Patent Publication Number: US-8120660-B2

Title: Image data up sampling

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     This invention relates in data processing and more specifically for a filter for image data up sampling. 
     2. Description of the Related Art 
     In some examples of spatial scalable video coding, predictions between different coding layers are utilized to improve performance. In particular, inter layer prediction from a lower resolution base layer to a higher resolution enhancement layer may be utilized for improving coding efficiency on the enhancement layer. Texture inter layer prediction may be done by up sampling using a low-pass filter to avoid aliasing and imaging. Currently, a fixed filter with a one dimensional (1D) 6-tap filter component may be used for low pass filtering. 
     A filter with a 1D 4-tap filter component may be utilized for re-sampling. However, these conventional filters do not provide the same or nearly the same coding efficiency as a filter with a 1D 6-tap filter component. 
     Extended Spatial Scalability (ESS) has been implemented to unite dyadic (power of 2) and non-dyadic image resolution re-sampling. To remove the phase shift of dyadic resampling, luma down-sampling is performed at ½ phase and up sampling is performed at +¼ phase and −¼ phase. The 1D 6-tap filter component utilizes coefficients of [1, −4, 28, 9, −2, 0] for the +¼ phase and coefficients of [0, −2, 9, 28, −4, 1] for −¼ for both the horizontal and vertical dimensions of two dimensional (2D) images. 
     What is desired is an improved filter for image data up sampling. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The present invention may be better understood, and its numerous objects, features, and advantages made apparent to those skilled in the art by referencing the accompanying drawings. 
         FIG. 1  is a block diagram of one embodiment of a video data encoding system and a video data decoding system communicatively coupled by a communications channel according to one embodiment of the present invention. 
         FIG. 2  is a block diagram of a filter according to one embodiment of the present invention. 
         FIG. 3  is a graphic representation of up sampling in a horizontal direction according to one embodiment of the present invention. 
         FIG. 4  is a graphic representation of up sampling in a horizontal and vertical direction according to one embodiment of the present invention. 
         FIGS. 5-9  show graphs of test results comparing a filter according to one embodiment of the present invention with a convention prior art filter. 
     
    
    
     The use of the same reference symbols in different drawings indicates identical items unless otherwise noted. The Figures are not necessarily drawn to scale. 
     DETAILED DESCRIPTION 
     The following sets forth a detailed description of a mode for carrying out the invention. The description is intended to be illustrative of the invention and should not be taken to be limiting. 
     In some embodiments, a filter with a 1D 4-tap filter component is utilized for inter-layer texture up sampling. This filter may be used in place of a conventional filter with a 1D 6-tap filter component for complexity reduction for scalable video coding. In some embodiments, the 1D 4-tap filter component provides for a lower complexity video image re-sampling filtering method with polyphase implementations. In some embodiments, the 1D 4-tap filter component may be used for image up-sampling for inter-layer texture predictions from the lower resolution base layer to the higher resolution enhancement layer. 
       FIG. 1  is a block diagram of one embodiment of a video data encoding system  103  and a video data decoding system  105  communicatively coupled by a communications channel  107 . Encoding system  103  and decoding system  105  may each be implemented in a communication device e.g. a cellular hand set, PDA, computer system, modem, base station, or set top box. Channel  107  may be wireless (WiFi, cellular, wireless broadband, satellite), wire line (e.g. phone, fiber, Ethernet, cable, LAN, WAN) or combination thereof. In one embodiment, systems  103  and  105  maybe implemented in electronic devices communicatively coupled to communication devices. 
     System  103  includes a base layer encoder  109  that receives lower resolution video sequences (Lower Resolution Video In). In one embodiment, the sequences have an image resolution of QCIF, 176×144 pixels. In some example, the lower resolution video sequence are provided by down sampling from an even higher resolution video sequence (e.g. CIF, 352×288 pixels). In one example, the sequence contains image information provided by a video camera. Base layer encoder  109  encodes the lower resolution video sequences into a lower resolution base layer bit stream. In one embodiment, the encoding complies with the H.264 ITU standard. 
     Encoding system  103  also includes a higher resolution video encoder  115  for encoding a higher image resolution video sequences (Video In) to an enhancement layer bit stream. In one embodiment, the higher resolution video sequences are produced by a video camera. In other embodiments, the higher resolution video sequences may be down sampled from a still higher resolution video sequences (e.g. produced by a video camera). In one embodiment, system  103  may include higher resolution encoders (not shown) for providing higher resolution enhancement layers. 
     System  103  also includes a controller  119  for bit rate control of the two encoders  109  and  115  and a MUX  121  for selecting between the base layer bit stream and the enhancement layer bit stream. System  103  also includes circuitry  123  for transforming, transmitting, and/or storing for transmitting the bit stream output of MUX  121  over channel  107 . In some embodiments, circuitry  123  includes a modulator, antenna, transmitter, and/or memory. In one embodiment, system  103  may transmit both the base layer and enhancement layer stream across channel  107 . 
     System  103  includes an interlayer predictor  111  that provides interlayer prediction information from base layer encoder  109  to enhancement layer encoder  115 . In one embodiment, the prediction information includes texture information. In one embodiment, this prediction information may allow for the encoding of a frame or macro block of image data by encoder  115  using the prediction information generated from the lower resolution video sequences. In one embodiment, base layer  109  reconstructs frames, macro blocks, or other image data units from the lower resolution video sequences and provides those to interlayer predictor  111 . 
     Interlayer predictor  111  performs interpolation of the lower resolution images to higher resolution images and provides these images to the higher resolution encoder. The enhancement layer encoder  115  uses these up sampled higher resolution images from predictor  111  as a prediction of the images conveyed by the higher resolution video sequences. 
     Interlayer predictor  111  includes a low pass filter  113  that is utilized for inter-layer texture up sampling. In one embodiment, low pass filter  113  is utilized for avoiding aliasing and imaging problems during interpolation (up sampling). In one embodiment, filter  113  is a Finite Impulse Response (FIR) filter. In one embodiment, filter  113  includes a 1D 4-tap filter component. In one embodiment, filter  113  is a poly phase (e.g. 2, 3, or 4 phase) filter. In one embodiment, filter  113  is a 2 dimensional (2D) separable filter. In one embodiment, the 2D separable filter has the identical one dimensional (1D) components for both the horizontal and vertical dimensions. In one embodiment, filter  113  is a 2D non separable 2D 16 tap filter. 
     System  105  includes bit stream extractor circuitry  125  for extracting a bit stream from the signals received from communication channel  107 . In one embodiment, circuitry  125  includes an antennae, receiver, modem, and/or channel decoder. The bit stream produced by circuitry  125  is provided to base layer decoder  133  and enhancement layer decoder  127 . 
     Base layer decoder  133  decodes the bit stream from circuitry  125  and reconstructs the lower resolution video. The lower resolution video is also provided to interlayer predictor  129 . Predictor  129  processes the lower resolution video and generates higher resolution video images and provides that information to enhancement layer decoder  127 . Enhancement layer decoder  127  uses the image information for prediction of the video frame, macro block, or other image data unit of the enhancement layer. The enhancement layer decoder  127  provides reconstructed higher resolution video. Either the lower or higher resolution video may be displayed on a display device (not shown). 
     Interlayer predictor  129  includes a low pass filter  131 . In one embodiment, low pass filter  131  is utilized for avoiding aliasing and imaging problems during interpolation (up sampling). In one embodiment, filter  131  is a Finite Impulse Response (FIR) filter. In one embodiment, filter  131  includes a 1D 4-tap filter component. In one embodiment, filter  131  is a poly phase (e.g. 2, 3, or 4 phase) filter. In one embodiment, filter  113  is a 2 dimensional (2D) separable filter. In one embodiment, the 2D separable filter has the identical one dimensional (1D) components for both the horizontal and vertical dimensions. In one embodiment, filter  131  is a 2D non separable 2D 16 tap filter. 
     In some embodiments, system  105  may include even higher resolution layers (not shown). In such embodiments, both system  103  and system  105  may include other interlayer predictors for providing images from lower layer encoders or decoders to higher layer encoders or decoders for e.g. prediction. 
     The various blocks shown in  FIG. 1  including filters  113  and  131  maybe implemented in hardware, software, firmware, or combinations thereof. Other encoding and decoding systems may have other configurations in other embodiments. 
       FIG. 2  shows a block diagram of one embodiment of filter  113 . In one embodiment, filter  131  is of the same configuration. In the embodiment shown, filter  113  receives data regarding lower resolution pixel information and outputs higher resolution pixel information. Filter  113  is a 1:X filter in that it interpolates information for more pixels per image than received at its input. In one embodiment, filter  113  is a 1:4 filter that outputs 4 pixels for each received pixel in a 2D dimension. 
     In one embodiment, the information for each pixel received by filter  113  includes an intensity value (e.g. Luma value) of the pixel. In other embodiments, the information for each pixel may include one or more color values (Croma U and Croma V). In some embodiments the pixel information includes all three values and/or other types of values. 
     Filter  113  includes a 2D convolution module  215  for convoluting the pixel information with coefficients (e.g. stored in registers  217 ) to form higher resolution pixel information. In one embodiment, convolution module  215  is a 2D, separable filter with identical 1D horizontal and vertical coefficients for both an X (e.g. horizontal), and a Y (e.g. vertical), dimension. In one embodiment, the filtering order is first on a horizontal dimension and then on a vertical dimension for both encoder and decoder. In another embodiment, the order is first on a vertical dimension and then on a horizontal dimension. 
     In one embodiment, filter  113  is a 2D, separable filter using the same tap of coefficients for both X (e.g. horizontal) and Y (e.g. vertical) dimensions. In other embodiments, to avoid mismatch between the encoder and decoder, the filtering order is the same. In one embodiment, the horizontal image data is up sampled first, the results are saved as a temporary image (e.g. in memory  216 ), and then the temporary image data are up sampled in a vertical dimension. The final resulting image is the up sampled higher resolution image. In other embodiments, the encoder and decoder may start up sample the 2D image data from the vertical dimension and then horizontal dimension. 
     In one embodiment, the 2D, separable convolution module  215  includes two convolutors, one for convoluting the information for the +¼ phase pixels and the other for convoluting the information for the −¼ phase pixels for both horizontal and vertical dimension, respectively. In other embodiments, the 2D, separable convoluting module  215  would include a convolutor that performs 1D convolution on the +¼ phase pixel information and then performs 1D convolution on the −¼ phase pixel information (or vice versa). In other embodiments, convolution module may perform convolution on information for alternating phase pixels. Convolution module  215  may have different configurations in other embodiments. 
     In one embodiment, the intensity and color values of a pixel are convoluted in filter  113  with the coefficients of the filter for up sampling. In other embodiments, only the intensity values would be convoluted by filter  113  with the other color components being up sampled by other methods. In other embodiments, the color values would be up sampled by convolution module  215  with different coefficients (or in a filter similar to filter  113 ). In still other embodiments, the color information (or intensity information in other embodiments) may be up sampled in a filter with a different number of taps. In other embodiment, the color information may be up sampled in different phases. 
       FIG. 3  sets forth a graphic example of 1D horizontal up sampling performed by one embodiment of filter  113  and filter  131 . In, one embodiment, TP(i) represents a sequence of higher resolution image units (the triangles in  FIG. 3 ) of a temporary higher resolution image that holds an intermediate results for following up sampling in the vertical dimension. In one embodiment, the higher resolution image units may have the same number of bits as pixels LP(j) or a different number of bits in other embodiments. LP(j) represents the a sequence of pixels of a lower resolution image used to generate the pixels of the higher resolution image where the two images are of the same time frame. In the embodiment shown, pixel  304  represents the first lower resolution pixel to be considered. 
     In  FIG. 3 , the higher resolution image units are labeled either +¼ or −¼ to represent two phases of up sampled image units where the +¼ represents the positive ¼ phase relative to a lower resolution pixel and the −¼ represents a negative ¼ phase relative to a lower resolution pixel. In the embodiment shown, the higher resolution image units are located ½ phase spacings apart from each other and each are located a ¼ phase spacing from a lower resolution pixel. The lower resolution pixels are spaced a whole phase (sampling interval of the lower layer) difference apart from each other. 
     In the embodiment shown, information for a pair of +¼ phase and −¼ phase higher resolution image units is produced by up sampling the information of four lower resolution pixels. Hence, filter  113  includes a 1D 4-tap filter component where information for one higher resolution image unit is produced from information of four lower resolution image unit. Filter  113  is a polyphase filter in the embodiment shown in that it produces higher resolution pixels at different phases with respect to the lower resolution pixels. 
     In one embodiment, the information for each pixel includes an intensity value (e.g. Luma value) of the pixel, a color value (Croma U), and another color value (Croma V). 
     In the embodiment shown, the information for the higher resolution image units are produced from the lower resolution pixels by first convoluting the information of four lower resolution pixels in the horizontal dimension with the coefficients of the four tap filters (f(j)) according to the following formula: 
                       TP   ⁡     (   i   )       higherres     =       ∑     j   =   1     4     ⁢       f   ⁡     (   j   )       *       LP   ⁡     (   j   )       lowerres                 (   1   )               
where i and j are both in the vertical direction
 
     The intermediate result image data TP(i) are saved (e.g. in memory  216 ). Then, a final higher resolution image data HP(i) is then generated by convoluting the information of four intermediate image units in the vertical dimension with the coefficients of the four tap filters (f(j)) according to the following formula: 
                     HP   ⁡     (   i   )       =       ∑     j   =   1     4     ⁢       f   ⁡     (   j   )       *     TP   ⁡     (   j   )                   (   2   )               
where i and j are both in the vertical direction.
 
       FIG. 4  sets forth a graphic example of 2D up sampling performed by one embodiment of filter  113  and filter  131  in both the horizontal and vertical direction. In  FIG. 4 , the squares are the high resolution pixels HP(i) that are the result of a second up sampling on the intermediate image units (e.g. the triangles). In the embodiment of  FIG. 4 , each square is formed by the convoluting the information of four immediate image units in the vertical direction according to  FIG. 2  above. 
     In one embodiment, the coefficients of f(j) are [−5, 31, 8, −2]/32 for the +¼ phase image units and [−2, 8, 31, −5]/32 for the −¼ phase pixels or image units for both the horizontal and vertical dimensions. The values of the four tap weights are represented here as normalized to 32 in both the horizontal and vertical dimension wherein the sum of the four weights equals 1. In other embodiments, the coefficient may be different in the horizontal direction from the vertical direction. 
     In the embodiment of  FIG. 3 , the information for +¼ phase image unit  311  and −¼ phase image unit  313  are produced by up sampling the information for pixels  307 ,  305 ,  303 , and  301 . In the embodiment with the coefficient weights given above, to produce the information for +¼ phase image unit  311 , the information of pixel  307  is multiplied by −5, the information of pixel  305  is multiplied by 31, the information of pixel  303  is multiplied by 8, and the information of pixel  301  is multiplied by −2. 
     To produce the information for −¼ phase image unit  313  as per the weights given above, the information for pixel  307  is multiplied by −2, the information of pixel  305  is multiplied by 8, the information of pixel  303  is multiplied by 31, and the information of pixel  301  is multiplied by −5. 
     To produce the information for higher resolution image units  321  and  322 , the information of lower resolution pixels  305 ,  303 ,  301 , and  323  are utilized. The information of image unit  321  is produced by multiplying the information of pixel  305  by −5, multiplying the information of pixel  303  by 31, multiplying the information of pixel  301  by 8, and multiplying the information of pixel  323  by −2. To produce the information of image unit  322 , the information of pixel  305  is multiplied by −2, the information of pixel  303  is multiplied by 8, the information of pixel  301  is multiplied by 31, and the information of pixel  323  is multiplied by −5. 
     Referring to  FIG. 4 , after higher resolution image units (TP(i)) are produced by up sampling in the horizontal direction, the higher resolution pixels (HP(i)) are produced by convoluting the higher resolution image units (TP(i)) in the vertical direction with the four coefficients [−5, 31, 8, −2]/32 for the +¼ phase pixels and [−2, 8, 31, −5]/32 for the −¼ phase pixels. 
     In the embodiment shown, the information of higher resolution +¼ phase pixel  431  is produced by multiplying the information of image unit  311  by −5, multiplying the information of image unit  401  by 31, multiplying the information of image unit  417  by 8, and multiplying the information of image unit  425  by −2. The information of higher resolution −¼ phase pixel  433  is produced by multiplying the information of image unit  311  by −2, multiplying the information of image unit  401  by 8, multiplying the information of image unit  417  by 31, and multiplying the information of image unit  425  by −5. 
     The information of higher resolution +¼ phase pixel  435  is produced by multiplying the information of image unit  313  by −5, multiplying the information of image unit  405  by 31, multiplying the information of image unit  418  by 8, and multiplying the information of image unit  427  by −2. The information of higher resolution −¼ phase pixel  437  is produced by multiplying the information of image unit  313  by −2, multiplying the information of image unit  405  by 8, multiplying the information of image unit  418  by 31, and multiplying the information of image unit  427  by −5. 
     In the embodiment shown, the resultant values for each higher resolution pixel (HP(i)) is then divided by 32×32 to obtain the final value for each higher resolution pixel. 
     In other embodiments, convoluting the information of the higher resolution pixels with the above given coefficients may be produced by other methods. For example, the information for each pixel may first be divided by 32 and then multiplied by the normalized numbers and added. Also in other embodiments, the nominator weights may first be divided by 32 before multiplication and then the four multiplied values are summed together. Also in other embodiments, the coefficients may be represented as scaled to another denominator value (e.g. 64). Also in other embodiments, the values of the intermediate image units (TP(i)) may be divided by 32 before being convoluted in the vertical direction with the four coefficients. In that embodiment, the sum from the final value for HP(i) would be divided by 32 instead of 32×32. 
     In one embodiment, the coefficients of filter  113  are convoluted with each of the intensity and two color values for a pixel. However, in other embodiments, the coefficients may be convoluted with less than all of the three values. For example, in one embodiment, the coefficients would be convoluted by the intensity value only. The other two color components would be up sampled by other up sampling methods. Also in other embodiments, other pixel values (e.g. other color values) may be convoluted with the coefficients. 
     In other embodiments, the 2D up sampling may be done concurrently e.g. by performing a 2D convolution. In one embodiment, filter  113  is a non-separable 2D filter. With a non separable filter, the filter would include four sets of 16 taps. 
     In one embodiment of a 2D non separable filter, for a +¼ phase horizontal and +¼ phase vertical, the 2D filter coefficients are defined as: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
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     For a +¼ phase horizontal and −¼ phase vertical, the 2D filter coefficients are defined as: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
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     For a −¼ phase horizontal and +¼ phase vertical, the 2D filter coefficients are defined as: 
     
       
         
           
             
               
                 
                   
                     
                       
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     For a −¼ phase horizontal and −¼ phase vertical, the 2D filter coefficients are defined as: 
     
       
         
           
             
               
                 
                   
                     
                       
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     Note that the four sets of 16 tap filter coefficients as given above are the resultant coefficients of a convolution of two 1D 4-tap filters in two dimensions. Thus, the resultant sets of coefficients filter given above are representative of a 2D separable filter (as described with the embodiments of  FIGS. 3 and 4  above) or a 2D non separable filter. 
     To produce the information for the +¼ phase, +¼ phase pixel  431  in  FIG. 4 , the 16 lower resolution pixels (LP(i)) would be convoluted with the 16 coefficients in equation 3. In such an example, the information for pixel  307  would be multiplied by 25, the information of pixel  301  would be multiplied by 10, the information of  451  would by multiplied by 10, and the information of pixel  453  would be multiplied by 4. Although not explicitly described, the other lower resolution pixels would also be multiplied by the coefficients of the respective locations and summed. The resulted sum is then divided by 32×32. 
     To produce the information for the +¼ phase, −¼ phase pixel  433  in  FIG. 4 , the 16 lower resolution pixels (LP(i)) shown would be convoluted with the 16 coefficients in equation 4. To produce the information for the −¼ phase, +¼ phase pixel  435  in  FIG. 4 , the 16 lower resolution pixels (LP(i)) shown would be convoluted with the 16 coefficients in equation 5. To produce the information for the −¼ phase, −¼ phase pixel  437  in  FIG. 4 , the 16 lower resolution pixels (LP(i)) shown would be convoluted with the 16 coefficients in equation 6. 
     In some embodiments where a 2D nonseparable filter is implemented, intermediate image units TP(i) are not formed. 
     Although the above describes a filter that convolutes information on a pixel bases, in other embodiments, information may be convoluted on the basis of other image units. 
     In the embodiment described, the X direction is set forth in as the horizontal direction and the Y direction is the vertical direction. In other embodiments, convoluting in other directions may be utilized. 
       FIGS. 5-9  set forth test results demonstrating the effectiveness of a 2D, separable filter with a 1D 4-tap filter component having tap coefficients of [−5, 31, 8, −2] for the +¼ phase image units and [−2, 8, 31, −5] for the −¼ phase image units in both the horizontal and vertical directions (“4-tap filter”) as compared to a conventional 2D separable filter with a 1D 6-tap filter having tap coefficients of [1, −4, 28, 9, 2, 0] for the +¼ phase image units and [0, −2, 9, 28, −4, 1] for −¼ phase image units in the horizontal and vertical directions (“6-tap filter”). 
       FIG. 5  sets forth coding efficiency results for simulations using a 4-tap filter described above with respect to the 6-tap filter described above for a first test case (Case #1). The images used for Case #1 had relatively complex scenes. The X-axis of  FIG. 5  is the bit rate of the enhancement layer encoder (encoder  115 ) in kilobits per second (kbps). The Y axis of  FIG. 5  is a quality assessment metric in terms of peak signal to noise ratio (PSNR) of the enhancement layer.  FIG. 5  sets forth two simulations. One is where the quantization parameter of the base layer encoder  109  is 20 which corresponds to a bit rate of 1722.7536 kbps of the base layer encoder (BL Rate). The other simulation is where the quantization parameter of the base layer encoder  109  is 35 which corresponds to a bit rate of 353.0848 kbps of the base layer encoder. The data underlying the graph of  FIG. 5  is shown in Appendix Table 1. The frame rate for the simulations was 30 frames per second. For the simulations, a 10-tap low pass filter was used for down sampling to produce the lower resolution base layer data. 
     As shown in  FIG. 5 , the PSNR line  505  (4-tap BL20) for the 4-tap filter simulation is very close to the PSNR line  501  (6-tap BL20) for the 6-tap filter simulation for the quantization parameter of 20 for the base layer encoder  109 . Also shown in the graph of  FIG. 5 , the PSNR line  507  (4-tap BL35) for the 4-tap filter simulation is very close to the PSNR line  503  (6-tap BL35) for the 6-tap filter simulation for the quantization parameter of 35 for the base layer encoder  109 . 
     Appendix Table 1 also shows results for quantization parameters of 25 and 30 for the base layer encoder  109 . A quantization parameter of 25 corresponds to a base layer bit rate (BL R) of 1122.448 kbps and a quantization parameter of 30 corresponds to a base layer bite rate of 654.336 kbps. Appendix table 1 also sets forth the total bit rate (Total R) which is the enhancement layer (EL Rate) bit rate plus the base layer bit rate (BL Rate). 
       FIG. 6  sets forth a graph of the results for Case #2. The data underlying the graph of  FIG. 6  is set forth in Appendix Table 2. In Case #2, the content of the video used is less complex than the video used for Case #1. The video for Case #2 included images with areas having less contrast. 
     As shown in  FIG. 6 , the PSNR line  607  for the 4-tap filter simulation is very close to the PSNR line  603  for the 6-tap filter simulation for the quantization parameter of 35 of the base layer encoder  109 . The results for base layer quantization parameter of 20 show that the PSNR line  605  for the 4-tap filter simulation is very close to the PSNR line  601  for the 6-tap filter simulation at the lower enhancement layer bit rates (e.g. less than 2000 kbps). At higher enhancement layer bit rates (e.g. above 3000 kbps), there is an up to 0.2 dB degradation for the performance of the 4-tap filter from the performance of the 6-tap filter. 
       FIG. 7  sets forth the results for Case #3. The video for Case #3 used images with a background scene closer to that of Case #1 than Case #2. The data underlying the graph of  FIG. 7  is set forth in Appendix Table 3. As shown in  FIG. 7 , the PSNR line  705  for the 4-tap filter simulation is very close to the PSNR line  701  for the 6-tap filter simulation for the quantization parameter of 20 of the base layer encoder  109 . Also shown in the graph of  FIG. 7 , the PSNR line  707  for the 4-tap filter simulation is very close to the PSNR line  703  for the 6-tap filter simulation for the quantization parameter of 35 of the base layer encoder  109 . 
       FIG. 8  sets forth a graph of the results for Case #4. The data underlying the graph of  FIG. 8  is set forth in Appendix Table 4. In Case #4, the content of the video is less complex than the video used for Cases #1, #2, and #3. The video for Case #4 includes images with an area having less contrast than the video for Cases #1, #2, and #3. 
     As shown in  FIG. 8 , the PSNR line  807  for the 4-tap filter simulation is very close to the PSNR line  803  for the 6-tap filter simulation for the quantization parameter of 35 of the base layer encoder  109 . The results for base layer quantization parameter of 20 show that there is an up to 0.2 dB degradation with the 4-tap filter (line  805 ) as opposed to the 6-tap filter (line  801 ). 
       FIG. 9  sets forth a graph of the results of Case #5. The data underlying the graph of  FIG. 9  is set forth in Appendix Table 5. In Case #5, the content of the video is similar in complexity to that of Case #1 and Case #3. As shown in  FIG. 9 , the PSNR line  905  for the 4-tap filter simulation is very close to the PSNR line  901  for the 6-tap filter simulation for the quantization parameter of 20 of the base layer encoder  109 . Also shown in the graph of  FIG. 9 , the PSNR line  907  for the 4-tap filter simulation is very close to the PSNR line  903  for the 6-tap filter simulation for the quantization parameter of 35 of the base layer encoder  109 . 
     As shown from the results set forth in  FIGS. 5-9  and Appendix Tables 1-5, the differences in performance between the conventional filter and the 4-tap filter is negligible for Cases #1, # 3, and #5. For Cases #2 and #4 which have images with a less complex scene, there was a 0.2 dB degradation or less for simulations with the base layer quantization parameter of 20 for the higher enhancement layer bit rates. For cases #2 and #4, the performance between the two filters was negligible for simulations having a base layer quantization parameter of 35. Accordingly, the poly phase 2D filter with two 1D 4-tap filtering used above provides performance that is very comparable if not negligible in most instances from a conventional 2D filter with 2 1D 6-tap filtering. 
     Furthermore, utilizing a filter a 1D 4-tap filter component in each direction instead of a conventional 6-tap filter reduces the complexity of the codec of the interlayer prediction module (e.g.  11  and  129 ). In some embodiments where the up sampling filter is implemented in hardware, this reduction in complexity may result in a reduction in total circuitry and/or a reduction in power consumption due to the filter having less taps. In some embodiments where the up sampling filter is implemented in software, the reduction in taps of the filter may result in a reduction in processor computations, a reduction in power, and/or a reduction in memory requirements. 
     Accordingly, the embodiments set forth above provide for a filter that obtains the benefits of less complexity with a negligible or nearly negligible reduction in performance from a filter having a greater number of taps in the 1D dimension. 
     Furthermore, using a multiphase filter in one embodiment, may provide for better coding efficiency over single phase up sampling filters. 
     While particular embodiments of the present invention have been shown and described, it will be recognized to those skilled in the art that, based upon the teachings herein, further changes and modifications may be made without departing from this invention and its broader aspects, and thus, the appended claims are to encompass within their scope all such changes and modifications as are within the true spirit and scope of this invention. 
     
       
         
           
               
             
               
                 APPENDIX TABLE 1 
               
               
                   
               
               
                 Case #1 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                 BL QP = 20 
                 BL Rate 
                   
                 BL PSNR 
                   
                   
                   
               
               
                 6-tap 
                 1722.7536 
                   
                 42.7226 
                 4-tap 
               
               
                 EL QP 
                 Total R 
                 EL Rate 
                 EL PSNR 
                 Total R 
                 EL Rate 
                 EL PSNR 
               
               
                   
               
               
                 20 
                 7295.136 
                 5572.382 
                 42.4133 
                 7300.562 
                 5577.808 
                 42.4087 
               
               
                 25 
                 5081.4128 
                 3358.659 
                 38.5501 
                 5084.363 
                 3361.61 
                 38.5417 
               
               
                 30 
                 3298.3936 
                 1575.64 
                 34.7004 
                 3288.998 
                 1566.245 
                 34.6829 
               
               
                 35 
                 2286.6976 
                 563.944 
                 31.8166 
                 2270.338 
                 547.584 
                 31.8205 
               
               
                 40 
                 1857.52 
                 134.7664 
                 29.9303 
                 1851.104 
                 128.3504 
                 29.9923 
               
               
                   
               
               
                   
                 BL R 
                   
                 BL PSNR 
                   
                   
                   
               
               
                 BL QP = 25 
                 1122.448 
                   
                 38.5447 
               
               
                 EL QP 
                 Total R 
                 EL Rate 
                 EL PSNR 
                 Total R 
                 EL Rate 
                 EL PSNR 
               
               
                   
               
               
                 20 
                 6964.1808 
                 5841.733 
                 42.4551 
                 6977.746 
                 5855.298 
                 42.455 
               
               
                 25 
                 4720.6208 
                 3598.173 
                 38.5468 
                 4732.899 
                 3610.451 
                 38.5426 
               
               
                 30 
                 2872.256 
                 1749.808 
                 34.565 
                 2874.997 
                 1752.549 
                 34.5466 
               
               
                 35 
                 1742.8112 
                 620.3632 
                 31.4812 
                 1731.093 
                 608.6448 
                 31.4587 
               
               
                 40 
                 1266.0768 
                 143.6288 
                 29.5495 
                 1259.214 
                 136.7664 
                 29.5807 
               
               
                   
               
               
                   
                 BL R 
                   
                 BL PSNR 
                   
                   
                   
               
               
                 BL QP = 30 
                 654.336 
                   
                 34.386 
               
               
                 EL QP 
                 Total R 
                 EL Rate 
                 EL PSNR 
                 Total R 
                 EL Rate 
                 EL PSNR 
               
               
                   
               
               
                 20 
                 6793.488 
                 6139.152 
                 42.5313 
                 6804.171 
                 6149.835 
                 42.5331 
               
               
                 25 
                 4533.1824 
                 3878.846 
                 38.5814 
                 4546.682 
                 3892.346 
                 38.5832 
               
               
                 30 
                 2676.5808 
                 2022.245 
                 34.528 
                 2684.555 
                 2030.219 
                 34.518 
               
               
                 35 
                 1455.5728 
                 801.2368 
                 31.1246 
                 1453.008 
                 798.672 
                 31.0989 
               
               
                 40 
                 834.344 
                 180.008 
                 28.8088 
                 828.1872 
                 173.8512 
                 28.8002 
               
               
                   
               
               
                   
                 BL R 
                   
                 BL PSNR 
                   
                   
                   
               
               
                 BL QP = 35 
                 353.0848 
                   
                 30.8376 
               
               
                 EL QP 
                 Total R 
                 EL Rate 
                 EL PSNR 
                 Total R 
                 EL Rate 
                 EL PSNR 
               
               
                   
               
               
                 20 
                 6726.9232 
                 6373.838 
                 42.6274 
                 6735.158 
                 6382.074 
                 42.6317 
               
               
                 25 
                 4465.0784 
                 4111.994 
                 38.6459 
                 4472.483 
                 4119.398 
                 38.6481 
               
               
                 30 
                 2628.9952 
                 2275.91 
                 34.5506 
                 2637.29 
                 2284.205 
                 34.5466 
               
               
                 35 
                 1392.1744 
                 1039.09 
                 31.0584 
                 1396.72 
                 1043.635 
                 31.0458 
               
               
                 40 
                 664.5472 
                 311.4624 
                 28.148 
                 662.8384 
                 309.7536 
                 28.126 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
             
               
                 APPENDIX TABLE 2 
               
               
                   
               
               
                 Case #2 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                 BL = 20 
                 BL Rate 
                   
                 BL PSNR 
                   
                   
                   
               
               
                 6-tap 
                 1730.736 
                   
                 43.1295 
                 4-tap 
               
               
                 EL QP 
                 Total R 
                 EL Rate 
                 EL PSNR 
                 Total R 
                 EL Rate 
                 EL PSNR 
               
               
                   
               
               
                 20 
                 5337.7136 
                 3606.978 
                 43.628 
                 5386.0704 
                 3655.334 
                 43.5688 
               
               
                 25 
                 3748.2672 
                 2017.531 
                 40.293 
                 3773.5968 
                 2042.861 
                 40.2351 
               
               
                 30 
                 2597.0048 
                 866.2688 
                 37.0695 
                 2600.632 
                 869.896 
                 37.0132 
               
               
                 35 
                 2045.7328 
                 314.9968 
                 34.8507 
                 2040.1424 
                 309.4064 
                 34.7786 
               
               
                 40 
                 1839.2352 
                 108.4992 
                 33.3553 
                 1832.0048 
                 101.2688 
                 33.2971 
               
               
                   
               
               
                 BL = 25 
                 BL Rate 
                   
                 BL PSNR 
                   
                   
                   
               
               
                 6-tap 
                 1142.6496 
                   
                 39.1136 
               
               
                 EL QP 
                 Total R 
                 EL Rate 
                 EL PSNR 
                 Total R 
                 EL Rate 
                 EL PSNR 
               
               
                   
               
               
                 20 
                 5092.7248 
                 3950.075 
                 43.6025 
                 5144.8176 
                 4002.168 
                 43.5517 
               
               
                 25 
                 3471.1824 
                 2328.533 
                 40.2411 
                 3499.1168 
                 2356.467 
                 40.182 
               
               
                 30 
                 2193.264 
                 1050.614 
                 36.6128 
                 2202.0528 
                 1059.403 
                 36.5587 
               
               
                 35 
                 1497.016 
                 354.3664 
                 34.1084 
                 1492.8368 
                 350.1872 
                 34.0281 
               
               
                 40 
                 1256.088 
                 113.4384 
                 32.6665 
                 1248.7328 
                 106.0832 
                 32.6099 
               
               
                   
               
               
                 BL = 30 
                 BL Rate 
                   
                 BL PSNR 
                   
                   
                   
               
               
                 6-tap 
                 677.3088 
                   
                 35.1289 
               
               
                 EL QP 
                 Total R 
                 EL Rate 
                 EL PSNR 
                 Total R 
                 EL Rate 
                 EL PSNR 
               
               
                   
               
               
                 20 
                 5002.248 
                 4324.939 
                 43.5989 
                 5052.7088 
                 4375.4 
                 43.57 
               
               
                 25 
                 3347.3408 
                 2670.032 
                 40.219 
                 3373.3952 
                 2696.086 
                 40.167 
               
               
                 30 
                 2035.448 
                 1358.139 
                 36.5194 
                 2045.3968 
                 1368.088 
                 36.4709 
               
               
                 35 
                 1194.112 
                 516.8032 
                 33.2693 
                 1195.584 
                 518.2752 
                 33.2199 
               
               
                 40 
                 816.6208 
                 139.312 
                 31.4499 
                 810.2736 
                 132.9648 
                 31.3825 
               
               
                   
               
               
                 BL = 35 
                 BL Rate 
                   
                 BL PSNR 
                   
                   
                   
               
               
                 6-tap 
                 381.376 
                   
                 31.9458 
               
               
                 EL QP 
                 Total R 
                 EL Rate 
                 EL PSNR 
                 Total R 
                 EL Rate 
                 EL PSNR 
               
               
                   
               
               
                 20 
                 5000.7264 
                 4619.35 
                 43.623 
                 5044.808 
                 4663.432 
                 43.6193 
               
               
                 25 
                 3319.9696 
                 2938.594 
                 40.2125 
                 3347.2608 
                 2965.885 
                 40.1843 
               
               
                 30 
                 1989.8928 
                 1608.517 
                 36.4832 
                 2002.888 
                 1621.512 
                 36.4442 
               
               
                 35 
                 1124.2608 
                 742.8848 
                 33.1729 
                 1129.7552 
                 748.3792 
                 33.1323 
               
               
                 40 
                 625.512 
                 244.136 
                 30.4503 
                 625.2256 
                 243.8496 
                 30.3922 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
             
               
                 APPENDIX TABLE 3 
               
               
                   
               
               
                 Case #3 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                 6-tap 
                 BL Rate 
                   
                 BL PSNR 
                   
                   
                   
               
               
                 QP = 20 
                 2141.912 
                   
                 42.552 
                 4-tap 
               
               
                 EL QP 
                 Total R 
                 EL Rate 
                 EL PSNR 
                 Total R 
                 EL Rate 
                 EL PSNR 
               
               
                   
               
               
                 20 
                 8659.694 
                 6517.782 
                 42.38 
                 8652.486 
                 6510.574 
                 42.3754 
               
               
                 25 
                 6363.418 
                 4221.506 
                 38.471 
                 6351.462 
                 4209.55 
                 38.4695 
               
               
                 30 
                 4466.595 
                 2324.683 
                 34.4025 
                 4445.773 
                 2303.861 
                 34.402 
               
               
                 35 
                 3260.453 
                 1118.541 
                 31.0774 
                 3237.52 
                 1095.608 
                 31.0952 
               
               
                 40 
                 2606.333 
                 464.4208 
                 28.4701 
                 2587.797 
                 445.8848 
                 28.5029 
               
               
                   
               
               
                   
                 BL Rate 
                   
                 BL PSNR 
                   
                   
                   
               
               
                 BL QP = 25 
                 1472.646 
                   
                 38.1964 
               
               
                 EL QP 
                 Total R 
                 EL Rate 
                 EL PSNR 
                 Total R 
                 EL Rate 
                 EL PSNR 
               
               
                   
               
               
                 20 
                 8218.371 
                 6745.725 
                 42.3969 
                 8216.155 
                 6743.509 
                 42.3949 
               
               
                 25 
                 5907.581 
                 4434.934 
                 38.4688 
                 5904.106 
                 4431.459 
                 38.4646 
               
               
                 30 
                 3958.371 
                 2485.725 
                 34.2929 
                 3948.221 
                 2475.574 
                 34.2843 
               
               
                 35 
                 2650.024 
                 1177.378 
                 30.827 
                 2631.131 
                 1158.485 
                 30.8216 
               
               
                 40 
                 1950.758 
                 478.112 
                 28.2202 
                 1932.578 
                 459.9312 
                 28.2362 
               
               
                   
               
               
                   
                 BL Rate 
                   
                 BL PSNR 
                   
                   
                   
               
               
                 BL QP = 30 
                 920.1568 
                   
                 33.8248 
               
               
                 EL QP 
                 Total R 
                 EL Rate 
                 EL PSNR 
                 Total R 
                 EL Rate 
                 EL PSNR 
               
               
                   
               
               
                 20 
                 7934.419 
                 7014.262 
                 42.4412 
                 7935.8 
                 7015.643 
                 42.4402 
               
               
                 25 
                 5603.566 
                 4683.41 
                 38.487 
                 5604.949 
                 4684.792 
                 38.4831 
               
               
                 30 
                 3645.854 
                 2725.698 
                 34.2721 
                 3643.862 
                 2723.706 
                 34.2629 
               
               
                 35 
                 2272.112 
                 1351.955 
                 30.5708 
                 2262.997 
                 1342.84 
                 30.553 
               
               
                 40 
                 1447.093 
                 526.936 
                 27.7337 
                 1432.688 
                 512.5312 
                 27.7223 
               
               
                   
               
               
                   
                 BL Rate 
                   
                 BL PSNR 
                   
                   
                   
               
               
                 BL QP = 35 
                 534.8912 
                   
                 30.112 
               
               
                 EL QP 
                 Total R 
                 EL Rate 
                 EL PSNR 
                 Total R 
                 EL Rate 
                 EL PSNR 
               
               
                   
               
               
                 20 
                 7787.963 
                 7253.072 
                 42.5152 
                 7795.155 
                 7260.264 
                 42.5141 
               
               
                 25 
                 5438.282 
                 4903.39 
                 38.526 
                 5445.547 
                 4910.656 
                 38.5263 
               
               
                 30 
                 3479.202 
                 2944.31 
                 34.2801 
                 3484.038 
                 2949.147 
                 34.2767 
               
               
                 35 
                 2095.776 
                 1560.885 
                 30.5344 
                 2097.658 
                 1562.766 
                 30.5203 
               
               
                 40 
                 1200.469 
                 665.5776 
                 27.3382 
                 1195.89 
                 660.9984 
                 27.3149 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
             
               
                 APPENDIX TABLE 4 
               
               
                   
               
               
                 Case #4 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                 BL QP = 20 
                 BL R 
                   
                 BL PSNR 
                   
                   
                   
               
               
                 6-tap 
                 1403.013 
                   
                 43.1379 
                 4-tap 
               
               
                 EL QP 
                 Total R 
                 EL Rate 
                 EL PSNR 
                 Total R 
                 EL Rate 
                 EL PSNR 
               
               
                   
               
               
                 20 
                 5234.427 
                 3831.414 
                 43.0616 
                 5323.443 
                 3920.43 
                 43.0391 
               
               
                 25 
                 3379.555 
                 1976.542 
                 39.7105 
                 3450.586 
                 2047.573 
                 39.6571 
               
               
                 30 
                 2203.397 
                 800.384 
                 36.7146 
                 2247.488 
                 844.4752 
                 36.5816 
               
               
                 35 
                 1700.968 
                 297.9552 
                 34.4438 
                 1692.619 
                 289.6064 
                 34.1679 
               
               
                 40 
                 1480.099 
                 77.0864 
                 32.7471 
                 1476.149 
                 73.136 
                 32.6568 
               
               
                   
               
               
                   
                 BL R 
                   
                 BL PSNR 
                   
                   
                   
               
               
                 BL QP = 25 
                 903.9136 
                   
                 39.4748 
               
               
                 EL QP 
                 Total R 
                 EL Rate 
                 EL PSNR 
                 Total R 
                 EL Rate 
                 EL PSNR 
               
               
                   
               
               
                 20 
                 5005.998 
                 4102.085 
                 43.0723 
                 5077.426 
                 4173.512 
                 43.0588 
               
               
                 25 
                 3132.779 
                 2228.866 
                 39.665 
                 3187.741 
                 2283.827 
                 39.6288 
               
               
                 30 
                 1853.926 
                 950.0128 
                 36.3912 
                 1893.64 
                 989.7264 
                 36.3211 
               
               
                 35 
                 1233.184 
                 329.2704 
                 33.9246 
                 1232.254 
                 328.3408 
                 33.7132 
               
               
                 40 
                 985.5856 
                 81.672 
                 32.2408 
                 981.0144 
                 77.1008 
                 32.1505 
               
               
                   
               
               
                   
                 BL R 
                   
                 BL PSNR 
                   
                   
                   
               
               
                 BL QP = 30 
                 550.7504 
                   
                 35.7681 
               
               
                 EL QP 
                 Total R 
                 EL Rate 
                 EL PSNR 
                 Total R 
                 EL Rate 
                 EL PSNR 
               
               
                   
               
               
                 20 
                 4897.35 
                 4346.6 
                 43.1013 
                 4939.6 
                 4388.85 
                 43.0982 
               
               
                 25 
                 3017.968 
                 2467.218 
                 39.6836 
                 3051.616 
                 2500.866 
                 39.6624 
               
               
                 30 
                 1732.76 
                 1182.01 
                 36.3348 
                 1757.203 
                 1206.453 
                 36.2886 
               
               
                 35 
                 1001.093 
                 450.3424 
                 33.3478 
                 1005.758 
                 455.008 
                 33.243 
               
               
                 40 
                 655.0752 
                 104.3248 
                 31.3399 
                 649.6336 
                 98.8832 
                 31.2577 
               
               
                   
               
               
                   
                 BL R 
                   
                 BL PSNR 
                   
                   
                   
               
               
                 BL QP = 35 
                 334.9952 
                   
                 32.2926 
               
               
                 EL QP 
                 Total R 
                 EL Rate 
                 EL PSNR 
                 Total R 
                 EL Rate 
                 EL PSNR 
               
               
                   
               
               
                 20 
                 4830.565 
                 4495.57 
                 43.1345 
                 4845.571 
                 4510.576 
                 43.14 
               
               
                 25 
                 2969.75 
                 2634.755 
                 39.7229 
                 2985.416 
                 2650.421 
                 39.7212 
               
               
                 30 
                 1702.907 
                 1367.912 
                 36.3584 
                 1719.035 
                 1384.04 
                 36.3367 
               
               
                 35 
                 968.1936 
                 633.1984 
                 33.2666 
                 975.8944 
                 640.8992 
                 33.2145 
               
               
                 40 
                 528.6912 
                 193.696 
                 30.4095 
                 526.8912 
                 191.896 
                 30.3446 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
             
               
                 APPENDIX TABLE 5 
               
               
                   
               
               
                 Case #5 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                 6-tap 
                   
                 BL Rate 
                 BL PSNR 
                   
                   
                   
               
               
                 BL QP 20 
                   
                 3349.24 
                 42.5441 
                 4-tap 
               
               
                 EL QP 
                 Total R 
                 EL Rate 
                 EL PSNR 
                 Total R 
                 EL Rate 
                 EL PSNR 
               
               
                   
               
               
                 20 
                 13708.256 
                 10359.02 
                 42.3147 
                 13727.68 
                 10378.44 
                 42.318 
               
               
                 25 
                 10556.0144 
                 7206.774 
                 38.0392 
                 10569.54 
                 7220.304 
                 38.0386 
               
               
                 30 
                 7762.7056 
                 4413.466 
                 33.4536 
                 7762.549 
                 4413.309 
                 33.4399 
               
               
                 35 
                 5719.384 
                 2370.144 
                 29.3448 
                 5702.638 
                 2353.398 
                 29.3186 
               
               
                 40 
                 4389.8992 
                 1040.659 
                 25.9553 
                 4358.186 
                 1008.946 
                 25.9345 
               
               
                   
               
               
                   
                   
                 BL Rate 
                 BL PSNR 
                   
                   
                   
               
               
                 BL QP 25 
                   
                 2458.88 
                 37.8697 
               
               
                 EL QP 
                 Total R 
                 EL Rate 
                 EL PSNR 
                 Total R 
                 EL Rate 
                 EL PSNR 
               
               
                   
               
               
                 20 
                 13006.2608 
                 10547.38 
                 42.3442 
                 13022.97 
                 10564.09 
                 42.3523 
               
               
                 25 
                 9860.6448 
                 7401.765 
                 38.0584 
                 9875.55 
                 7416.67 
                 38.0619 
               
               
                 30 
                 7032.9296 
                 4574.05 
                 33.4329 
                 7038.981 
                 4580.101 
                 33.4244 
               
               
                 35 
                 4914.0496 
                 2455.17 
                 29.2794 
                 4898.218 
                 2439.338 
                 29.2516 
               
               
                 40 
                 3533.4464 
                 1074.566 
                 25.8627 
                 3501.502 
                 1042.622 
                 25.8376 
               
               
                   
               
               
                   
                   
                 BL Rate 
                 BL PSNR 
                   
                   
                   
               
               
                 BL QP 30 
                   
                 1656.936 
                 33.0581 
               
               
                 EL QP 
                 Total R 
                 EL Rate 
                 EL PSNR 
                 Total R 
                 EL Rate 
                 EL PSNR 
               
               
                   
               
               
                 20 
                 12451.744 
                 10794.81 
                 42.3967 
                 12472.23 
                 10815.3 
                 42.4034 
               
               
                 25 
                 9313.9088 
                 7656.973 
                 38.0969 
                 9326.706 
                 7669.77 
                 38.102 
               
               
                 30 
                 6496.1584 
                 4839.222 
                 33.445 
                 6506.048 
                 4849.112 
                 33.4436 
               
               
                 35 
                 4319.0672 
                 2662.131 
                 29.2015 
                 4314.717 
                 2657.781 
                 29.1798 
               
               
                 40 
                 2835.8144 
                 1178.878 
                 25.6563 
                 2811.155 
                 1154.219 
                 25.6206 
               
               
                   
               
               
                   
                   
                 BL Rate 
                 BL PSNR 
                   
                   
                   
               
               
                 BL QP 35 
                   
                 1030.901 
                 28.7886 
               
               
                 EL QP 
                 Total R 
                 EL Rate 
                 EL PSNR 
                 Total R 
                 EL Rate 
                 EL PSNR 
               
               
                   
               
               
                 20 
                 12131.6512 
                 11100.75 
                 42.4776 
                 12153.51 
                 11122.6 
                 42.4874 
               
               
                 25 
                 8992.1632 
                 7961.262 
                 38.1609 
                 9011.858 
                 7980.957 
                 38.1682 
               
               
                 30 
                 6194.5744 
                 5163.674 
                 33.4987 
                 6213.43 
                 5182.53 
                 33.5009 
               
               
                 35 
                 4016.7408 
                 2985.84 
                 29.2012 
                 4025.827 
                 2994.926 
                 29.1931 
               
               
                 40 
                 2451.4032 
                 1420.502 
                 25.445 
                 2446.87 
                 1415.97 
                 25.411