Abstract:
Described is a technology in which a low resolution image is processed into a high-resolution image, including by a two interpolation passes. In the first pass, missing in-block pixels, which are the pixels within a block formed by four neighboring original pixels, are given values by gradient diffusion based upon interpolation of the surrounding original pixels. In the second interpolation pass, missing on-block pixels, which are the pixels on a block edge formed by two adjacent original pixels, are given values by gradient diffusion based upon interpolation of the values of those adjacent original pixels and the previously interpolated values of their adjacent in-block pixels. Also described is a difference projection process that varies the values of the interpolated pixels according to a computed difference projection.

Description:
BACKGROUND 
     Increasing the resolution of images is useful in providing viewers with a better observation experience. To this end, image interpolation is used in many real world applications to fill in missing pixels generally based on surrounding information. In general, two criteria are used to evaluate the performance of an image interpolator, namely perceptual quality and computational complexity. 
     Conventional linear operators like bilinear and bicubic image interpolation are relatively simple and fast, but often introduce annoying “baggy” artifacts around the edges, primarily because local features in images are not taken into consideration. Therefore, various adaptive image interpolators have been implemented in an attempt to better preserve the edges, by utilizing more accurate models. 
     However, such models suffer from a number of drawbacks, including computational inefficiency. For example, due to the iterative property and/or significant complexity of reliable estimation of adaptive coefficients, the overall computational cost may be much higher than that of linear interpolators, even when hybrid algorithms are used to reduce the complexity. 
     Another drawback is that some models limit edge orientations to several predefined choices, which affects the accuracy of the imposed model. Other interpolators have a limited interpolation ratio, that is, many interpolators are restricted to a ratio of 2 n ; interpolation to another ratio requires re-sampling from a higher 2 n  image. 
     SUMMARY 
     This Summary is provided to introduce a selection of representative concepts in a simplified form that are further described below in the Detailed Description. This Summary is not intended to identify key features or essential features of the claimed subject matter, nor is it intended to be used in any way that would limit the scope of the claimed subject matter. 
     Briefly, various aspects of the subject matter described herein are directed towards a technology by which a low resolution image is processed into a high-resolution image, including by a two pass interpolation. In a first pass, a first set of pixel values are interpolated for a first set of missing pixels based on original pixel values. In a second pass, a second set of pixel values are interpolated for a second set of missing pixels based on the original pixel values and the interpolated first set of pixel values. 
     In one aspect, the first set of pixels contains in-block pixels, in which each in-block pixel corresponds to a pixel location within a block formed by four of the original pixels. The second set of pixels contains on-block pixels, in which each on-block pixel corresponds to a pixel location on an edge formed by two of the original pixels. 
     In one aspect, a difference projection is performed on the interpolated pixels. The difference projection may be performed by computing virtual interpolation values for the original pixels from the surrounding interpolated pixels, computing the differences between the original pixels and the virtual interpolation values, and then adjusting values of interpolated pixels by the projected differences, e.g., adding the projected differences. 
     Other advantages may become apparent from the following detailed description when taken in conjunction with the drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The present invention is illustrated by way of example and not limited in the accompanying figures in which like reference numerals indicate similar elements and in which: 
         FIG. 1  is a block diagram showing an example directional image interpolator with difference projection that interpolates a low resolution image into a high resolution image. 
         FIG. 2  is a representation of pixels including original pixels, in-block pixels to be interpolated and on-block pixels to be interpolated. 
         FIG. 3  is a representation of interpolation of an in-block pixel based on its relationship to original pixels. 
         FIG. 4  is a representation of interpolation of an on-block pixel based on its relationship to original pixels and in-block pixels. 
         FIG. 5  is a representation of a virtual original interpolation layout. 
         FIG. 6  is a representation of difference projection. 
         FIG. 7  is a flow diagram representing example steps for interpolating a low resolution image into a higher resolution image. 
         FIG. 8  shows an illustrative example of a computing environment into which various aspects of the present invention may be incorporated. 
     
    
    
     DETAILED DESCRIPTION 
     Various aspects of the technology described herein are generally directed towards a directional interpolator in which gradients are diffused for the high-resolution image to determine the orientations of local minimum variation at missing pixels, along which linear interpolation is performed. In this manner, arbitrary edge orientations can be detected and utilized. Then, the continuities between original and interpolated pixels are enforced by “difference projection”, which can be viewed as a reapplication of the described interpolator. 
     While some of the examples described herein are directed towards an interpolator based on gradient diffusion and bilinear interpolation, it is understood that these are only examples. Other types of interpolation may be performed. As such, the present invention is not limited to any particular embodiments, aspects, concepts, structures, functionalities or examples described herein. Rather, any of the embodiments, aspects, concepts, structures, functionalities or examples described herein are non-limiting, and the present invention may be used in various ways that provide benefits and advantages in computing and image processing in general. 
     Turning to  FIG. 1 , there is shown a directional interpolator  102  comprising an algorithm that interpolates a low-resolution image X of size H×W into a high-resolution image Y of size nH×nW. For purposes of concise description herein, and without loss of generality, the examples use n=3; thus, 3× interpolation is used as an example, which may be easily extended to other, n× interpolation. 
     As illustrated in  FIG. 2 , the pixels in image Y (thirty-six are shown) are divided into three categories. The black dots represent copies of original the pixels from X; the gray and white dots are the missing pixels to be interpolated into the pixel space. With the four neighboring black dots forming a square block (marked via the dashed line), the gray dots within the dashed block are referred to as “in-block” pixels. The white dots are on edges formed by two adjacent original pixels, (on a dashed line) and are referred to as “on-block” pixels. 
     In one method, the “in-block” interpolation is first performed in the square to generate pixel values for the gray dots, (as generally represented via steps  702 - 704  of  FIG. 7 ) and then the “on-block” interpolation is carried out in two hexagons (marked via the dotted lines and shown in  FIG. 4  and via steps  706 - 708  of  FIG. 7 ) to generate pixel values for the white dots. This two-pass strategy exploits the correlations between neighboring pixels, compared with one-pass algorithms that in essence only use the original pixels (the black dots herein). The fixed positions of available neighbors facilitate the implementation. 
     In one implementation, gradient diffusion is used for interpolation. More particularly, the weights of available pixels for interpolation are determined by the estimated gradient of the missing pixel. In general, the orientation of local minimum variation at a certain pixel can be derived from the direction of its gradient, as they are perpendicular to each other. Thus the interpolator makes use of various edge orientations with gradient-adaptive interpolation weights. 
     To obtain the gradients of missing pixels in Y, the gradients on X are first calculated, and then bilinearly interpolated to the high-resolution. This process is referred to as “gradient diffusion”. More particularly, there is a kind of edge pixel referred as a “ridge” whose gradient value is near zero. The process marks these pixels separately and estimates the edge orientations from their neighborhood. 
     After the high-resolution gradients are obtained, the four in-block pixels are interpolated as indexed by I 0 , I 1 , I 2  and I 3  in the right portion of  FIG. 3 . Because they are symmetric in geometry, only the interpolation of I 0  is shown as an example for purposes of brevity herein. 
     As shown in the left portion of  FIG. 3 , there is a line/perpendicular to the gradient direction (denoted by the arrow) at I 0 , along which the local variation is minimum. The line/has two intersections, P and Q, with the square formed by the original pixels A, B, C and D. There are three kinds of distributions of P and Q according to different slopes of I (denoted as k). With the gradient of I 0 , k is calculated and P and Q located. 
     P and Q are generated by linear interpolation with the two vertices of the square side on which they are situated. Then P and Q are used to interpolate I 0 , linearly. The following interpolation formula is deduced, in which the coefficients of available pixels are directly calculated from k:
 
 I   0 =(α A ,α B ,α C ,α D )( A,B,C,D ) T   (1)
 
     Here the indices of pixels also refer to their intensity value. Due to the geometric symmetry, the same weights can be used for I 1 , I 2  and I 3  by mirroring A, B, C, D and I. 
     Table 1 gives corresponding weights with respect to k. The permutation of available pixels and the transformed slope are shown in Table 2. 
     
       
         
               
             
               
               
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 In-block interpolation weights: 
               
             
          
           
               
                   
                 k 
                 (−∞, −1]∪[2, +∞) 
                 [−1, ½] 
                 [½, 2] 
               
               
                   
                   
               
               
                   
                 α A   
                 (4k − 2)/9k 
                 (4 − 2k)/9 
                 ⅓ 
               
               
                   
                 α B   
                 (2k + 2)/9k 
                 (2 + 2k)/9 
                 ⅓ 
               
               
                   
                 α C   
                 (k − 2)/9k 
                 (1 − 2k)/9 
                 0 
               
               
                   
                 α D   
                 (2k + 2)/9k 
                 (2 + 2k)/9 
                 ⅓ 
               
               
                   
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
               
               
               
               
             
           
               
                 TABLE 2 
               
               
                   
               
               
                 In-block interpolation mirroring 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 I 0   
                 A 
                 B 
                 C 
                 D 
                 k 
               
               
                   
                 I 1   
                 B 
                 A 
                 D 
                 C 
                 −k 
               
               
                   
                 I 2   
                 C 
                 D 
                 A 
                 B 
                 k 
               
               
                   
                 I 3   
                 D 
                 C 
                 B 
                 A 
                 −k 
               
               
                   
                   
               
             
          
         
       
     
     When the in-block pixels are generated, the on-block pixels in the two hexagons formed by the original and in-block pixels are interpolated, as illustrated in  FIG. 4  and in steps  706 - 708  of  FIG. 7 . The procedure of on-block interpolation is similar to that of in-block interpolation, which can be formulated as:
 
 J   0 =(α A ,α B ,α C ,α D ,α E ,α F )( A,B,C,D,E,F ) T   (2)
 
     The interpolation weights, the available pixel permutation and the transformed slope are listed in Table 3 and Table 4: 
     
       
         
               
             
               
               
               
               
               
             
           
               
                 TABLE 3 
               
             
             
               
                   
               
               
                 On-block interpolation weights: 
               
             
          
           
               
                 k 
                 (−∞, −1] 
                 [−1, 0] 
                 [0, 1] 
                 [1, +∞) 
               
               
                   
               
               
                 αA 
                 1/(1 − 2k) 
                 2/(3 − 3k) 
                 2/(3 + 3k) 
                 1/(1 + 2k) 
               
               
                 αB 
                 −(1 + k)/(1 − 2k) 
                 0 
                 2k/(3 + 3k) 
                 k/(1 + 2k) 
               
               
                 αC 
                 1/(1 − 2k) 
                 −2k/(3 − 3k) 
                 0 
                 0 
               
               
                 αD 
                 0 
                 (1 + k)/(3 − 3k) 
                 (1 − k)/(3 + 3k) 
                 0 
               
               
                 α E   
                 0 
                 0 
                 2k/(3 + 3k) 
                 1/(1 + 2k) 
               
               
                 α F   
                 −k/(1 − 2k) 
                 −2k/(3 − 3k) 
                 0 
                 (k − 1)/ 
               
               
                   
                   
                   
                   
                 (1 + 2k) 
               
               
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 4 
               
               
                   
               
               
                 On-block interpolation mirroring: 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 J 0   
                 A 
                 B 
                 C 
                 D 
                 E 
                 F 
                 k 
               
               
                   
                 J 1   
                 D 
                 C 
                 B 
                 A 
                 F 
                 E 
                 −k 
               
               
                   
                 J 2   
                 A 
                 B 
                 C 
                 D 
                 E 
                 F 
                 1/k 
               
               
                   
                 J 3   
                 D 
                 C 
                 B 
                 A 
                 F 
                 E 
                 −1/k 
               
               
                   
                   
               
             
          
         
       
     
     As can be seen, both the in-block and on-block interpolations are performed in a uniform manner, which are favorable to simple and fast implementation. 
     Compared with high-order interpolators, there is a disadvantage to first-order interpolation such as bilinear and other methods, namely that the continuities between original and interpolated pixels are not well preserved, because fewer pixels are involved for interpolation. In order to solve this problem, a “difference projection” process is used, as generally represented by step  710  of  FIG. 7 . 
     More particularly, because the original pixels from low resolution image X are reliable, they are not directly modified to enforce the continuity of high resolution image Y. Instead, a “virtual” interpolation is performed at each original pixel according to equation (3) and (with  FIG. 5  and Table 5), calculating the difference between its previous and interpolated value. This difference is then propagated to other pixels by employing the interpolator again, for which the adopted gradients are the same as used before.
 
 K =(α A ,α B ,α C ,α D ,α E ,α F ,α G ,α H )( A,B,C,D,E,F,G,H ) T   (3)
 
     
       
         
               
             
               
               
               
               
               
             
           
               
                 TABLE 5 
               
             
             
               
                   
               
               
                 Virtual original interpolation weights: 
               
             
          
           
               
                 k 
                 (−∞, −1] 
                 [−1, 0] 
                 [0, 1] 
                 [1, +∞) 
               
               
                   
               
               
                 α A   
                 −½k 
                 −k/2 
                 0 
                 0 
               
               
                 α B   
                 0 
                 (1 + k)/2 
                 (1 − k)/2 
                 0 
               
               
                 α C   
                 0 
                 0 
                 k/2 
                 ½k 
               
               
                 α D   
                 (1 + k)/2k 
                 0 
                 0 
                 (k − 1)/2k 
               
               
                 α E   
                 −½k 
                 −k/2 
                 0 
                 0 
               
               
                 α F   
                 0 
                 (1 + k)/2 
                 (1 − k)/2 
                 0 
               
               
                 α G   
                 0 
                 0 
                 k/2 
                 ½k 
               
               
                 α H   
                 (1 + k)/2k 
                 0 
                 0 
                 (k − 1)/2k 
               
               
                   
               
             
          
         
       
     
       FIG. 6  illustrates the difference projection in a one-dimensional case, where the black dots indicate the original pixels, the white diamonds the interpolated pixels, the white dots the virtual interpolated originals, and the black diamonds the rectified interpolated ones. The dashed line represents the signal before difference projection, and the solid line after difference projection. 
     It can be observed that once the projected differences are added to the interpolated pixels, the continuities at original pixels are enforced while their values remain unchanged. The interpolation algorithm can be formulated as:
 
 Y=F   + ( {tilde over (X)} )+ F ( F ( {tilde over (X)} )− F   + ( {tilde over (X)} ))  (4)
 
     F denotes the combination of in-block and on-block interpolation, and F +  includes the virtual original interpolation as well as F. {tilde over (X)} is a high-resolution image with original copies from X at the black dots and zero at the gray and white dots. 
     With respect to computational complexity of the described algorithm, in terms of the multiplication times per missing pixel, three procedures are included, namely gradient diffusion, in-block and on-block interpolation, and difference projection. The multiplications used are 4.0, 4.0 and 4.5 (0.5 for virtual original interpolation), respectively. The interpolation weights only cost 3.5 multiplications (including 1 for calculating k) on average due to their similarity. Therefore, the complexity of the interpolator described herein is relatively low in implementation. It is even faster by clipping k to zero at the locations with small gradients (i.e. smooth image regions, which often cover the majority of an image), where the interpolation weights degrade to the predefined coefficients of bilinear interpolation. The computation for these weights can thus be saved. Also, difference projection is typically not necessary in these regions. 
     There is thus described is a new directional image interpolator, which aims at achieving high perceptual quality with low computational complexity. The missing pixels in a high-resolution image are generated with their available neighbors in certain fixed positions, whose weights are determined by the gradients diffused from the low-resolution image. Afterwards, the continuity of the interpolated image is enforced by a difference projection process. Due to its adaptiveness and uniformity, the described interpolator preserves edges in various orientations. Further, the interpolation ratio can be of any integer. Experimental results show that this interpolator achieves better perceptual and objective quality compared with bilinear and bicubic interpolation, as well as known existing adaptive methods. 
     Exemplary Operating Environment 
       FIG. 8  illustrates an example of a suitable computing and networking environment  800  on which the examples of  FIGS. 1-7  may be implemented. The computing system environment  800  is only one example of a suitable computing environment and is not intended to suggest any limitation as to the scope of use or functionality of the invention. Neither should the computing environment  800  be interpreted as having any dependency or requirement relating to any one or combination of components illustrated in the exemplary operating environment  800 . 
     The invention is operational with numerous other general purpose or special purpose computing system environments or configurations. Examples of well known computing systems, environments, and/or configurations that may be suitable for use with the invention include, but are not limited to: personal computers, server computers, hand-held or laptop devices, tablet devices, multiprocessor systems, microprocessor-based systems, set top boxes, programmable consumer electronics, network PCs, minicomputers, mainframe computers, distributed computing environments that include any of the above systems or devices, and the like. 
     The invention may be described in the general context of computer-executable instructions, such as program modules, being executed by a computer. Generally, program modules include routines, programs, objects, components, data structures, and so forth, which perform particular tasks or implement particular abstract data types. The invention may also be practiced in distributed computing environments where tasks are performed by remote processing devices that are linked through a communications network. In a distributed computing environment, program modules may be located in local and/or remote computer storage media including memory storage devices. 
     With reference to  FIG. 8 , an exemplary system for implementing various aspects of the invention may include a general purpose computing device in the form of a computer  810 . Components of the computer  810  may include, but are not limited to, a processing unit  820 , a system memory  830 , and a system bus  821  that couples various system components including the system memory to the processing unit  820 . The system bus  821  may be any of several types of bus structures including a memory bus or memory controller, a peripheral bus, and a local bus using any of a variety of bus architectures. By way of example, and not limitation, such architectures include Industry Standard Architecture (ISA) bus, Micro Channel Architecture (MCA) bus, Enhanced ISA (EISA) bus, Video Electronics Standards Association (VESA) local bus, and Peripheral Component Interconnect (PCI) bus also known as Mezzanine bus. 
     The computer  810  typically includes a variety of computer-readable media. Computer-readable media can be any available media that can be accessed by the computer  810  and includes both volatile and nonvolatile media, and removable and non-removable media. By way of example, and not limitation, computer-readable media may comprise computer storage media and communication media. Computer storage media includes volatile and nonvolatile, removable and non-removable media implemented in any method or technology for storage of information such as computer-readable instructions, data structures, program modules or other data. Computer storage media includes, but is not limited to, RAM, ROM, EEPROM, flash memory or other memory technology, CD-ROM, digital versatile disks (DVD) or other optical disk storage, magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices, or any other medium which can be used to store the desired information and which can accessed by the computer  810 . Communication media typically embodies computer-readable instructions, data structures, program modules or other data in a modulated data signal such as a carrier wave or other transport mechanism and includes any information delivery media. The term “modulated data signal” means a signal that has one or more of its characteristics set or changed in such a manner as to encode information in the signal. By way of example, and not limitation, communication media includes wired media such as a wired network or direct-wired connection, and wireless media such as acoustic, RF, infrared and other wireless media. Combinations of any of the above may also be included within the scope of computer-readable media. 
     The system memory  830  includes computer storage media in the form of volatile and/or nonvolatile memory such as read only memory (ROM)  831  and random access memory (RAM)  832 . A basic input/output system  833  (BIOS), containing the basic routines that help to transfer information between elements within computer  810 , such as during start-up, is typically stored in ROM  831 . RAM  832  typically contains data and/or program modules that are immediately accessible to and/or presently being operated on by processing unit  820 . By way of example, and not limitation,  FIG. 8  illustrates operating system  834 , application programs  835 , other program modules  836  and program data  837 . 
     The computer  810  may also include other removable/non-removable, volatile/nonvolatile computer storage media. By way of example only,  FIG. 8  illustrates a hard disk drive  841  that reads from or writes to non-removable, nonvolatile magnetic media, a magnetic disk drive  851  that reads from or writes to a removable, nonvolatile magnetic disk  852 , and an optical disk drive  855  that reads from or writes to a removable, nonvolatile optical disk  856  such as a CD ROM or other optical media. Other removable/non-removable, volatile/nonvolatile computer storage media that can be used in the exemplary operating environment include, but are not limited to, magnetic tape cassettes, flash memory cards, digital versatile disks, digital video tape, solid state RAM, solid state ROM, and the like. The hard disk drive  841  is typically connected to the system bus  821  through a non-removable memory interface such as interface  840 , and magnetic disk drive  851  and optical disk drive  855  are typically connected to the system bus  821  by a removable memory interface, such as interface  850 . 
     The drives and their associated computer storage media, described above and illustrated in  FIG. 8 , provide storage of computer-readable instructions, data structures, program modules and other data for the computer  810 . In  FIG. 8 , for example, hard disk drive  841  is illustrated as storing operating system  844 , application programs  845 , other program modules  846  and program data  847 . Note that these components can either be the same as or different from operating system  834 , application programs  835 , other program modules  836 , and program data  837 . Operating system  844 , application programs  845 , other program modules  846 , and program data  847  are given different numbers herein to illustrate that, at a minimum, they are different copies. A user may enter commands and information into the computer  810  through input devices such as a tablet, or electronic digitizer,  864 , a microphone  863 , a keyboard  862  and pointing device  861 , commonly referred to as mouse, trackball or touch pad. Other input devices not shown in  FIG. 8  may include a joystick, game pad, satellite dish, scanner, or the like. These and other input devices are often connected to the processing unit  820  through a user input interface  860  that is coupled to the system bus, but may be connected by other interface and bus structures, such as a parallel port, game port or a universal serial bus (USB). A monitor  891  or other type of display device is also connected to the system bus  821  via an interface, such as a video interface  890 . The monitor  891  may also be integrated with a touch-screen panel or the like. Note that the monitor and/or touch screen panel can be physically coupled to a housing in which the computing device  810  is incorporated, such as in a tablet-type personal computer. In addition, computers such as the computing device  810  may also include other peripheral output devices such as speakers  895  and printer  896 , which may be connected through an output peripheral interface  894  or the like. 
     The computer  810  may operate in a networked environment using logical connections to one or more remote computers, such as a remote computer  880 . The remote computer  880  may be a personal computer, a server, a router, a network PC, a peer device or other common network node, and typically includes many or all of the elements described above relative to the computer  810 , although only a memory storage device  881  has been illustrated in  FIG. 8 . The logical connections depicted in  FIG. 8  include one or more local area networks (LAN)  871  and one or more wide area networks (WAN)  873 , but may also include other networks. Such networking environments are commonplace in offices, enterprise-wide computer networks, intranets and the Internet. 
     When used in a LAN networking environment, the computer  810  is connected to the LAN  871  through a network interface or adapter  870 . When used in a WAN networking environment, the computer  810  typically includes a modem  872  or other means for establishing communications over the WAN  873 , such as the Internet. The modem  872 , which may be internal or external, may be connected to the system bus  821  via the user input interface  860  or other appropriate mechanism. A wireless networking component  874  such as comprising an interface and antenna may be coupled through a suitable device such as an access point or peer computer to a WAN or LAN. In a networked environment, program modules depicted relative to the computer  810 , or portions thereof, may be stored in the remote memory storage device. By way of example, and not limitation,  FIG. 8  illustrates remote application programs  885  as residing on memory device  881 . It may be appreciated that the network connections shown are exemplary and other means of establishing a communications link between the computers may be used. 
     An auxiliary subsystem  899  (e.g., for auxiliary display of content) may be connected via the user interface  860  to allow data such as program content, system status and event notifications to be provided to the user, even if the main portions of the computer system are in a low power state. The auxiliary subsystem  899  may be connected to the modem  872  and/or network interface  870  to allow communication between these systems while the main processing unit  820  is in a low power state. 
     Conclusion 
     While the invention is susceptible to various modifications and alternative constructions, certain illustrated embodiments thereof are shown in the drawings and have been described above in detail. It should be understood, however, that there is no intention to limit the invention to the specific forms disclosed, but on the contrary, the intention is to cover all modifications, alternative constructions, and equivalents falling within the spirit and scope of the invention.