;pFrameIn[k1]: 
               
               
                 memcpy(es1.es0.sz01): 
               
               
                 //////////////////////////////// 
               
               
                 ix=sz-2: 
               
               
                 i=sz2-2: 
               
               
                 while (i&gt;0) { 
               
               
                  j=0: 
               
               
                  while (j&lt;sz01) { 
               
               
                 k1=j+1*sz01: 
               
               
                 k2=j+ix*sz01: 
               
               
                 es1=&amp;Img[k2]: 
               
               
                 es0=&amp;pFrameIn[k1]: 
               
               
                 //////////////////////////////////// 
               
               
                 temp=((float)(2* ( *es0)+(*(es1+sz01))))/3.0: 
               
               
                 if (temp&gt;255.0) temp=255.0: 
               
               
                 *es1=(unsigned char) temp: 
               
               
                 temp=((float) (2* ( * (es0+1)) + (* (es1+1+sz01))))/3.0: 
               
               
                 if (temp&gt;255.0) temp=255.0: 
               
               
                 *(es1+1)=(unsigned char) temp: 
               
               
                 temp=((float)(2* ( * (es0+2))+(*(es1+2+sz01))))/3.0: 
               
               
                 if (temp&gt;255.0) temp=255.0: 
               
               
                 *(es1+2)=(unsigned char) temp: 
               
               
                 temp=((float)(2*(*(es0+3))+(*(es1+3+sz01))))/3.0: 
               
               
                 if (temp&gt;255.0) temp=255.0: 
               
               
                 *(es1+3)=(unsigned char) temp: 
               
               
                 ///////////////////////////////////////// 
               
               
                 j+=4: } 
               
               
                 ///////////////////////////////////////// 
               
               
                 ix--; 
               
               
                  j=0 
               
               
                  while (j&lt;sz01) { 
               
               
                 k1=j+i*sz01: 
               
               
                 k2=j+(ix+1)*sz01: 
               
               
                 k3=j+ix*sz01: 
               
               
                 es1=&amp;Img[k2]: 
               
               
                 es0=&amp;pFrameIn[k1]: 
               
               
                 es3=&amp;Img[k3]: 
               
               
                 //////////////////////////////////// 
               
               
                 temp=((float) (4*(es0)−(*(es1+sz01))))/3.0: 
               
               
                 if (temp&lt;0.0) temp=0.0: 
               
               
                 else if (temp&gt;255.0) temp=255.0: 
               
               
                 *es3(unsigned char) temp: 
               
               
                 temp=((float)(4*(*(es0+1))−(*(es1+1+sz01))))/3.0: 
               
               
                 if (temp&lt;0.0) temp=0.0: 
               
               
                 else if (temp&gt;255.0) temp=255.0: 
               
               
                 *(es3+1) = (unsigned char) temp: 
               
               
                 temp = ((float) (4*(*(es0+2)) − (*(es1+2+sz01))))/3.0: 
               
               
                 if (temp&lt;0.0) temp=0.0: 
               
               
                 else if (temp&gt;255.0) temp=255.0: 
               
               
                 *(es3+2)=(unsigned char) temp: 
               
               
                 temp=((float)(4*(*(es0+3)) − (*(es1+3+sz01))))3.0: 
               
               
                 if (temp&lt;0.0) temp=0.0: 
               
               
                 else if (temp&gt;255.0) temp=255.0: 
               
               
                 *(es3+3)=(unsigned char) temp: 
               
               
                 ////////////////////////////////////////////// 
               
               
                 j+=4: } 
               
               
                 ////////////////////////////////////////////// 
               
               
                 jx - - : 
               
               
                 i - - : } 
               
               
                 /////////////////////////////////////////// 
               
               
                 j=0: 
               
               
                 while (j&lt;sz01) { 
               
               
                  Img[j]=(5*Img[j+sz01] ) &gt;&gt; 1 − (3*Img[j+3*sz01])&gt;&gt;1: 
               
               
                 ///////////////////////////////////////////// 
               
               
                 } 
               
               
                 /////////////////////////////VERTICAL/////////////////////////////// 
               
               
                 //////////////////////////////////////////////////////////////////////////// 
               
               
                 //////////////////////////////HORIZONTAL///////////////////////////////////////// 
               
               
                 //////////////////////////////////////////////////////////////////////////////////////////// 
               
               
                 #pragma omp parallel shared(sz.sz01.Img.pFrameOut) 
               
               
                 private(i.k01.k02.jx.j.k1.k2.es2.es1.temp.es3) 
               
               
                 { 
               
               
                 //////////////////////////////////////////////////////////////////////////////////////// 
               
               
                 i=01: 
               
               
                 while (i&lt;sz) { 
               
               
                  k0l=i*sz01: 
               
               
                  k02=i*2*sz01: 
               
               
                 jx=2*sz01-4: 
               
               
                 j=sz01-4: 
               
               
                 k1=k01+j: 
               
               
                 k2=k02+jx: 
               
               
                 es1=&amp;Img[k1]: 
               
               
                 es2=&amp;pFrameOut[k2]: 
               
               
                 memcpy(es2.es1.4): 
               
               
                 j−=4: 
               
               
                 jx−=4: 
               
               
                 while (j&gt;0) { 
               
               
                 k1=k01+j: 
               
               
                 k2=k02+jx: 
               
               
                 es1=&amp;Img[k1]: 
               
               
                 es2=&amp;pFrameOut[k2]: 
               
               
                 //////////////////////////////////////////////////////////////// 
               
               
                 temp=((float) (2*(*es1) + (*(es2+4))))/3.0: 
               
               
                 if (temp&gt;255.0) temp=255.0: 
               
               
                 *es2=(unsigned char) temp: 
               
               
                 temp=((float) (2*(*(es1+1))+(*(es2+5))))/3.0: 
               
               
                 if (temp&gt;255.0) temp=255.0: 
               
               
                 *(es2+1)=(unsigned char) temp: 
               
               
                 temp=((float) (2*(*(es1+2))+(*(es2+6))))/3.0: 
               
               
                 if (temp&gt;255.0) temp = 255.0: 
               
               
                 *(es2+2)=(unsigned char) temp: 
               
               
                 temp=((float) (2*(*(es1+3))+(*(es2+7))))/3.0: 
               
               
                 if (temp&gt;255.0) temp=255.0: 
               
               
                 *(es2+3)=(unsigned char) temp: 
               
               
                 //////////////////////////////////////////////////////////////////// 
               
               
                 jx−=4: 
               
               
                 k3=k02+jx: 
               
               
                 es3=&amp;pFrameOut[k3]: 
               
               
                 //////////////////////////////////////////////////////////////////// 
               
               
                 temp=((float) (4*(*es1) − (*(es3+4))))/3.0: 
               
               
                 if (temp&lt;0.0) temp=0.0: 
               
               
                 else if (temp&gt;255.0) temp=255.0: 
               
               
                 es3=(unsigned char) temp: 
               
               
                 temp=((float) (4*(*(es1+1))−(*(es3+5))))/3.0: 
               
               
                 if (temp&lt;0.0) temp=0.0: 
               
               
                 else if (temp&gt;255.0) temp=255.0: 
               
               
                 *(es3+1)=(unsigned char) temp: 
               
               
                 temp=((float) (4*(*(es1+2)) − (*(es3+6))))/3.0: 
               
               
                 if (temp&lt;0.0) temp=0.0: 
               
               
                 else if (temp&gt;255.0) temp=255.0: 
               
               
                 *(es3+2)=(unsigned char) temp: 
               
               
                 temp=((float) (4*(*(es1+3)) − (*(es3+7))))/3.0: 
               
               
                 if (temp&lt;0.0) temp=0.0: 
               
               
                 else if (temp&gt;255.0) temp=255.0: 
               
               
                 *(es3+3)=(unsigned char) temp: 
               
               
                 /////////////////////////////////////////////////////////////// 
               
               
                 jx−=4: 
               
               
                 j−=4: } 
               
               
                 //////////////////////////////////////////////////////////// 
               
               
                 k2=k02: 
               
               
                 es2&amp;pFrameOut[k2]: 
               
               
                 *es2=(5*(*(es2+4)))&gt;&gt;1−(3*(*(es2+12)))&gt;&gt;1: 
               
               
                 *(es2+1)=(5*(*(es2+5)))&gt;&gt;1−(3*(*(es2+12)))&gt;&gt;1: 
               
               
                 *(es2+2)=(5*(*(es2+6)))&gt;&gt;1−(3*(*(es2+13)))&gt;&gt;1: 
               
               
                 *(es2+2)=(5*(*(es2+7)))&gt;&gt;1−(3*(*(es2+14)))&gt;&gt;1: 
               
               
                 ?????????????????????????????????????????????????????? 
               
               
                 i++: } 
               
               
                 ////////////////////////////HORIZONTAL////////////////////////////////// 
               
               
                 } 
               
               
                 /////////////////////////////////////////////////////////////////////////////////// 
               
               
                 if(Img) delete Img: 
               
               
                 ////////////////////////////////////////////////////////////////////////////////// 
               
               
                 } 
               
               
                 ///////////////////////////////////////////////////////////////////////////////// 
               
               
                   
               
             
          
         
       
     
       Example 6 
     Comparison of Some Video Codecs Compressed Using Methods of this Invention 
       [0217]    Table 3 below presents working examples of video compression improvements of some widely used standard video codecs through application of methods of this invention, without loss of video quality for any given codec. Note that the methods of this invention are applicable to HD 1080i and 1080p videos, HD 720p videos, and SD480i videos, as examples, as well as any other formats. 
         [0000]    
       
         
               
             
               
               
               
               
               
               
             
           
               
                 TABLE 3 
               
               
                   
               
               
                 Improvement of Video Codec Compression Capabilities 
               
               
                 Typical Results for MPEG-4, H-264, VC-1 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                   
                 Codec- 
                 Codec + 
                 Codec + 
                 Codec + 
               
               
                 HD 1080i 
                 Original 
                 Compressed 
                 Invention 
                 Invention 
                 Invention 
               
               
                 Videos 
                 Size 
                 (6 Mbps) 
                 Level 1 
                 Level 2 
                 Level 3 
               
               
                   
               
               
                 Living 
                  37.5 GB 
                  232.3 MB 
                  118.8 MB 
                  81.3 MB 
                  62.5 MB 
               
               
                 Landscapes 
               
               
                 Yellowstone 
                  39.6 GB 
                 244.75 MB 
                  125.4 MB 
                  85.7 MB 
                  65.9 MB 
               
               
                 Planet Earth 
                   75 GB 
                  462.6 MB 
                  237.3 MB 
                 162.3 MB 
                 124.75 MB 
               
               
                 Over 
                   42 GB 
                  240.2 MB 
                  132.6 MB 
                  90.8 MB 
                  69.8 MB 
               
               
                 America 
               
               
                   
               
               
                   
                   
                 Codec- 
               
               
                 HD 720p 
                 Original 
                 Compressed 
                 Codec + ADC2 
                 Codec + ADC2 
                 Codec + ADC2 
               
               
                 Videos 
                 Size 
                 (5 Mbps) 
                 Level 1 
                 Level 2 
                 Level 3 
               
               
                   
               
               
                 KABC 
                 48.76 GB 
                 267.86 MB 
                 139.94 MB 
                 98.36 MB 
                  77.7 MB 
               
               
                   
               
               
                   
                   
                 Codec- 
               
               
                 SD 480i 
                 Original 
                 Compressed 
                 Codec + ADC2 
                 Codec + ADC2 
                 Codec + ADC2 
               
               
                 Videos 
                 Size 
                 (4 Mbps) 
                 Level 1 
                 Level 2 
                 Level 3 
               
               
                   
               
               
                 Football 
                  6.5 GB 
                  157.3 MB 
                  81.7 MB 
                  56.5 MB 
                  43.9 MB 
               
               
                 Basketball 
                  6.5 GB 
                   157 MB 
                  73.9 MB 
                  55.6 MB 
                  43.8 MB 
               
               
                 Tennis 
                  8.1 GB 
                  196.9 MB 
                  92.55 MB 
                  70.8 MB 
                    55 MB 
               
               
                 King Kong 
                  1.3 GB 
                  31.3 MB 
                  16.25 MB 
                  11.2 MB 
                   8.7 MB 
               
               
                 clip 
               
               
                 Line of Fire 
                   20 GB 
                  482.3 MB 
                  250.5 MB 
                 173.3 MB 
                  134.7 MB 
               
               
                 clip 
               
               
                   
               
             
          
         
       
     
       Example 7 
     Further Levels of Size Reduction and Expansion with Wavelet-Based Methods 
       [0218]    The results described in Examples 4 through 6 can be further improved by additional levels of frame size reduction by using for expansion to the previous level any of the filters obtained from the calculations disclosed in U.S. Pat. No. 7,317,840. For example, using a biorthogonal wavelet, the two resulting filters 
         [0000]    
       
         
           
             
               
                 
                   - 
                   0.0575 
                 
               
               
                 1.1151 
               
               
                 
                   - 
                   0.0575 
                 
               
               
                 0.0 
               
             
           
         
       
       
         
           and 
         
       
       
         
           
             
               
                 
                   - 
                   0.0912 
                 
               
               
                 0.591 
               
               
                 0.591 
               
               
                 
                   - 
                   0.0912 
                 
               
             
           
         
       
     
         [0000]    are convolved consecutively with the rows of a given image or frame to convert every pixel to two pixels of each new expanded row. The process is then repeated vertically column by column to obtain an expanded frame that is four times larger than the original frame. 
         [0219]    In additional embodiments, other wavelet based expansion filters can be used, for example, the expansion filters of Table 2. 
         [0220]    Such wavelet-based filters can be applied to the data of Table 3 to further reduce the video file sizes to about ⅛ and about 1/16 of the size produced by any of the codecs of the Table 3 with little or no perceptible loss in video quality. 
       Example 8 
     Results of Application of Methods of this Invention 
       [0221]    It can be seen from the preceding Table 3 that the size reductions of video files compressed using the techniques of this invention are about ½, ⅓, ¼ (depending on the level of size reduction) of the compressed files using the codecs alone. The perceived qualities of the decompressed videos for the different reduction levels are indistinguishable from that of the decompressed videos produced by the codec alone for all the codecs. 
         [0222]      FIGS. 10 through 21  show examples of frame quality produced by a given codec and by the same codec enhanced by pre-processing and post-processing according to methods of this invention. Any differences in quality are clearly imperceptible. 
         [0223]      FIGS. 10 through 21  are arranged in sets of three each, wherein the first figure of each set (i.e.,  FIG. 10 ,  FIG. 13 ,  FIG. 16 , and  FIG. 19  represent photographs of video frames that have been compressed using only a codec to a compression of 6 Mbps. 
         [0224]    The second figure of each set (i.e.,  FIG. 11 ,  FIG. 14 ,  FIG. 17  and  FIG. 20 ) represent photographs of video frames shown in  FIGS. 10 ,  13 ,  16  and  19 , respectively, that have been pre-processed using methods of this invention, then compressed by the codec, decompressed by the codec and finally post-processed using methods of this invention to provide a compression to 3 Mbps. 
         [0225]    The third figure in each set (i.e.,  FIG. 12 ,  FIG. 15 ,  FIG. 18 , and  FIG. 21 ) represent photographs of video frames shown in  FIGS. 10 ,  13 ,  16  and  19 , respectively, that have been pre-processed using methods of this invention, then compressed by the codec, decompressed by the codec and finally post-processed using methods of this invention to provide a compression to 1.5 Mbps. 
         [0226]    It can be readily appreciated that the quality of the images of the second and third figure of each of the above sets are of high quality, and show little, if any, perceptible degradation of image quality. 
       Example 9 
     System for Implementing Pre-Processing, Codecs, and Post-Processing 
       [0227]      FIG. 22  depicts a schematic drawing  2200  of a computer-based system for implementing frame pre-processing, codec compression and decompression, and frame post-processing of this invention. An image of Object  2210  is captured as Frame  2214  by Camera  2212 . Frame  2214  is transferred at step  2216  to First Device  2220 , which contains Buffer  2225  to store Frame  2214 , Memory Device  2230  containing instructions for pre-processing and Pre-Processing Module  2235 . Frame  2214  is transferred to Pre-Processing Module  2235 , and pre-processing steps  2232  of this invention are carried out in Pre-Processing Module  2235 , thereby producing a pre-processed frame. The pre-processed frame is transferred to Codec Compression Module  2240 , where the pre-processed frame is compressed. The compressed frame is transferred at step  2243  to Receiver  2245 , containing Codec Decompression Module  2250 , where the compressed frame is decompressed. The decompressed frame is transferred at step  2253  to Device  2260 , which contains Memory Device  2270  containing instructions for post-processing. Device  2260  also contains Post-Processing Module  2265 , where the decompressed frame is post-processed according to embodiments of this invention. The post-processed frame may be stored in buffer  2275  or transferred directly via step  2277  to Display Monitor  2280 , where Post-Processed Image  2290  is displayed. 
         [0228]      FIG. 22  also shows that optionally, the compressed frame is transferred at step  2244  to Receiver  2246 , containing Storage Device  2251 , where the compressed frame is kept for further use. When the frame is needed to be displayed, it is transferred at step  2254  to device  2245  where it is processed as described above. 
         [0229]    It can be appreciated that similar systems can be constructed in which different codecs are incorporated, each of which receives a pre-processed frame according to methods of this invention, but which are compressed and decompressed using the particular codec. Then, after decompression, post-processing of this invention can be accomplished such that the monitor devices of different systems display images that are similar in quality to each other. 
       Example 10 
     Pre-Processing and Post-Processing Devices 
       [0230]    This invention includes integrated devices for pre-processing and post-processing of frames according to methods disclosed herein.  FIG. 23A  depicts a schematic diagram  2300  of Pre-Processing Device  2301  of this invention. Pre-Processing Device  2301  contains a Memory Area  2302  containing instructions for pre-processing, and Processor  2303  for carrying out instructions contained in Memory Area  2302 . Such combined memory and pre-processing devices may be integrated circuits that can be manufactured separately and then incorporated into video systems. Connection of Pre-Processing Device  2301  into a video system is indicated at Input  2304 , where a frame from an image capture device (e.g., camera) can be input into Pre-Processing Device  2301 . Output of the Pre-Processing Device  2301  is shown at Output  2305 , which can be connected to a codec (not shown). Optionally, a buffer area (not shown) may be included in Pre-Processing Device  2301 . 
         [0231]    Similarly,  FIG. 23B  depicts a schematic diagram  2320  of Post-Processing Device  2321  of this invention. Post-Processing Device  2321  contains a Memory Area  2322  containing instructions for post-processing according to methods of this invention, and also includes Processor  2323  for carrying out instructions contained in Memory Area  2322 . Such combined memory and post-processing devices may be integrated circuits that can be manufactured separately and then incorporated into video systems. Connection of Post-Processing Device  2321  to a video system is indicated at Input  2324 , where a decompressed frame from a codec (not shown) can be input into Post-Processing Device  2321 . Output of the Post-Processing Device  2321  is shown at Output  2325 , which can be attached to an output device, such as a video monitor (not shown). Optionally, a buffer area (not shown) may be included in Post-Processing Device  2321 . 
       Example 11 
     Computer-Readable Devices Containing Instructions for Pre-Processing and Post-Processing 
       [0232]      FIG. 24A  depicts a schematic drawing  2400  of an embodiment of a computer readable device  2401  of this invention. Device  2401  contains Memory Area  2402 , which contains instructions for frame pre-processing according to methods of this invention. Such a device may be a diskette, flash memory, tape drive or other hardware component. Instructions contained on Device  2401  can be transferred at step  2403  to an external pre-processor (not shown) for execution of the instructions contained in Memory Area  2402 . 
         [0233]      FIG. 24B  depicts a schematic drawing  2420  of an embodiment of a computer readable device  2421  of this invention. Device  2421  contains Memory Area  2422 , which contains instructions for frame post-processing according to methods of this invention. Such a device may be a diskette, flash memory, tape drive or other hardware component. Instructions contained on Device  2421  can be transferred at step  2423  to an external post-processor (not shown) for execution of the instructions contained in Memory Area  2422 . 
       Example 12 
     Calculation of Inverse Wavelet Transform (IWT) Using Haar Wavelets 
       [0234]      FIG. 25  is a graphical depiction of a method for calculating the inverse wavelet transform (IWT) using Haar wavelets. The top row of  FIG. 25  represents the x elements of a series of pixels in sequence. The vertical dashed lines represent alignment of the y elements of a series of pixels and the x elements above, and with the z elements of a series of pixels below. Values of the low pass wavelet coefficients are indicated in the three pairs of cells immediately below the top row (x elements). Values of the high pass wavelet coefficients are indicated as the three pairs of cells immediately above the bottom row (z elements). Equations used to calculate the y values corresponding to the pixels are indicated between the cells depicting low pass and high pass wavelet coefficients. 
         [0235]    Also shown in  FIG. 25  are the equations used to calculate y values (y 0 , y 1 , etc.) from the corresponding x values (x 0 , x 1 , etc.) and the corresponding z values (z0, z1, etc.). 
       Example 13 
     Calculation of Wavelet Transform (WT) Using Haar Wavelets 
       [0236]      FIG. 26  is a graphical depiction of a method for calculating wavelet transform (WT) low frequency (LF) and high frequency (HF) values process using Haar wavelets. A series of y elements is shown (y 2n , y 2n−1 , y 0 , etc.). Above the row of y elements are three pairs of cells, each indicating x values of corresponding pixels, along with equations used to calculate the x values shown to the right of each of the pairs of cells. 
         [0237]    Also shown in  FIG. 26  are three pairs of cells shown below the row of y values, each indicating z values of corresponding pixels, along with equations used to calculate the z values shown at the right of each of the pairs of cells. 
         [0238]    Also shown in  FIG. 26  are the equations used to calculate x values (x 0 , x 1 , etc.; on the left side), and the equations used to calculate z values (z 0 , z 1 , etc.; on the right side). 
       REFERENCES 
       [0239]    The following references are each expressly incorporated fully by reference.
       1. Ten Lectures on Wavelets by Ingrid Daubechies, Society for Industrial and Applied Mathematics, 1992.   2. U.S. Pat. No. 7,317,840 entitled: “Methods for Real-Time Software Video/Audio Compression, Transmission, Decompression, and Display,” Angel DeCegama, Inventor.       
 
       INDUSTRIAL APPLICABILITY 
       [0242]    Systems and methods of this invention can be used in the telecommunications and video industries to permit high-quality video to be stored, transmitted and replayed at reduced cost and with reduced requirements for computer storage capacity. The implications of aspects of this invention for the reduction of the current staggering costs of video storage and transmission are significant.