# Patent

## Title
Subpixel-based image down-sampling

## Abstract

Systems, methods, and apparatus for sampling images using minimum mean square error subpixel-based down-sampling are presented herein. A partition component can be configured to divide a first array of pixels into two-dimensional (2-D) blocks of pixels. A sampling component can be configured to receive a query from a device including information associated with a display of the device; diagonally down-sample subpixels of a block of the 2-D blocks and generate a second array of pixels based on the down-sampled subpixels and the query. The sampling component can communicate at least a portion of the second array to the device based on the query. The information associated with the display can include a dimension associated with the display. Further, sampling component can be associated with a web browser; and the query can include an internet based query.

## Background
PRIORITY CLAIM
This application is a continuation-in-part of U.S. patent application Ser. No. 13/043,703, filed on Mar. 9, 2011, entitled “SUBPIXEL-BASED IMAGE DOWN-SAMPLING,” which claims priority to U.S. Provisional Patent Application Ser. No. 61/282,620, filed on Mar. 9, 2010, entitled “NOVEL 2-D MMSE SUBPIXEL-BASED IMAGE DOWN-SAMPLING.” The entireties of the aforementioned applications are incorporated by reference herein.


TECHNICAL FIELD
This disclosure relates generally to image processing including, but not limited to, minimum mean square error subpixel-based down-sampling (MMSE-SD).
BACKGROUND
With the advance of portable technologies, down-sampling of high resolution image information is often required to display high resolution images(s) and/or video(s), e.g., high-definition (HD) television (HDTV) information, HD movies, etc. on a lower resolution display, e.g., included in a handheld device such as a cellular phones, a portable multimedia player (PMP), a personal data assistant (PDA), etc.
A color pixel of a high resolution matrix display, e.g. liquid crystal display (LCD), plasma display panel (PDP), etc. includes three subpixels, each subpixel representing one of three primary colors, i.e., red (R), green (G), and blue (B). Although the subpixels are not separately visible, they are perceived together as color(s). One conventional technique for down-sampling a high resolution, e.g., color, image is pixel-based down-sampling, which selects every third pixel of the high resolution image to display. Such down-sampling severely affects shapes and/or details of the image, as over 30% of information of the image is compressed (or lost). Further, pixel-based down-sampling causes aliasing, or distortion, of the image near shape edges.
Another conventional technique for down-sampling a high resolution image is subpixel-based down-sampling, which alternately selects red, green, and blue subpixels from consecutive pixels of a block of pixels of the high resolution image in a horizontal direction. As such, the (i,j) pixel in the downsampled image includes subpixels (Ri,j, Gi,j+1, Bi,j+2) of the block of pixels—the subscripts denoting pixel indices of the block of pixels. Although such subpixel-based down-sampling preserves the shapes of images more effectively than pixel-based down-sampling, resulting subpixel-based images incur more color fringing, i.e., artifacts, around non-horizontal edges than pixel-based downsampled images.
The above-described deficiencies of today's image down-sampling techniques and related technologies are merely intended to provide an overview of some of the problems of conventional technology, and are not intended to be exhaustive. Other problems with the state of the art, and corresponding benefits of some of the various non-limiting embodiments described herein, may become further apparent upon review of the following detailed description.
SUMMARY
The following presents a simplified summary to provide a basic understanding of some aspects described herein. This summary is not an extensive overview of the disclosed subject matter. It is not intended to identify key or critical elements of the disclosed subject matter, or delineate the scope of the subject disclosure. Its sole purpose is to present some concepts of the disclosed subject matter in a simplified form as a prelude to the more detailed description presented later.
To correct for the above identified deficiencies of today's image processing environments and other drawbacks of conventional image down-sampling environments, various systems, methods, and apparatus described herein sample images using MMSE-SD.
For example, a method can include receiving a request from a device including information associated with a display of the device; alternately sampling subpixels of a two-dimensional (2-D) block of pixels in a diagonal direction; and generating an image based on a result of the alternately sampling the subpixels of the 2-D block of pixels and the information associated with the display of the device. Further, the method can include communicating the image to the device based on the request. The method can include receiving information associated with a dimension associated with the display, and generating the image based on the dimension associated with the display. Further, the method can include receiving, by one or more computing devices via a cloud based computing environment, a query including the information from the device; and communicating, via the cloud based computing environment, the image to the device based on the request. For example, the method can include receiving an internet based query via a web browser.
In another example, a system can include a partition component configured to divide a first array of pixels into two-dimensional (2-D) blocks of pixels. Further, the system can include a sampling component configured to: receive a query from a device including information associated with a display of the device; diagonally down-sample subpixels of a block of the 2-D blocks; and generate a second array of pixels based on the down-sampled subpixels and the query. The sampling component can be configured to communicate at least a portion of the second array of pixels to the device based on the query. In one example, the sampling component can be configured to: alternately sample a red subpixel, a green subpixel, and a blue subpixel of adjacent pixels of the block in a diagonal direction; and generate the second array based on the red subpixel, the green subpixel, the blue subpixel, and the information associated with the display. Further, the sampling component can be configured to receive a dimension associated with the display, and generate the second array based on the dimension associated with the display.
For example, the sampling component can be configured to: receive, by one or more computing devices via a cloud based computing environment, the query; and communicate, via the cloud based computing environment, the second array to the device based on the query. Further, the sampling component can be associated with a web browser, and the query can include an internet based query.
In yet another example, an apparatus can include means for communicating a network based query including information associated with a means for displaying an image. Further, the means for displaying the image can include a means for receiving the image based on the network based query. Furthermore, the network based query can include an internet based query; and the information can include a dimension associated with the means for displaying the image.
The following description and the annexed drawings set forth in detail certain illustrative aspects of the disclosed subject matter. These aspects are indicative, however, of but a few of the various ways in which the principles of the innovation may be employed. The disclosed subject matter is intended to include all such aspects and their equivalents. Other advantages and distinctive features of the disclosed subject matter will become apparent from the following detailed description of the innovation when considered in conjunction with the drawings.



BRIEF DESCRIPTION OF THE DRAWINGS
Non-limiting and non-exhaustive embodiments of the subject disclosure are described with reference to the following figures, wherein like reference numerals refer to like parts throughout the various views unless otherwise specified.
FIG. 1 illustrates a block diagram of a subpixel-based down-sampling environment, in accordance with an embodiment.
FIG. 2 illustrates a block diagram of a subpixel based down-sampling system, in accordance with an embodiment.
FIG. 3 illustrates a block diagram of a two-dimensional high resolution image, in accordance with an embodiment.
FIG. 4 illustrates a block diagram of a pixel, in accordance with an embodiment.
FIG. 5 illustrates a block diagram of a subpixel-based down-sampling model, in accordance with an embodiment.
FIG. 6 illustrates an example of down-sampling a block of pixels, in accordance with an embodiment.
FIG. 7 illustrates another example of down-sampling a block of pixels, in accordance with an embodiment.
FIG. 8 illustrates a block diagram of a system for reconstructing a high resolution image, in accordance with an embodiment.
FIG. 9 illustrates a block diagram of a subpixel-based reconstruction model, in accordance with an embodiment.
FIG. 10 illustrates a block diagram of a minimum mean square error down-sampling (MMSE-SD) system, in accordance with an embodiment.
FIG. 11 illustrates a block diagram of another subpixel-based reconstruction model, in accordance with an embodiment.
FIG. 12 illustrates a block diagram of another subpixel-based down-sampling environment, in accordance with an embodiment.
FIG. 13 illustrates a block diagram of yet another subpixel-based down-sampling environment, in accordance with an embodiment.
FIGS. 14-16 illustrate various processes associated with minimum mean square error subpixel-based down-sampling, in accordance with an embodiment.
FIG. 17 illustrates a block diagram of a computing system operable to execute the disclosed systems and methods, in accordance with an embodiment.



DETAILED DESCRIPTION
Various non-limiting embodiments of systems, methods, and apparatus presented herein sample images using minimum mean square error subpixel-based down-sampling (MMSE-SD).
In the following description, numerous specific details are set forth to provide a thorough understanding of the embodiments. One skilled in the relevant art will recognize, however, that the techniques described herein can be practiced without one or more of the specific details, or with other methods, components, materials, etc. In other instances, well-known structures, materials, or operations are not shown or described in detail to avoid obscuring certain aspects.
Reference throughout this specification to “one embodiment,” or “an embodiment,” means that a particular feature, structure, or characteristic described in connection with the embodiment is included in at least one embodiment. Thus, the appearances of the phrase “in one embodiment,” or “in an embodiment,” in various places throughout this specification are not necessarily all referring to the same embodiment. Furthermore, the particular features, structures, or characteristics may be combined in any suitable manner in one or more embodiments.
As utilized herein, terms “component,” “system,” “interface,” and the like are intended to refer to a computer-related entity, hardware, software (e.g., in execution), and/or firmware. For example, a component can be a processor, a process running on a processor, an object, an executable, a program, a storage device, and/or a computer. By way of illustration, an application running on a server and the server can be a component. One or more components can reside within a process, and a component can be localized on one computer and/or distributed between two or more computers.
Further, these components can execute from various computer readable media having various data structures stored thereon. The components can communicate via local and/or remote processes such as in accordance with a signal having one or more data packets (e.g., data from one component interacting with another component in a local system, distributed system, and/or across a network, e.g., the Internet, a local area network, a wide area network, etc. with other systems via the signal).
As another example, a component can be an apparatus with specific functionality provided by mechanical parts operated by electric or electronic circuitry; the electric or electronic circuitry can be operated by a software application or a firmware application executed by one or more processors; the one or more processors can be internal or external to the apparatus and can execute at least a part of the software or firmware application. As yet another example, a component can be an apparatus that provides specific functionality through electronic components without mechanical parts; the electronic components can include one or more processors therein to execute software and/or firmware that confer(s), at least in part, the functionality of the electronic components. In an aspect, a component can emulate an electronic component via a virtual machine, e.g., within a cloud computing system.
The word “exemplary” and/or “demonstrative” is used herein to mean serving as an example, instance, or illustration. For the avoidance of doubt, the subject matter disclosed herein is not limited by such examples. In addition, any aspect or design described herein as “exemplary” and/or “demonstrative” is not necessarily to be construed as preferred or advantageous over other aspects or designs, nor is it meant to preclude equivalent exemplary structures and techniques known to those of ordinary skill in the art. Furthermore, to the extent that the terms “includes,” “has,” “contains,” and other similar words are used in either the detailed description or the claims, such terms are intended to be inclusive—in a manner similar to the term “comprising” as an open transition word—without precluding any additional or other elements.
Artificial intelligence based systems, e.g., utilizing explicitly and/or implicitly trained classifiers, can be employed in connection with performing inference and/or probabilistic determinations and/or statistical-based determinations as in accordance with one or more aspects of the disclosed subject matter as described herein. For example, an artificial intelligence system can be used to automatically partition, e.g., via partition component 210, a 2-D array of pixels into 2-D blocks of pixels. Further, the artificial intelligence system can be used to automatically alternately sample, e.g., via sampling component 220, subpixels of a block of pixels of the 2-D blocks of pixels in a diagonal direction; and generate an image based on such sampling and a dimension associated with a display device. Furthermore, the artificial intelligence system can derive, e.g., via reconstruction component 810, a virtual image according to a size of another image associated with the 2-D array of pixels.
As used herein, the term “infer” or “inference” refers generally to the process of reasoning about, or inferring states of, the system, environment, user, and/or intent from a set of observations as captured via events and/or data. Captured data and events can include user data, device data, environment data, data from sensors, sensor data, application data, implicit data, explicit data, etc. Inference can be employed to identify a specific context or action, or can generate a probability distribution over states of interest based on a consideration of data and events, for example.
Inference can also refer to techniques employed for composing higher-level events from a set of events and/or data. Such inference results in the construction of new events or actions from a set of observed events and/or stored event data, whether the events are correlated in close temporal proximity, and whether the events and data come from one or several event and data sources. Various classification schemes and/or systems (e.g., support vector machines, neural networks, expert systems, Bayesian belief networks, fuzzy logic, and data fusion engines) can be employed in connection with performing automatic and/or inferred action in connection with the disclosed subject matter.
In addition, the disclosed subject matter can be implemented as a method, apparatus, or article of manufacture using standard programming and/or engineering techniques to produce software, firmware, hardware, or any combination thereof to control a computer to implement the disclosed subject matter. The term “article of manufacture” as used herein is intended to encompass a computer program accessible from any computer-readable device, computer-readable carrier, or computer-readable media. For example, computer-readable media can include, but are not limited to, a magnetic storage device, e.g., hard disk; floppy disk; magnetic strip(s); an optical disk (e.g., compact disk (CD), a digital video disc (DVD), a Blu-ray DiscTM (BD)); a smart card; a flash memory device (e.g., card, stick, key drive); and/or a virtual device that emulates a storage device and/or any of the above computer-readable media.
Conventional downsampling techniques negatively affect shapes and/or details of a sampled image, causing aliasing of the sampled image near shape edges, and/or causing increased color fringing around non-horizontal edges of the sampled image. Compared to such technology, various systems, methods, and apparatus described herein in various embodiments can improve sampling of images by using MMSE-SD.
Referring now to FIG. 1, a block diagram of a subpixel-based down-sampling environment 100 is illustrated, in accordance with an embodiment. Aspects of down-sampling environment 100, and systems, networks, other apparatus, and processes explained herein can constitute machine-executable instructions embodied within machine(s), e.g., embodied in one or more computer readable mediums (or media) associated with one or more machines. Such instructions, when executed by the one or more machines, e.g., computer(s), computing device(s), virtual machine(s), etc. can cause the machine(s) to perform the operations described.
Additionally, the systems and processes explained herein can be embodied within hardware, such as an application specific integrated circuit (ASIC) or the like. Further, the order in which some or all of the process blocks appear in each process should not be deemed limiting. Rather, it should be understood by a person of ordinary skill in the art having the benefit of the instant disclosure that some of the process blocks can be executed in a variety of orders not illustrated.
System 110 can be configured to receive, store, process, etc. a high resolution image (L) including a two-dimensional (2-D) array of pixels. Further, system 110 can be communicatively coupled to a computing device 120, e.g., a single-processor computer system, a multi-processor computer system, a personal computer, a remote processing device, a hand-held computing device, e.g., a mobile phone, a cellular phone, a camera, a portable multimedia player (PMP), a personal data assistant (PDA), a wrist watch, etc. via a communications network, e.g., a wired communication network, an internet protocol (IP) based network, a wireless communication network, a wireless protocol associated with, e.g., wireless fidelity (Wi-Fi), Worldwide Interoperability for Microwave Access (WiMAX), Enhanced General Packet Radio Service (Enhanced GPRS), 3GPP Long Term Evolution (LTE), Third Generation Partnership Project 2 (3GPP2) Ultra Mobile Broadband (UMB), 3GPP UMTS, High Speed Packet Access (HSPA), High Speed Downlink Packet Access (HSDPA), High Speed Uplink Packet Access (HSUPA), LTE Advanced, Third Generation Partnership Project (3GPP) Universal Mobile Telecommunication System (UMTS), Global System for Mobile Communication (GSM), etc.
In an aspect, system 110 can receive, from computing device 120, a request including information associated with a display 125 coupled to computing device 120. In one aspect, display 125 can include an LCD display, a lower resolution display, etc. in which a resolution of display 125 can be based on, e.g., a pixel resolution of the display, a spatial resolution of the display, a spectral resolution of the display, a temporal resolution of the display, a radiometric resolution of the display, etc. In another aspect, the information can include a dimension associated with display 125, a dimension associated with an image to be displayed via display 125, a resolution of the image, a pixel resolution of the image, a spatial resolution of the image, a spectral resolution of the image, a temporal resolution of the image, a resolution associated with display 125, a pixel resolution associated with display 125, a spatial resolution associated with display 125, a spectral resolution associated with display 125, a temporal resolution of display 125, etc. In an aspect, the dimension associated with the image can be lower than the resolution of the image.
In one aspect, system 110 can include, or be communicatively coupled to, an internet (or World Wide Web) based server, e.g., via a cloud computing environment. Further, computing device 120 can include features supporting a display, via display 125, of a web browser. In an aspect, system 110 can receive the request in response to a communication associated with the web browser, e.g., via a Uniform Resource Identifier (URI), a hyperlink, etc. For example, computing device 120 can be configured to derive the request and/or the information based on software and/or firmware, e.g., a display driver, that can be configured to generate the request in response to receiving input associated with the web browser, e.g., a request to display information and/or a web page, e.g., associated with an internet query, etc. via display 125.
In another aspect, system 110 can generate, based on the request and/or the information received from computing device 120, e.g., based on the dimension associated with display 125, a low resolution image to be displayed via display 125. Further, system 110 can communication the low resolution image to computing device 120, e.g., in response to the internet query, etc. As such, computing device 120 can be configured to display the low resolution image via display 125.
Now referring to FIG. 2, a subpixel-based down-sampling environment 200 is illustrated, in accordance with an embodiment. In one aspect, components and/or features of system 200 can be included in system 110. In an aspect, partition component 210 and/or sampling component 220 can be configured to receive a query, request, etc. from computing device 120 including information associated with display 125, e.g., a dimension associated with display 125. Further, partition component 210 can partition, divide, etc. the 2-D array of pixels of high-resolution image L into 2-D blocks of pixels, e.g., into blocks including a 3×3 array of pixels. Further, sampling component 220 can down-sample each block, or 3×3 array of pixels, by selecting, sampling, etc. subpixels of pixels of the block in a diagonal direction to generate a low resolution image (S) based on the request and/or the information, e.g., the dimension associated with display 125. Furthermore, system 200 can communicate low resolution image S to computing device 120, which can be configured to display low resolution image S via display 125.
In one aspect, partition component 210 can be configured to partition, divide, etc. the 2-D array of pixels of high-resolution image L into the 2-D blocks of pixels based on the request and/or the information, e.g., the dimension associated with display 125. Further, sampling component 220 can down-sample each 3×3 array of pixels by selecting, sampling, etc. subpixels of pixels of the block in the diagonal direction to generate low resolution image S.
FIG. 3 illustrates a block diagram of a 2-D high resolution image L (300) including pixels 310, in accordance with an embodiment. Pixels 310 are addressable screen elements of a display, arranged in a 2-D grid. Each pixel 310 is addressed by coordinates (not shown), which can be arbitrarily assigned and/or re-assigned during image processing. As illustrated by FIG. 4, pixel 310 can include three subpixels: red subpixel 410, green subpixel 420, and blue subpixel 430. Subpixels 410, 420, and 430, which together represent color when perceived at a distance, are also addressed by coordinates.
Referring now to FIG. 3, partition component 210 can be configured to divide high resolution image 300 into at least two blocks 305 and 315 that include respective 3×3 arrays of pixels 310. In one embodiment, high resolution image 300 can include 3M×2N pixels. Further, in another aspect illustrated by FIG. 5, partition component 210 can associate coordinates (i,j) with pixels 310 of blocks 305 and 315. As such, as illustrated by FIG. 6, sampling component 220 can be configured to diagonally select, copy, sample, etc. red subpixel 410 at coordinate R3i−2,3j−2, green subpixel 420 at coordinate G3i−1,3j−1, and blue subpixel 430 at coordinate B3i,3j from pixels 310 of blocks 305/315 in a first diagonal direction 610, based on the request and/or the information, e.g., the dimension associated with display 125, to create sample 620 corresponding to a pixel of the low resolution, down-sampled, image S (e.g. see FIG. 2). As such, down-sampled, image S can include M×N pixels corresponding to the 3M×2N pixels of high resolution image 300. Further, system 200 can communicate image S to computing device 120, e.g., which can display the information and/or the web page, e.g., associated with the internet query, etc. via display 125.
In another embodiment illustrated by FIG. 7, sampling component 220 can be configured to diagonally select, copy, sample, etc. red subpixel 410 at coordinate R3i,3j−2, green subpixel 420 at coordinate G3i−1,3j−1, and blue subpixel 430 at coordinate B3i−2,3j from pixels 310 of blocks 305/315 in a second diagonal direction 710, based on the information received from computing device 120, to create sample 720 corresponding to a (i, j)th pixel (ri,j, gi,j, bi,j) of image S. As such, system 200 can more effectively preserve shape details of a high resolution image than conventional down-sampling techniques. Further, system 200 can be configured to communicate the image S to computing device 120, e.g., which can display the information and/or the web page, e.g., associated with the internet query, etc. via display 125.
Now referring to FIG. 8, a block diagram of a system 800 for reconstructing a high resolution image L is illustrated, in accordance with an embodiment. System 800 can include a reconstruction component 810 that can be configured to receive a low resolution image, e.g., low resolution image S generated by sampling system 110, 200, etc. In an aspect, reconstruction component 810 can be configured to derive a virtual image (L′), approximating high resolution image L, based on, at least in part, a reconstruction model 900 for red component (or subpixel) generation illustrated by FIG. 9. As illustrated by 905, each square of reconstruction model 900 represents a pixel in virtual image L′, in which αk, βk, k=1, 2, 3, 4; and αk+βk=1 for each k is satisfied.
As such, the (i, j)th pixel (ri,j, gi,j, bi,j) in S corresponds to a 3×3 block, or array 910, of pixels in L′ at locations (k, l), with k=3i−2, 3i−1, or 3i; and l=3j−2, 3j−1, or 3j. Reconstruction component 810 can copy red subpixel ri,j of the (i, j)th pixel of S to a first location (3i−2, 3j−2), at 920, of L′; green subpixel gi,j of the (i, j)th pixel of S to a second location (3i−1, 3j−1), at 930, of L′; and blue subpixel bi,j of the (i, j)th pixel of S to a third location (3i, 3j), at 940, of L′. Further, reconstruction component 810 can generate red components (or red subpixels) neighboring the first location using a directional weighted average.
For example, reconstruction component 810 can generate red components missing from locations (3i−3, 3j−3) of L′ and (3i−4, 3j−4) of L′ based on equations (1) and (2), respectively, as follows:

α3ri,j+β3ri−1,j−1,  (1)

β3ri,j+α3ri−1,j−1.  (2)

Further, reconstruction component 810 can generate red components missing from locations (3i−2, 3j−1) of L′ and (3i−2, 3j) of L′ based on equations (3) and (4), respectively, as follows:

α1ri,j+β1ri,j+1,  (3)

β1ri,j+α1ri,j+1.  (4)

In another aspect, reconstruction component 810 can generate green components missing from locations (3i−2, 3j−2) of L′ and (3i−3, 3j−3) of L′ based on equations (5) and (6), respectively, as follows:

α3gi,j+β3gi−1,j−1,  (5)

α3gi,j+α3gi−1,j−1.  (6)

Further, reconstruction component 810 can generate green components missing from locations (3i−1, 3j) of L′ and (3i−1, 3j+1) of L′ based on equations (7) and (8), respectively, as follows:

α1gi,j+β1gi,j+1,  (7)

β1gi,j+α1gi,j+1.  (8)

In yet another aspect, reconstruction component 810 can generate blue components missing from locations (3i−1, 3j−1) of L′ and (3i−2, 3j−2) of L′ based on equations (9) and (10), respectively, as follows:

α3bi,j+β3bi−1,j−1,  (9)

β3bi,j+α3bi−1,j−1.  (10)

Further, reconstruction component 810 can generate blue components missing from locations (3i, 3j+1) of L′ and (3i, 3j+2) of L′ based on equations (11) and (12), respectively, as follows:

α1bi,j+β1bi,j+1,  (11)

β1bi,j+α1bi,j+1.  (12)

In one aspect, reconstruction component 810 can copy red subpixel r1−1,j−1 of the (i−1, j−1)th pixel of S to location (3i−5, 3j−5), at 950, of L′; green subpixel gi−1,j−1 of the (i−1, j−1)th pixel of S to a second location (3i−4, 3j−4), at 960, of L′; and blue subpixel bi−1,j−1 of the (i−1, j)th pixel of S to a third location (3i, 3j), at 970, of L′. Further, reconstruction component 810 can generate red components (or red subpixels) neighboring ri−1,j−1 using a directional weighted average of respective neighboring components as described above.
Further, reconstruction component 810 can copy red subpixels ri−1,j, ri−1,j+1, ri,j+1, ri+1,j+1, ri+1,j, ri+1,j−1, and ri,j−1 of corresponding pixels of S to locations 980, 982, 984, 988, 990, and 992, respectively. Further, reconstruction component 810 can copy green and blue subpixels of the corresponding pixels of S to locations of L′ in a manner similar to the description above. As such, reconstruction component 810 can generate neighboring red, green, and blue components (or subpixels) using a directional weighted average of respective neighboring components as described above to reconstruct virtual image L′.
FIG. 10 illustrates a minimum mean square error (MMSE) subpixel-based down-sampling (MMSE-SD) system 1000, which various features and/or components can be included in one or more components of systems (e.g., 100, 200), methods, and/or apparatus described herein, in accordance with an embodiment. MMSE-SD system 1000 can include a MMSE-SD component 1010 that can be configured to receive a request, e.g., from computing device 120, including information (see above) associated with display 125. Further, MMSE-SD component 1010 can receive information associated with high resolution image L, including 3M×3N color components, or subpixels, R, G, and B of high resolution image L. Further, MMSE-SD can generate 3M×3N color components, or subpixels, R′, G′, and B′ of a derived, virtual image L′ based on reconstruction function ƒr(r) associated with Equations 1-4 and 13, reconstruction function ƒg(g) associated with Equations 5-8 and 13, and reconstruction ƒb(b) associated with Equations 9-12 and 13, respectively.
In an aspect, MMSE-SD component 1010 can separately derive subpixels r, g, and b of low resolution image S by minimizing a mean square error (MSE) between L and L′, according to Equation 13 as follows:




  
    
      
        
          
            
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As such, MMSE-SD component 1010 can derive subpixel r of low resolution image S according to Equations 14-16 as follows:




  
    
      
        
          
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          r
          =
          
            
              
                (
                
                  
                    H
                    r
                    
                      -
                      1
                    
                  
                  ⁢
                  
                    H
                    R
                  
                
                )
              
              ⁢
              R
            
            =
            HR
          
        
        ,
      
    
    
      
        (
        16
        )
      
    
  




in which Hr and HR are block-circulant matrices of size MN×MN and MN×9MN, respectively, according to Equations 17-26 as follows; in which R is a row-ordered vector of size 9MN×1 from the red component of L, and r is the row-ordered vector of size MN×1 from the red component of S, respectively, and in which H=Hr−1 HR is a block circulant matrix of size MN×9MN, including blocks of size N×9N that are block-tri-circulant (see below). Each of the N×9N sized blocks include three sub-blocks of size N×3N, and each of the three sub-blocks is block-circulant (see below) with a block size of 1×3:





  
    
      
        
          
            H
            r
          
          =
          
            
            
              
                
                  A
                
                
                  
                    B
                    2
                  
                
                
                  0
                
                
                  ...
                
                
                  
                    B
                    1
                  
                
              
              
                
                  
                    B
                    1
                  
                
                
                  A
                
                
                  
                    B
                    2
                  
                
                
                  ...
                
                
                  0
                
              
              
                
                  0
                
                
                  
                    B
                    1
                  
                
                
                  A
                
                
                  
                    B
                    2
                  
                
                
                  ...
                
              
              
                
                  ⋮
                
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋮
                
              
              
                
                  
                    B
                    2
                  
                
                
                  0
                
                
                  ...
                
                
                  
                    B
                    1
                  
                
                
                  A
                
              
            
            
          
        
        ,
      
    
    
      
        (
        17
        )
      
    
  
  
    
      
        
          
            H
            R
          
          =
          
            
            
              
                
                  C
                
                
                  
                    D
                    2
                  
                
                
                  
                    E
                    2
                  
                
                
                  0
                
                
                  0
                
                
                  0
                
                
                  0
                
                
                  ...
                
                
                  0
                
                
                  
                    E
                    1
                  
                
                
                  
                    D
                    1
                  
                
              
              
                
                  0
                
                
                  
                    E
                    1
                  
                
                
                  
                    D
                    1
                  
                
                
                  C
                
                
                  
                    D
                    2
                  
                
                
                  
                    E
                    2
                  
                
                
                  0
                
                
                  0
                
                
                  0
                
                
                  ...
                
                
                  0
                
              
              
                
                  ⋮
                
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋮
                
                
                  ⋮
                
                
                  ⋮
                
              
              
                
                  0
                
                
                  ...
                
                
                  0
                
                
                  
                    E
                    1
                  
                
                
                  
                    D
                    1
                  
                
                
                  C
                
                
                  
                    D
                    2
                  
                
                
                  
                    E
                    2
                  
                
                
                  0
                
                
                  0
                
                
                  0
                
              
              
                
                  0
                
                
                  0
                
                
                  0
                
                
                  0
                
                
                  ...
                
                
                  0
                
                
                  
                    E
                    1
                  
                
                
                  
                    D
                    1
                  
                
                
                  C
                
                
                  
                    D
                    2
                  
                
                
                  
                    E
                    2
                  
                
              
            
            
          
        
        ,
      
    
    
      
        (
        18
        )
      
    
  
  
    
      
        
          A
          =
          
            
            
              
                
                  
                    k
                    0
                  
                
                
                  
                    k
                    1
                  
                
                
                  0
                
                
                  ...
                
                
                  
                    k
                    1
                  
                
              
              
                
                  
                    k
                    1
                  
                
                
                  
                    k
                    0
                  
                
                
                  
                    k
                    1
                  
                
                
                  0
                
                
                  ...
                
              
              
                
                  ⋮
                
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋮
                
              
              
                
                  0
                
                
                  ...
                
                
                  
                    k
                    1
                  
                
                
                  
                    k
                    0
                  
                
                
                  
                    k
                    1
                  
                
              
              
                
                  
                    k
                    1
                  
                
                
                  0
                
                
                  ...
                
                
                  
                    k
                    1
                  
                
                
                  
                    k
                    2
                  
                
              
            
            
          
        
        ,
      
    
    
      
        (
        19
        )
      
    
  
  
    
      
        
          
            B
            1
          
          =
          
            
            
              
                
                  
                    k
                    2
                  
                
                
                  
                    k
                    4
                  
                
                
                  0
                
                
                  ...
                
                
                  
                    k
                    3
                  
                
              
              
                
                  
                    k
                    3
                  
                
                
                  
                    k
                    2
                  
                
                
                  
                    k
                    4
                  
                
                
                  0
                
                
                  ...
                
              
              
                
                  ⋮
                
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋮
                
              
              
                
                  0
                
                
                  ...
                
                
                  
                    k
                    3
                  
                
                
                  
                    k
                    2
                  
                
                
                  
                    k
                    4
                  
                
              
              
                
                  
                    k
                    4
                  
                
                
                  0
                
                
                  ...
                
                
                  
                    k
                    3
                  
                
                
                  
                    k
                    2
                  
                
              
            
            
          
        
        ,
      
    
    
      
        (
        20
        )
      
    
  
  
    
      
        
          
            B
            2
          
          =
          
            
            
              
                
                  
                    k
                    2
                  
                
                
                  
                    k
                    3
                  
                
                
                  0
                
                
                  ...
                
                
                  
                    k
                    4
                  
                
              
              
                
                  
                    k
                    4
                  
                
                
                  
                    k
                    2
                  
                
                
                  
                    k
                    3
                  
                
                
                  0
                
                
                  ...
                
              
              
                
                  ⋮
                
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋮
                
              
              
                
                  0
                
                
                  ...
                
                
                  
                    k
                    4
                  
                
                
                  
                    k
                    2
                  
                
                
                  
                    k
                    3
                  
                
              
              
                
                  
                    k
                    3
                  
                
                
                  0
                
                
                  ...
                
                
                  
                    k
                    4
                  
                
                
                  
                    k
                    2
                  
                
              
            
            
          
        
        ,
      
    
    
      
        (
        21
        )
      
    
  




in which k0=1+2α12+2β12+2α22+2β22+2α32+2β32+2α42+2β42, k1=2α1β1, k2=2α2β2, k3=2α3β3, and k4=2α4β4.

HR includes M×3M blocks (e.g., either C, D1, D2, E1, E2, or 0), wherein each block of the M×3M blocks is a matrix of size N×3N as described by Equations 22-26 as follows:




  
    
      
        
          C
          =
          
            
            
              
                
                  1
                
                
                  
                    α
                    1
                  
                
                
                  
                    β
                    1
                  
                
                
                  0
                
                
                  0
                
                
                  ...
                
                
                  
                    β
                    1
                  
                
                
                  
                    α
                    1
                  
                
              
              
                
                  0
                
                
                  
                    β
                    1
                  
                
                
                  
                    α
                    1
                  
                
                
                  1
                
                
                  
                    α
                    1
                  
                
                
                  
                    β
                    1
                  
                
                
                  ...
                
                
                  0
                
              
              
                
                  ⋮
                
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋮
                
              
              
                
                  0
                
                
                  ...
                
                
                  0
                
                
                  
                    β
                    1
                  
                
                
                  
                    α
                    1
                  
                
                
                  1
                
                
                  
                    α
                    1
                  
                
                
                  
                    β
                    1
                  
                
              
            
            
          
        
        ,
      
    
    
      
        (
        22
        )
      
    
  
  
    
      
        
          
            D
            1
          
          =
          
            
            
              
                
                  
                    α
                    2
                  
                
                
                  
                    α
                    4
                  
                
                
                  0
                
                
                  0
                
                
                  0
                
                
                  ...
                
                
                  0
                
                
                  
                    α
                    3
                  
                
              
              
                
                  0
                
                
                  0
                
                
                  
                    α
                    3
                  
                
                
                  
                    α
                    2
                  
                
                
                  
                    α
                    4
                  
                
                
                  0
                
                
                  ...
                
                
                  0
                
              
              
                
                  ⋮
                
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋮
                
              
              
                
                  0
                
                
                  ...
                
                
                  0
                
                
                  0
                
                
                  
                    α
                    3
                  
                
                
                  
                    α
                    2
                  
                
                
                  
                    α
                    4
                  
                
                
                  0
                
              
            
            
          
        
        ,
      
    
    
      
        (
        23
        )
      
    
  
  
    
      
        
          
            D
            2
          
          =
          
            
            
              
                
                  
                    α
                    2
                  
                
                
                  
                    α
                    3
                  
                
                
                  0
                
                
                  0
                
                
                  0
                
                
                  ...
                
                
                  0
                
                
                  
                    α
                    4
                  
                
              
              
                
                  0
                
                
                  0
                
                
                  
                    α
                    4
                  
                
                
                  
                    α
                    2
                  
                
                
                  
                    α
                    3
                  
                
                
                  0
                
                
                  ...
                
                
                  0
                
              
              
                
                  ⋮
                
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋮
                
              
              
                
                  0
                
                
                  ...
                
                
                  0
                
                
                  0
                
                
                  
                    α
                    4
                  
                
                
                  
                    α
                    2
                  
                
                
                  
                    α
                    3
                  
                
                
                  0
                
              
            
            
          
        
        ,
      
    
    
      
        (
        24
        )
      
    
  
  
    
      
        
          
            E
            1
          
          =
          
            
            
              
                
                  
                    β
                    2
                  
                
                
                  0
                
                
                  
                    β
                    4
                  
                
                
                  0
                
                
                  ...
                
                
                  0
                
                
                  
                    β
                    3
                  
                
                
                  0
                
              
              
                
                  0
                
                
                  
                    β
                    3
                  
                
                
                  0
                
                
                  
                    β
                    2
                  
                
                
                  0
                
                
                  
                    β
                    4
                  
                
                
                  ...
                
                
                  0
                
              
              
                
                  ⋮
                
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋮
                
              
              
                
                  0
                
                
                  ...
                
                
                  0
                
                
                  
                    β
                    3
                  
                
                
                  0
                
                
                  
                    β
                    2
                  
                
                
                  0
                
                
                  
                    β
                    4
                  
                
              
            
            
          
        
        ,
      
    
    
      
        (
        25
        )
      
    
  
  
    
      
        
          E
          2
        
        =
        
          
            
            
              
                
                  
                    β
                    2
                  
                
                
                  0
                
                
                  
                    β
                    3
                  
                
                
                  0
                
                
                  ...
                
                
                  0
                
                
                  
                    β
                    4
                  
                
                
                  0
                
              
              
                
                  0
                
                
                  
                    β
                    4
                  
                
                
                  0
                
                
                  
                    β
                    2
                  
                
                
                  0
                
                
                  
                    β
                    3
                  
                
                
                  ...
                
                
                  0
                
              
              
                
                  ⋮
                
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋮
                
              
              
                
                  0
                
                
                  ...
                
                
                  0
                
                
                  
                    β
                    4
                  
                
                
                  0
                
                
                  
                    β
                    2
                  
                
                
                  0
                
                
                  
                    β
                    3
                  
                
              
            
            
          
          .
        
      
    
    
      
        (
        26
        )
      
    
  




Consider 3 elements in the first row of C as a block, such as [1α1β1]. Such a block is repeatedly shifted to the right by 3 positions in the subsequent rows. Thus, each of C, D1, D2, E1, E2, or 0 is block-circulant. Similarly, considering 3 sub-blocks of HR in the horizontal direction (e.g. [C D2 E2]) as a block, which appears in the first row in HR and is repeatedly shifted to the right by 3 block positions in the subsequent rows, HR is a “block-circulant” matrix.
Here, we call a matrix of size N×9N “block-tri-circulant” if it contains three sub-blocks of size N×3N, each of which is block-circulant (e.g. [C D2 E2] is block-tri-circulant). Therefore, HR is a block circulant matrix, with blocks of size N×9N that are block-tri-circulant.
In another aspect, reconstruction component 810 can be configured to derive L′ based on a reconstruction model 1100 for red component (or subpixel) generation illustrated by FIG. 11. As illustrated by 1110, each square of reconstruction model 1100 represents a pixel in virtual image L′ in which αk; βk; and k=1, 2, 3, 4 are weighting coefficients in horizontal, vertical, diagonal (or first diagonal), and anti-diagonal (or second diagonal) directions. Further, γ is a weighting coefficient for other directions. Furthermore, 1120 illustrates an equivalent interpolation filter Hinterp.
For example, reconstruction component 810 can generate red components missing from locations (3i−3, 3j−3) of L′ and (3i−4, 3j−4) of L′ based on equations (27) and (28), respectively, as follows:

α3ri,j+β3ri−1,j−1+γri,j−1+γri−1,j,  (27)

β3ri,j+α3ri−1,j−1+γri,j−1+γr1−1,j.  (28)

Further, reconstruction component 810 can generate red components missing from locations (3i−2, 3j−1) of L′ and (3i−2, 3j) of L′ based on equations (29) and (30), respectively, as follows:

α1ri,j+β1ri,j+1,  (29)

β1ri,j+α1ri,j+1.  (30)

In another aspect, reconstruction component 810 can generate green components missing from locations (3i−2, 3j−2) of L′ and (3i−3, 3j−3) of L′ based on equations (31) and (32), respectively, as follows:

α3gi,j+β3gi−1,j−1+γgi,j−1+γgi−1,j,  (31)

β3gi,j+α3gi−1,j−1+γgi,j−1+γgi−1,j.  (32)

Further, reconstruction component 810 can generate green components missing from locations (3i−1, 3j) of L′ and (3i−1, 3j+1) of L′ based on equations (33) and (34), respectively, as follows:

α1gi,j+β1gi,j+1,  (33)

β1gi,j+α1gi,j+1.  (34)

In yet another aspect, reconstruction component 810 can generate blue components missing from locations (3i−1, 3j−1) of L′ and (3i−2, 3j−2) of L′ based on equations (35) and (36), respectively, as follows:

α3bi,j+β3bi−1,j−1+γbi,j−1+γbi−1,j,  (35)

β3bi,j+α3bi−1,j−1+γbi,j−1+γbi−1,j.  (36)

Further, reconstruction component 810 can generate blue components missing from locations (3i, 3j+1) of L′ and (3i, 3j+2) of L′ based on equations (37) and (38), respectively, as follows:

α1bi,j+β1bi,j+1,  (37)

β1bi,j+α1bi,j+1.  (38)

In one aspect, a bilinear interpolation filter HBI and a directional interpolation filter HDI can be obtained by approximating a 22.5° diagonal direction contribution, and treating the diagonal and the anti-diagonal interpolation in the same way as horizontal and vertical interpolation, as defined by equation (39) as follows:




  
    
      
        
          
            H
            BI
          
          =
          
            [
            
              
                
                  
                    
                      1
                      9
                    
                    ⁢
                    
                      2
                      9
                    
                    ⁢
                    
                      1
                      3
                    
                    ⁢
                    
                      2
                      9
                    
                    ⁢
                    
                      1
                      9
                    
                  
                
              
              
                
                  
                    
                      2
                      9
                    
                    ⁢
                    
                      4
                      9
                    
                    ⁢
                    
                      2
                      3
                    
                    ⁢
                    
                      4
                      9
                    
                    ⁢
                    
                      2
                      9
                    
                  
                
              
              
                
                  
                    
                      1
                      3
                    
                    ⁢
                    
                      2
                      3
                    
                    ⁢
                    1
                    ⁢
                    
                      2
                      3
                    
                    ⁢
                    
                      1
                      3
                    
                  
                
              
              
                
                  
                    
                      2
                      9
                    
                    ⁢
                    
                      4
                      9
                    
                    ⁢
                    
                      2
                      3
                    
                    ⁢
                    
                      4
                      9
                    
                    ⁢
                    
                      2
                      9
                    
                  
                
              
              
                
                  
                    
                      1
                      9
                    
                    ⁢
                    
                      2
                      9
                    
                    ⁢
                    
                      1
                      3
                    
                    ⁢
                    
                      2
                      9
                    
                    ⁢
                    
                      1
                      9
                    
                  
                
              
            
            ]
          
        
        ,
        
          
            H
            DI
          
          =
          
            
              [
              
                
                  
                    
                      
                        1
                        3
                      
                      ⁢
                      0
                      ⁢
                      
                        1
                        3
                      
                      ⁢
                      0
                      ⁢
                      
                        1
                        3
                      
                    
                  
                
                
                  
                    
                      0
                      ⁢
                      
                        2
                        3
                      
                      ⁢
                      
                        2
                        3
                      
                      ⁢
                      
                        2
                        3
                      
                      ⁢
                      0
                    
                  
                
                
                  
                    
                      
                        1
                        3
                      
                      ⁢
                      
                        2
                        3
                      
                      ⁢
                      1
                      ⁢
                      
                        2
                        3
                      
                      ⁢
                      
                        1
                        3
                      
                    
                  
                
                
                  
                    
                      0
                      ⁢
                      
                        2
                        3
                      
                      ⁢
                      
                        2
                        3
                      
                      ⁢
                      
                        2
                        3
                      
                      ⁢
                      0
                    
                  
                
                
                  
                    
                      
                        1
                        3
                      
                      ⁢
                      0
                      ⁢
                      
                        1
                        3
                      
                      ⁢
                      0
                      ⁢
                      
                        1
                        3
                      
                    
                  
                
              
              ]
            
            .
          
        
      
    
    
      
        (
        39
        )
      
    
  




In another aspect, MMSE-SD component 1010 can generate 3M×3N color components, or subpixels, R′, G′, and B′ of a derived, virtual image L′ based on reconstruction function ƒr(r) associated with Equations 1-4 and 40, reconstruction function ƒg(g) associated with Equations 5-8 and 40, and reconstruction ƒb(b) associated with equations 9-12 and 40, respectively.
In an aspect, MMSE-SD component 1010 can separately derive subpixels r, g, and b of low resolution image S by minimizing a MSE between L and L′ according to Equation 40 as follows:




  
    
      
        
          
            
              min
              
                r
                ,
                g
                ,
                b
              
            
            ⁢
            
              
                
                
                  R
                  -
                  
                    R
                    ′
                  
                
                
              
              2
              2
            
          
          +
          
            
              
              
                G
                -
                
                  G
                  ′
                
              
              
            
            2
            2
          
          +
          
            
              
              
                B
                -
                
                  B
                  ′
                
              
              
            
            2
            2
          
        
        ⁢
        
          

        
        ⁢
        
          
            s
            .
            t
            .
            
              
            
            ⁢
            
              R
              ′
            
          
          =
          
            
              H
              interp
              *
            
            ⁡
            
              (
              
                r
                ⊗
                
                  I
                  r
                
              
              )
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            G
            ′
          
          =
          
            
              H
              interp
              *
            
            ⁡
            
              (
              
                g
                ⊗
                
                  I
                  g
                
              
              )
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              B
              ′
            
            =
            
              
                H
                interp
                *
              
              ⁡
              
                (
                
                  b
                  ⊗
                  
                    I
                    b
                  
                
                )
              
            
          
          ,
        
      
    
    
      
        (
        40
        )
      
    
  




in which r, g, and b are M×N color components of S; R, G, and B are the 3M×3N color components of L; and R′, G′, and B′ are the 3M×3N color components of L′. Further, Ir, Ig, and Ib are 3×3 matrices given by equation (41) as follows:





  
    
      
        
          
            I
            r
          
          =
          
            [
            
              
                
                  100
                
              
              
                
                  000
                
              
              
                
                  000
                
              
            
            ]
          
        
        ,
        
          
            I
            g
          
          =
          
            [
            
              
                
                  000
                
              
              
                
                  010
                
              
              
                
                  000
                
              
            
            ]
          
        
        ,
        
          
            I
            b
          
          ⁡
          
            [
            
              
                
                  000
                
              
              
                
                  000
                
              
              
                
                  001
                
              
            
            ]
          
        
        ,
      
    
    
      
        (
        41
        )
      
    
  




in which  is the Kronecker product that expands r, g, and b to be 3M×3 N matrices. Further, Hinterp is the interpolation filter (HBI or HDI) defined above, and * is a convolution operation.

As such, with Hinterp=HBI, MMSE-SD component 1010 can derive subpixel r of low resolution image S according to equation (42) as follows:




  
    
      
        
          
            min
            r
          
          ⁢
          
            
              
              
                R
                -
                
                  R
                  ′
                
              
              
            
            2
            2
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              s
              .
              t
              .
              
                
              
              ⁢
              
                R
                ′
              
            
            =
            
              
                H
                BI
                *
              
              ⁡
              
                (
                
                  r
                  ⊗
                  
                    I
                    r
                  
                
                )
              
            
          
          ,
        
      
    
    
      
        (
        42
        )
      
    
  




Let E∥R−R′∥22 with R′=HBI*(rI3r). Then E would contain 3M×3N terms. Consider a particular (i,j). We are interested in those terms containing ri,j. While there are altogether 25 such terms (inside the red square of FIG. 11), we have limited space and will thus only show the 9 terms at location (k,l), with k=3i−3,3i−2,3i−1 and l=3j−3,3j−2,3j−1.





  E
  =
  
    ...
    +
    
      [
      
        
          
            (
            
              
                R
                
                  
                    
                      3
                      ⁢
                      
                        
                      
                      ⁢
                      i
                    
                    -
                    3
                  
                  ;
                  
                    
                      3
                      ⁢
                      j
                    
                    -
                    3
                  
                
              
              -
              
                (
                
                  
                    
                      α
                      3
                    
                    ⁢
                    
                      r
                      
                        i
                        ,
                        j
                      
                    
                  
                  +
                  
                    
                      β
                      3
                    
                    ⁢
                    
                      r
                      
                        i
                        ;
                        j
                      
                    
                  
                  +
                  
                    
                      β
                      3
                    
                    ⁢
                    
                      r
                      
                        
                          i
                          -
                          1
                        
                        ;
                        
                          j
                          -
                          1
                        
                      
                    
                  
                  +
                  
                    γ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      r
                      
                        i
                        ;
                        
                          j
                          -
                          1
                        
                      
                    
                  
                  +
                  
                    γ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      r
                      
                        
                          i
                          -
                          1
                        
                        ;
                        j
                      
                    
                  
                
                )
              
            
            )
          
          2
        
        +
        
          
            (
            
              
                R
                
                  
                    
                      3
                      ⁢
                      i
                    
                    -
                    3
                  
                  ;
                  
                    
                      3
                      ⁢
                      j
                    
                    -
                    2
                  
                
              
              -
              
                (
                
                  
                    
                      α
                      2
                    
                    ⁢
                    
                      r
                      
                        i
                        ;
                        j
                      
                    
                  
                  +
                  
                    
                      β
                      2
                    
                    ⁢
                    
                      r
                      
                        
                          i
                          -
                          1
                        
                        ;
                        j
                      
                    
                  
                
                )
              
            
            )
          
          2
        
        +
        
          
            (
            
              
                R
                
                  
                    
                      3
                      ⁢
                      i
                    
                    -
                    3
                  
                  ;
                  
                    
                      3
                      ⁢
                      j
                    
                    -
                    1
                  
                
              
              -
              
                (
                
                  
                    
                      α
                      4
                    
                    ⁢
                    
                      r
                      
                        i
                        ;
                        j
                      
                    
                  
                  +
                  
                    
                      β
                      4
                    
                    ⁢
                    
                      r
                      
                        
                          i
                          -
                          1
                        
                        ;
                        
                          j
                          +
                          1
                        
                      
                    
                  
                  +
                  
                    γ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      r
                      
                        
                          i
                          -
                          1
                        
                        ;
                        j
                      
                    
                  
                  +
                  
                    γ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      r
                      
                        i
                        ;
                        
                          j
                          +
                          1
                        
                      
                    
                  
                
                )
              
            
            )
          
          2
        
        +
        
          
            (
            
              
                R
                
                  
                    
                      3
                      ⁢
                      i
                    
                    -
                    2
                  
                  ;
                  
                    
                      3
                      ⁢
                      j
                    
                    -
                    3
                  
                
              
              -
              
                (
                
                  
                    
                      α
                      1
                    
                    ⁢
                    
                      r
                      
                        i
                        ,
                        j
                      
                    
                  
                  +
                  
                    
                      β
                      1
                    
                    ⁢
                    
                      r
                      
                        i
                        ;
                        
                          j
                          -
                          1
                        
                      
                    
                  
                
                )
              
            
            )
          
          2
        
        +
        
          
            (
            
              
                R
                
                  
                    
                      3
                      ⁢
                      i
                    
                    -
                    2
                  
                  ;
                  
                    
                      3
                      ⁢
                      j
                    
                    -
                    2
                  
                
              
              -
              
                r
                
                  i
                  ;
                  j
                
              
            
            )
          
          2
        
        +
        
          
            (
            
              
                R
                
                  
                    
                      3
                      ⁢
                      i
                    
                    -
                    2
                  
                  ;
                  
                    
                      3
                      ⁢
                      j
                    
                    -
                    1
                  
                
              
              -
              
                (
                
                  
                    
                      α
                      1
                    
                    ⁢
                    
                      r
                      
                        i
                        ;
                        j
                      
                    
                  
                  +
                  
                    
                      β
                      1
                    
                    ⁢
                    
                      r
                      
                        i
                        ;
                        
                          j
                          +
                          1
                        
                      
                    
                  
                
                )
              
            
            )
          
          2
        
        +
        
          
            (
            
              
                R
                
                  
                    
                      3
                      ⁢
                      i
                    
                    -
                    1
                  
                  ;
                  
                    
                      3
                      ⁢
                      j
                    
                    -
                    3
                  
                
              
              -
              
                (
                
                  
                    
                      α
                      4
                    
                    ⁢
                    
                      r
                      
                        i
                        ;
                        j
                      
                    
                  
                  +
                  
                    
                      β
                      4
                    
                    ⁢
                    
                      r
                      
                        
                          i
                          +
                          1
                        
                        ;
                        
                          j
                          -
                          1
                        
                      
                    
                  
                  +
                  
                    γ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      r
                      
                        i
                        ;
                        
                          j
                          -
                          1
                        
                      
                    
                  
                  +
                  
                    γ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      r
                      
                        
                          i
                          +
                          1
                        
                        ;
                        j
                      
                    
                  
                
                )
              
            
            )
          
          2
        
        +
        
          
            (
            
              
                R
                
                  
                    3
                    ⁢
                    i1
                  
                  ;
                  
                    
                      3
                      ⁢
                      j
                    
                    -
                    2
                  
                
              
              -
              
                (
                
                  
                    
                      α
                      2
                    
                    ⁢
                    
                      r
                      
                        i
                        ;
                        j
                      
                    
                  
                  +
                  
                    
                      β
                      2
                    
                    ⁢
                    
                      r
                      
                        
                          i
                          +
                          1
                        
                        ;
                        j
                      
                    
                  
                
                )
              
            
            )
          
          2
        
        +
        
          
            (
            
              
                R
                
                  
                    
                      3
                      ⁢
                      i
                    
                    -
                    1
                  
                  ;
                  
                    
                      3
                      ⁢
                      j
                    
                    -
                    1
                  
                
              
              -
              
                (
                
                  
                    
                      α
                      3
                    
                    ⁢
                    
                      r
                      
                        i
                        ;
                        j
                      
                    
                  
                  +
                  
                    
                      β
                      3
                    
                    ⁢
                    
                      r
                      
                        
                          i
                          +
                          1
                        
                        ;
                        
                          j
                          +
                          1
                        
                      
                    
                  
                  +
                  
                    γ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      r
                      
                        
                          i
                          +
                          1
                        
                        ;
                        j
                      
                    
                  
                  +
                  
                    γ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      r
                      
                        
                          j
                          +
                          1
                        
                        ;
                        
                          j
                          +
                          1
                        
                      
                    
                  
                
                )
              
            
            )
          
          2
        
      
      ]
    
    +
    ...
  




Differentiating E with respect to ri,j and setting it to zero, we obtain





  
    
      
        k
        3
      
      ⁢
      
        r
        
          
            i
            -
            1
          
          ;
          
            j
            -
            1
          
        
      
    
    +
    
      
        k
        2
      
      ⁢
      
        r
        
          
            i
            -
            1
          
          ;
          j
        
      
    
    +
    
      
        k
        4
      
      ⁢
      
        r
        
          
            i
            -
            1
          
          ;
          
            j
            +
            1
          
        
      
    
    +
    
      
        k
        1
      
      ⁢
      
        r
        
          i
          ;
          
            j
            -
            1
          
        
      
    
    +
    
      
        k
        0
      
      ⁢
      
        r
        
          i
          ;
          j
        
      
    
    +
    
      
        k
        1
      
      ⁢
      
        r
        
          i
          ;
          
            j
            +
            1
          
        
      
    
    +
    
      
        k
        4
      
      ⁢
      
        r
        
          
            i
            +
            1
          
          ;
          
            j
            -
            1
          
        
      
    
    +
    
      
        k
        2
      
      ⁢
      
        r
        
          
            i
            +
            1
          
          ;
          j
        
      
    
    +
    
      
        k
        3
      
      ⁢
      
        r
        
          
            i
            +
            1
          
          ;
          
            j
            +
            1
          
        
      
    
  
  =
  
    
      
        β
        3
      
      ⁢
      
        R
        
          
            
              3
              ⁢
              i
            
            -
            4
          
          ;
          
            
              3
              ⁢
              j
            
            -
            4
          
        
      
    
    +
    
      γ
      ⁢
      
        
      
      ⁢
      
        R
        
          
            
              3
              ⁢
              i
            
            -
            4
          
          ;
          
            
              3
              ⁢
              j
            
            -
            3
          
        
      
    
    +
    
      
        β
        2
      
      ⁢
      
        R
        
          
            
              3
              ⁢
              i
            
            -
            4
          
          ;
          
            
              3
              ⁢
              j
            
            -
            2
          
        
      
    
    +
    
      γ
      ⁢
      
        
      
      ⁢
      
        R
        
          
            
              3
              ⁢
              i
            
            -
            4
          
          ;
          
            
              3
              ⁢
              j
            
            -
            1
          
        
      
    
    +
    
      
        β
        4
      
      ⁢
      
        R
        
          
            
              3
              ⁢
              i
            
            -
            4
          
          ,
          
            3
            ⁢
            j
          
        
      
    
    +
    
      γ
      ⁢
      
        
      
      ⁢
      
        R
        
          
            
              3
              ⁢
              i
            
            -
            3
          
          ;
          
            
              3
              ⁢
              j
            
            -
            4
          
        
      
    
    +
    
      
        α
        3
      
      ⁢
      
        R
        
          
            
              3
              ⁢
              i
            
            -
            3
          
          ;
          
            
              3
              ⁢
              j
            
            -
            3
          
        
      
    
    +
    
      
        α
        2
      
      ⁢
      
        R
        
          
            
              3
              ⁢
              i
            
            -
            3
          
          ;
          
            
              3
              ⁢
              j
            
            -
            2
          
        
      
    
    +
    
      
        α
        4
      
      ⁢
      
        R
        
          
            
              3
              ⁢
              i
            
            -
            3
          
          ;
          
            
              3
              ⁢
              j
            
            -
            1
          
        
      
    
    +
    
      γ
      ⁢
      
        
      
      ⁢
      
        R
        
          
            
              3
              ⁢
              i
            
            -
            3
          
          ;
          
            3
            ⁢
            j
          
        
      
    
    +
    
      
        β
        1
      
      ⁢
      
        R
        
          
            
              3
              ⁢
              i
            
            -
            2
          
          ;
          
            
              3
              ⁢
              j
            
            -
            4
          
        
      
    
    +
    
      
        α
        1
      
      ⁢
      
        R
        
          
            
              3
              ⁢
              i
            
            -
            2
          
          ;
          
            
              3
              ⁢
              j
            
            -
            3
          
        
      
    
    +
    
      R
      
        
          
            3
            ⁢
            i
          
          -
          2
        
        ;
        
          
            3
            ⁢
            j
          
          -
          2
        
      
    
    +
    
      
        α
        1
      
      ⁢
      
        R
        
          
            
              3
              ⁢
              i
            
            -
            2
          
          ;
          
            
              3
              ⁢
              j
            
            -
            1
          
        
      
    
    +
    
      
        β
        1
      
      ⁢
      
        R
        
          
            
              3
              ⁢
              i
            
            -
            2
          
          ;
          
            3
            ⁢
            j
          
        
      
    
    +
    
      γ
      ⁢
      
        
      
      ⁢
      
        R
        
          
            
              3
              ⁢
              i
            
            -
            1
          
          ;
          
            
              3
              ⁢
              j
            
            -
            4
          
        
      
    
    +
    
      
        α
        4
      
      ⁢
      
        R
        
          
            
              3
              ⁢
              i
            
            -
            1
          
          ;
          
            
              3
              ⁢
              j
            
            -
            3
          
        
      
    
    +
    
      
        α
        2
      
      ⁢
      
        R
        
          
            
              3
              ⁢
              i
            
            -
            1
          
          ;
          
            
              3
              ⁢
              j
            
            -
            2
          
        
      
    
    +
    
      
        α
        3
      
      ⁢
      
        R
        
          
            
              3
              ⁢
              i
            
            -
            1
          
          ;
          
            
              3
              ⁢
              j
            
            -
            1
          
        
      
    
    +
    
      γ
      ⁢
      
        
      
      ⁢
      
        R
        
          
            
              3
              ⁢
              i
            
            -
            1
          
          ;
          
            3
            ⁢
            j
          
        
      
    
    +
    
      
        β
        4
      
      ⁢
      
        R
        
          
            3
            ⁢
            i
          
          ;
          
            
              3
              ⁢
              j
            
            -
            4
          
        
      
    
    +
    
      γ
      ⁢
      
        
      
      ⁢
      
        R
        
          
            3
            ⁢
            i
          
          ;
          
            
              3
              ⁢
              j
            
            -
            3
          
        
      
    
    +
    
      
        β
        2
      
      ⁢
      
        R
        
          
            3
            ⁢
            i
          
          ;
          
            
              3
              ⁢
              j
            
            -
            2
          
        
      
    
    +
    
      γ
      ⁢
      
        
      
      ⁢
      
        R
        
          
            3
            ⁢
            i
          
          ;
          
            
              3
              ⁢
              j
            
            -
            1
          
        
      
    
    +
    
      
        β
        3
      
      ⁢
      
        R
        
          
            3
            ⁢
            i
          
          ;
          
            3
            ⁢
            j
          
        
      
    
  




in which k=1+2α12+2β12+2α22+2β22+2α32+2β32+2α42+2β42+8γ2, k1=2α1β1+2α3γ+2β3γ+2α4γ+2β4γ, k2=2α2β2+2α3γ+2β3γ+2α4γ+2β4γ, k3=2α3β3+2γ2, and k4=2α4β4+2γ2.

Considering all the i=1, 2, . . . , M, j=1, 2, . . . , N, we have a total of M×N equations which can be represented in matrix form as shown by equation 15 above. As described above, H is a block circulant matrix of size MN×9MN, including N×9N blocks that are block-tri-circulant. Each of the N×9N blocks include three sub-blocks of size N×3N, and each of the three sub-blocks is block-circulant with a block size of 1×3. In other words, each row of a (k,l)th sub-block, k=1, . . . , M, and l=1, . . . , 3M, of the N×9 N blocks of block circulant matrix H has 3N coefficients and is equal to the previous row rotated, or shifted, to the right by 3 sub-block positions. The mth row of the (k,l)th sub-block of H is associated with an inner-product with the lth row of L and adds a term to the mth element of the kth row of S, i.e., the (k,m)th pixel of S.
MMSE-SD component 1010 can be configured to generate a 3N×3N circulant matrix using the first row of the (k,l)th sub-block, as the row has 3N coefficients and is equal to the previous row rotated to the right by 3 sub-block positions. Further, MMSE-SD component 1010 can perform a 3:1 down-sampling on the 3N×3N circulant matrix in a vertical direction utilizing equation (43) described below:




  
    
      
        
          
            I
            a
          
          ⁡
          
            (
            
              i
              ,
              j
            
            )
          
        
        =
        
          {
          
            
              
                
                  1
                  ,
                
              
              
                
                  
                    i
                    =
                    1
                  
                  ,
                  ...
                  ⁢
                  
                    
                  
                  ,
                  N
                  ,
                  
                    
                      j
                      =
                      
                        
                          3
                          ⁢
                          i
                        
                        -
                        2
                      
                    
                    ;
                  
                
              
            
            
              
                
                  0
                  ,
                
              
              
                
                  otherwise
                  .
                
              
            
          
        
      
    
    
      
        (
        43
        )
      
    
  




Thus, MMSE-SD component 1010 can be configured to (1) perform a 1-D convolution of the first row of the (k,l)th sub-block of block circulant matrix H with a periodic extension of the lth row of L to generate a row of size 3N; (2) perform a 3:1 down-sampling of the row of size 3N in a horizontal direction; and (3) add a result of the 3:1 down-sampling to the kth row of S.
In an aspect, MMSE-SD component 1010 can perform such operations of first rows of sub-blocks of block circulant matrix H on L (with k=1, l=1, . . . , 3M), effectively applying a 2-D spatial-invariant linear filter on a periodic extension of L to obtain a row of size 3N, in which MMSE-SD can obtain coefficients of the 2-D spatial-invariant filter from the first row of block circulant matrix H. Further, MMSE-SD component 1010 can 3:1 down-sample the row in a horizontal direction to obtain the 1st row of S, e.g., based on the information received from computing device 120.
As stated above, H is a block-circulant matrix including N×9N blocks that are block-tri-circulant (each block including three sub-blocks of size N×3N that are block-circulant). Further, each row of the N×9N blocks can be considered as a row of sub-blocks of size N×3N, in which each row of sub-blocks is equivalent to a previous row rotated to the right by 3 sub-block positions. As such, in an embodiment, MMSE-SD component 1010 can be configured to generate a derived block-circulant matrix of size 3MN×9MN that includes blocks of size N×3N. Further, MMSE-SD component 1010 can be configured to compute H as the product of a matrix Ib (defined by equation (44) below) of size MN×3MN and the derived block-circulant matrix:




  
    
      
        
          
            I
            b
          
          ⁡
          
            (
            
              i
              ,
              j
            
            )
          
        
        =
        
          {
          
            
              
                
                  
                    I
                    N
                  
                  ,
                
              
              
                
                  
                    i
                    =
                    1
                  
                  ,
                  ...
                  ⁢
                  
                    
                  
                  ,
                  M
                  ,
                  
                    
                      j
                      =
                      
                        
                          3
                          ⁢
                          i
                        
                        -
                        2
                      
                    
                    ;
                  
                
              
            
            
              
                
                  0
                  ,
                
              
              
                
                  otherwise
                  ..
                
              
            
          
        
      
    
    
      
        (
        44
        )
      
    
  




In one aspect, the product of matrix Ib and the derived block-circulant matrix is a 3:1 down-sampling on the sub-blocks of the derived block-circulant matrix in a vertical direction. As such, MMSE-SD component 1010 can apply a 2-D spatial-invariant filter, of size 3M×3N, on the periodic extension of L to obtain an image of size 3N×3N. Further, MMSE-SD component 1010 can select portions of the image, e.g., reduce the size of the image, based on the information received from computing device 120, e.g., the dimension associated with display 125. The 2-D spatial invariant filter can be independent of high resolution image (L), and thus pre-computed and stored, e.g., in a storage medium. In an aspect, the 2-D spatial-invariant filter can be set to a size k×k, e.g., k=15, for example, based on a dimension associated with display 125. Further MMSE-SD component 1010 can 3:1 down-sample the image in a horizontal direction and 3:1 down-sample the image in a vertical direction based on the information, e.g., the dimension associated with display 125, received from computing device 120. In another aspect, the 2-D spatial-invariant filter can be set to a size k×k in which k=9, and in which the 9×9 filter coefficients are symmetric in such a way that six coefficients appear 8 times, eight coefficients appear 4 times, and one coefficient appears 1 time. As such, only 6+8+1=15 different coefficients need to be stored, and only 15 multiplications are required for every down-sampled location.
Now referring to FIG. 12, a block diagram of a down-sampling environment 1200 including a low resolution display 1225 is illustrated, in accordance with an embodiment. Down-sampling environment 1200 can include system 120, which can receive, store, etc. a high resolution image associated with, e.g., HDTV information, HD movies, etc. In an aspect, system 120 can include and/or be communicatively coupled with a network, an information based server, e.g., web server, online service provider, etc. via a cloud computing environment and/or a remote storage environment.
In another aspect, computing device 1220 can be communicatively coupled with system 110, via a communication network, e.g., a wired communication network, an internet protocol (IP) based network, a wireless communication network, a wireless protocol associated with, e.g., wireless fidelity (Wi-Fi), Worldwide Interoperability for Microwave Access (WiMAX), Enhanced General Packet Radio Service (Enhanced GPRS), 3GPP Long Term Evolution (LTE), Third Generation Partnership Project 2 (3GPP2) Ultra Mobile Broadband (UMB), 3GPP UMTS, High Speed Packet Access (HSPA), High Speed Downlink Packet Access (HSDPA), High Speed Uplink Packet Access (HSUPA), LTE Advanced, Third Generation Partnership Project (3GPP) Universal Mobile Telecommunication System (UMTS), Global System for Mobile Communication (GSM), etc.
Further, computing device 1220 can include, comprise, etc. a single-processor computer system, a multi-processor computer system, a personal computer, a remote processing device, a hand-held computing device, e.g., a mobile phone, a cellular phone, a camera, a PMP, a PDA, a wrist watch, etc. Furthermore, computing device 1220 can include software, firmware, display driver(s), etc. that can be configured to display a network interface, e.g., web browser, etc. via display 1225. As such, in response to receiving, via an input interface (not shown) of computing device 1220, a request for information (e.g., an internet query, etc. associated with, e.g., the network, the information based server, the web browser, etc.), the software, the firmware, the display driver(s), etc. can be configured to generate, modify, etc. information associated with display 125 (see above).
Further, computing device 1220 can be configured to communicate the information associated with display 1225 to system 110. System 110 can be configured to generate an image S including subpixels sampled and/or generated from high resolution image (L), based on the information associated with display 1225. As such, computing device 1220 can be configured to receive the image S, e.g., communicated by system 110, and display the image, or portions thereof, via display 1225, which can include a lower resolution display, e.g., included in a handheld device such as a cellular phone, a PMP, a PDA, etc. In one aspect, display 1225 can include an LCD display, a lower resolution display, etc. in which a resolution of display 1225 can be based on, e.g., a pixel resolution of display 1225, a spatial resolution of display 1225, a spectral resolution of display 1225, a temporal resolution of display 1225, a radiometric resolution of display 1225, etc. In another aspect, the information can include a dimension associated with display 1225, a dimension associated with an image to be displayed via display 1225, a resolution of the image, a pixel resolution of the image, a spatial resolution of the image, a spectral resolution of the image, a temporal resolution of the image, a resolution associated with display 1225, a pixel resolution associated with display 1225, a spatial resolution associated with display 1225, a spectral resolution associated with display 1225, a temporal resolution of display 1225, etc. In an aspect, the dimension associated with the image can be lower than the resolution of the image.
FIG. 13 illustrates a block diagram of a down-sampling environment 1300 including a server 1310, in accordance with an embodiment. Server 1310 can include components and/or features associated with systems 110, 200, 1000, etc. described above. For example, server 1310 can include a network based server, an internet based server, a World Wide Web based server, etc. In an aspect, server 1310 can receive from devices, e.g., display device 1320, portable communication device 1322, personal computer 1324, etc. respective requests, e.g., REQ 1, REQ 2, REQ3, etc. that can be generated from respective devices of the devices, e.g., in response to receiving respective queries, respective inputs, etc. to display respective information and/or respective web pages, e.g., associated with respective internet queries, etc. via respective displays (not shown) of the respective devices.
The respective requests can include respective display information, e.g., respective dimensions associated with the respective displays. Further, server 1310 can generate respective images, e.g., IMAGE 1, IMAGE 2, IMAGE 3, etc. based on down-sampling 3×3 arrays of pixels of the 2-D array of pixels of high-resolution image L (see above), e.g., by selecting, sampling, etc. subpixels of pixels of an array of the 3×3 arrays of pixels in the diagonal direction to generate low resolution image S. Furthermore, server 1310 can communicate the respective images to the respective devices, which can display the respective images via the respective displays.
FIGS. 14-16 illustrate methodologies in accordance with the disclosed subject matter. For simplicity of explanation, the methodologies are depicted and described as a series of acts, e.g., that can be formed via one or more components, features, elements, etc. associated with systems described above. It is to be understood and appreciated that the subject innovation is not limited by the acts illustrated and/or by the order of acts. For example, acts can occur in various orders and/or concurrently, and with other acts not presented or described herein. Furthermore, not all illustrated acts may be required to implement the methodologies in accordance with the disclosed subject matter. In addition, those skilled in the art will understand and appreciate that the methodologies could alternatively be represented as a series of interrelated states via a state diagram or events. Additionally, it should be further appreciated that the methodologies disclosed hereinafter and throughout this specification are capable of being stored on an article of manufacture to facilitate transporting and transferring such methodologies to computers. The term article of manufacture, as used herein, is intended to encompass a computer program accessible from any computer-readable device, carrier, or media.
Referring now to FIG. 14, a process 1400 associated with minimum mean square error subpixel-based down-sampling (MMSE-SD) is illustrated, in accordance with an embodiment. At 1410, a high resolution image can be divided into at least two blocks, wherein each block of the at least two blocks includes a 3×3 array of pixels. At 1420, a request, e.g., network based query, internet based query, etc. can be received from a device, the request including information associated with a display of the device. Red, green, and blue subpixels can be diagonally selected from respective adjacent pixels of each block of the at least two blocks at 1430. At 1440, a low resolution, down-sampled image S can be created based on subpixels selected at 1430 and the information associated with the display of the device.
FIG. 15 illustrates another process (1500) for creating an optimal low resolution image (S), in accordance with an embodiment. At 1510, a virtual image can be derived, based on the low resolution image created at 1540 (see FIG. 14). Mean square error between the virtual image and the low resolution image can be minimized at 1520. At 1530, the optimal low resolution image (S) can be created, generated, etc. based on a result of step 1520 and the information associated with the display of the device.
FIG. 16 illustrates a process 1600 associated with a device including a display, in accordance with an embodiment. At 1610, process 1600 can determine whether a network query, e.g., internet based request, can be associated with an image to be displayed via the display of the device. If the request is not associated with the image, process 1600 can end, otherwise, process 1600 can continue to 1620, at which the network query, and a dimension associated with the display, can be communicated, e.g., to a web based server, network based server, etc. At 1630, process 1600 can receive graphical information, e.g., image(s), video content, etc. based on the network query. At 1640, process 1600 can display at least a portion of the graphical information using the display.
As it employed in the subject specification, the term “processor” can refer to substantially any computing processing unit or device comprising, but not limited to comprising, single-core processors; single-processors with software multithread execution capability; multi-core processors; multi-core processors with software multithread execution capability; multi-core processors with hardware multithread technology; parallel platforms; and parallel platforms with distributed shared memory. Additionally, a processor can refer to an integrated circuit, an application specific integrated circuit (ASIC), a digital signal processor (DSP), a field programmable gate array (FPGA), a programmable logic controller (PLC), a complex programmable logic device (CPLD), a discrete gate or transistor logic, discrete hardware components, or any combination thereof designed to perform the functions and/or processes described herein. Processors can exploit nano-scale architectures such as, but not limited to, molecular and quantum-dot based transistors, switches and gates, in order to optimize space usage or enhance performance of mobile devices. A processor may also be implemented as a combination of computing processing units.
In the subject specification, terms such as “store,” “data store,” “data storage,” “database,” “storage medium,” and substantially any other information storage component relevant to operation and functionality of a component and/or process, refer to “memory components,” or entities embodied in a “memory,” or components comprising the memory. It will be appreciated that the memory components described herein can be either volatile memory or nonvolatile memory, or can include both volatile and nonvolatile memory.
By way of illustration, and not limitation, nonvolatile memory, for example, can be included in storage systems described above, computing devices (e.g., 120, 1220), non-volatile memory 1722 (see below), disk storage 1724 (see below), and memory storage 1746 (see below). Further, nonvolatile memory can be included in read only memory (ROM), programmable ROM (PROM), electrically programmable ROM (EPROM), electrically erasable ROM (EEPROM), or flash memory. Volatile memory can include random access memory (RAM), which acts as external cache memory. By way of illustration and not limitation, RAM is available in many forms such as synchronous RAM (SRAM), dynamic RAM (DRAM), synchronous DRAM (SDRAM), double data rate SDRAM (DDR SDRAM), enhanced SDRAM (ESDRAM), Synchlink DRAM (SLDRAM), and direct Rambus RAM (DRRAM). Additionally, the disclosed memory components of systems or methods herein are intended to comprise, without being limited to comprising, these and any other suitable types of memory.
In order to provide a context for the various aspects of the disclosed subject matter, FIG. 17, and the following discussion, are intended to provide a brief, general description of a suitable environment in which the various aspects of the disclosed subject matter can be implemented, e.g., various processes associated with FIGS. 1-16. While the subject matter has been described above in the general context of computer-executable instructions of a computer program that runs on a computer and/or computers, those skilled in the art will recognize that the subject innovation also can be implemented in combination with other program modules. Generally, program modules include routines, programs, components, data structures, etc. that perform particular tasks and/or implement particular abstract data types.
Moreover, those skilled in the art will appreciate that the inventive systems can be practiced with other computer system configurations, including single-processor or multiprocessor computer systems, mini-computing devices, mainframe computers, as well as personal computers, hand-held computing devices (e.g., PDA, phone, watch), microprocessor-based or programmable consumer or industrial electronics, and the like. The illustrated aspects can also be practiced in distributed computing environments where tasks are performed by remote processing devices that are linked through a communications network; however, some if not all aspects of the subject disclosure can be practiced on stand-alone computers. In a distributed computing environment, program modules can be located in both local and remote memory storage devices.
With reference to FIG. 17, a block diagram of a computing system 1700 operable to execute the disclosed systems and methods is illustrated, in accordance with an embodiment. Computer 1712 includes a processing unit 1714, a system memory 1716, and a system bus 1718. System bus 1718 couples system components including, but not limited to, system memory 1716 to processing unit 1714. Processing unit 1714 can be any of various available processors. Dual microprocessors and other multiprocessor architectures also can be employed as processing unit 1714.
System bus 1718 can be any of several types of bus structure(s) including a memory bus or a memory controller, a peripheral bus or an external bus, and/or a local bus using any variety of available bus architectures including, but not limited to, Industrial Standard Architecture (ISA), Micro-Channel Architecture (MSA), Extended ISA (EISA), Intelligent Drive Electronics (IDE), VESA Local Bus (VLB), Peripheral Component Interconnect (PCI), Card Bus, Universal Serial Bus (USB), Advanced Graphics Port (AGP), Personal Computer Memory Card International Association bus (PCMCIA), Firewire (IEEE 1194), and Small Computer Systems Interface (SCSI).
System memory 1716 includes volatile memory 1720 and nonvolatile memory 1722. A basic input/output system (BIOS), containing routines to transfer information between elements within computer 1712, such as during start-up, can be stored in nonvolatile memory 1722. By way of illustration, and not limitation, nonvolatile memory 1722 can include ROM, PROM, EPROM, EEPROM, or flash memory. Volatile memory 1720 includes RAM, which acts as external cache memory. By way of illustration and not limitation, RAM is available in many forms such as SRAM, dynamic RAM (DRAM), synchronous DRAM (SDRAM), double data rate SDRAM (DDR SDRAM), enhanced SDRAM (ESDRAM), Synchlink DRAM (SLDRAM), Rambus direct RAM (RDRAM), direct Rambus dynamic RAM (DRDRAM), and Rambus dynamic RAM (RDRAM).
Computer 1712 can also include removable/non-removable, volatile/non-volatile computer storage media, networked attached storage (NAS), e.g., SAN storage, etc. FIG. 17 illustrates, for example, disk storage 1724. Disk storage 1724 includes, but is not limited to, devices like a magnetic disk drive, floppy disk drive, tape drive, Jaz drive, Zip drive, LS-100 drive, flash memory card, or memory stick. In addition, disk storage 1724 can include storage media separately or in combination with other storage media including, but not limited to, an optical disk drive such as a compact disk ROM device (CD-ROM), CD recordable drive (CD-R Drive), CD rewritable drive (CD-RW Drive) or a digital versatile disk ROM drive (DVD-ROM). To facilitate connection of the disk storage devices 1724 to system bus 1718, a removable or non-removable interface is typically used, such as interface 1726.
It is to be appreciated that FIG. 17 describes software that acts as an intermediary between users and computer resources described in suitable operating environment 1700. Such software includes an operating system 1728. Operating system 1728, which can be stored on disk storage 1724, acts to control and allocate resources of computer 1712. System applications 1730 take advantage of the management of resources by operating system 1728 through program modules 1732 and program data 1734 stored either in system memory 1716 or on disk storage 1724. It is to be appreciated that the disclosed subject matter can be implemented with various operating systems or combinations of operating systems.
A user can enter commands or information into computer 1712 through input device(s) 1736. Input devices 1736 include, but are not limited to, a pointing device such as a mouse, trackball, stylus, touch pad, keyboard, microphone, joystick, game pad, satellite dish, scanner, TV tuner card, digital camera, digital video camera, web camera, and the like. These and other input devices connect to processing unit 1714 through system bus 1718 via interface port(s) 1738. Interface port(s) 1738 include, for example, a serial port, a parallel port, a game port, and a universal serial bus (USB). Output device(s) 1740 use some of the same type of ports as input device(s) 1736.
Thus, for example, a USB port can be used to provide input to computer 1712 and to output information from computer 1712 to an output device 1740. Output adapter 1742 is provided to illustrate that there are some output devices 1740 like monitors, speakers, and printers, among other output devices 1740, which use special adapters. Output adapters 1742 include, by way of illustration and not limitation, video and sound cards that provide means of connection between output device 1740 and system bus 1718. It should be noted that other devices and/or systems of devices provide both input and output capabilities such as remote computer(s) 1744.
Computer 1712 can operate in a networked environment using logical connections to one or more remote computers, such as remote computer(s) 1744. Remote computer(s) 1744 can be a personal computer, a server, a router, a network PC, a workstation, a microprocessor based appliance, a peer device, or other common network node and the like, and typically includes many or all of the elements described relative to computer 1712.
For purposes of brevity, only a memory storage device 1746 is illustrated with remote computer(s) 1744. Remote computer(s) 1744 is logically connected to computer 1712 through a network interface 1748 and then physically connected via communication connection 1750. Network interface 1748 encompasses wire and/or wireless communication networks such as local-area networks (LAN) and wide-area networks (WAN). LAN technologies include Fiber Distributed Data Interface (FDDI), Copper Distributed Data Interface (CDDI), Ethernet, Token Ring and the like. WAN technologies include, but are not limited to, point-to-point links, circuit switching networks like Integrated Services Digital Networks (ISDN) and variations thereon, packet switching networks, and Digital Subscriber Lines (DSL).
Communication connection(s) 1750 refer(s) to hardware/software employed to connect network interface 1748 to bus 1718. While communication connection 1750 is shown for illustrative clarity inside computer 1712, it can also be external to computer 1712. The hardware/software for connection to network interface 1748 can include, for example, internal and external technologies such as modems, including regular telephone grade modems, cable modems and DSL modems, ISDN adapters, and Ethernet cards.
The above description of illustrated embodiments of the subject disclosure, including what is described in the Abstract, is not intended to be exhaustive or to limit the disclosed embodiments to the precise forms disclosed. While specific embodiments and examples are described herein for illustrative purposes, various modifications are possible that are considered within the scope of such embodiments and examples, as those skilled in the relevant art can recognize.
In this regard, while the disclosed subject matter has been described in connection with various embodiments and corresponding Figures, where applicable, it is to be understood that other similar embodiments can be used or modifications and additions can be made to the described embodiments for performing the same, similar, alternative, or substitute function of the disclosed subject matter without deviating therefrom. Therefore, the disclosed subject matter should not be limited to any single embodiment described herein, but rather should be construed in breadth and scope in accordance with the appended claims below.

## Claims

1. A method, comprising:
receiving, by a system comprising a processor, a request from a device including information associated with a display of the device;
dividing a two-dimensional (2-D) array of pixels into at least 3 pixel×at least 3 pixel blocks, wherein pixels of a block of the at least 3 pixel×at least 3 pixel blocks comprise at least 3 subpixels;
alternately down sampling selected subpixels of the pixels of the block in a diagonal direction; and
generating an image based on the information and a result of the alternately down sampling the selected subpixels.

2. The method of claim 1, further comprising:
communicating the image to the device based on the request.

3. The method of claim 1, wherein the alternately down sampling the selected subpixels comprises alternately selecting a red subpixel, a green subpixel, and a blue subpixel from consecutive pixels of the block in the diagonal direction.

4. The method of claim 1, wherein the receiving the request comprises receiving information associated with a dimension associated with the display, and wherein the generating the image comprises generating the image based on the dimension associated with the display.

5. The method of claim 1, wherein the receiving the request comprises:
receiving, via a network device of a cloud based computing environment, a query including the information from the device; and
communicating, via the network device of the cloud based computing environment, the image to the device based on the request.

6. The method of claim 1, wherein the receiving the query comprises receiving an internet based query via a web browser.

7. The method of claim 1, wherein the generating the image comprises determining at least one color component of a defined low resolution image associated with the result based on a block circulant matrix.

8. The method of claim 7, wherein the determining the at least one color component comprises determining the at least on color component based on a block circulant matrix of size MN×9MN including N×9N arrays of blocks that are block-tri-circulant.

9. A system, comprising:
a memory to store instructions; and
a processor, coupled to the memory, that facilitates execution of the instructions to at least:
divide a first array of pixels into 3 or more pixel×3 or more pixel blocks, wherein pixels of a block of the 3 or more pixel×3 or more pixel blocks comprise at least 3 subpixels;
receive a query from a device including information associated with a display of the device;
alternately down-sample selected subpixels of the pixels of the block in a diagonal direction; and
generate a second array of pixels based on the selected subpixels and the query.

10. The system of claim 9, wherein the processor further facilitates the execution of the instructions to communicate at least a portion of the second array of pixels to the device based on the query.

11. The system of claim 9, wherein the processor further facilitates the execution of the instructions to:
alternately sample a red subpixel, a green subpixel, and a blue subpixel of adjacent pixels of the block in the diagonal direction; and
generate the second array based on the red subpixel, the green subpixel, the blue subpixel, and the information associated with the display.

12. The system of claim 9, wherein the processor further facilitates the execution of the instructions to receive a dimension associated with the display, and generate the second array based on the dimension associated with the display.

13. The system of claim 9, wherein the processor further facilitates the execution of the instructions to:
receive via a network device of a cloud based computing environment, the query; and
communicate, via the network device of the cloud based computing environment, the second array to the device based on the query.

14. The system of claim 9, wherein the processor further facilitates the execution of the instructions to receive an internet based query via a web browser.

15. The system of claim 9, wherein the processor further facilitates the execution of the instructions to create a virtual image utilizing a directional weighted average of neighboring subpixels of the second array.

16. The system of claim 9, wherein the processor further facilitates the execution of the instructions to determine a low resolution image that satisfies a defined criterion based on respective color components of a virtual image and a high resolution image associated with the first array, wherein the high resolution image is defined to have a higher resolution than the low resolution.

17. The system of claim 16, wherein the processor further facilitates the execution of the instructions to:
minimize a mean square error between the virtual image and the high resolution image; and
determine the low resolution image based on the low resolution image satisfying a condition with respect to the mean square error.

18. The system of claim 16, wherein the processor further facilitates the execution of the instructions to determine color components of the low resolution image according to a block circulant matrix.

19. The system of claim 18, wherein the block circulant matrix is an MN×9MN matrix including N×9N arrays of blocks that are block-tri-circulant.

20. An apparatus, comprising:
means for communicating a network based query comprising information associated with a display of an image, wherein the image corresponds to diagonally down-sampled subpixels of pixels of an n×n block of pixels, wherein n is an integer and is at least 3, and wherein the pixels comprise 3 or more subpixels;
means for receiving the image based on the network based query; and
means for displaying the image.

21. The apparatus of claim 20, wherein the network based query is an internet based query, and wherein the information comprises a dimension associated with the means for displaying the image.

22. A non-transitory computer-readable medium including computer-executable instructions that, in response to execution, cause a computing system to perform operations, comprising:
dividing a first array of pixels into at least 3 pixel-by-at least 3 pixel blocks comprise at least 3 subpixels;
receiving a query from a device comprising information associated with a display of the device;
diagonally down-sampling subpixels of the pixels of the block; and
generating a second array of pixels based on the information associated with the display and a result of the diagonally down-sampling the subpixels of the pixels.

23. The non-transitory computer-readable medium of claim 22, wherein the operations further comprise:
communicating at least a portion of the second array of pixels to the device based on the query.

24. A non-transitory computer-readable medium including computer-executable instructions that, in response to execution, cause a computing system to perform operations comprising:
evaluating information associated with a display of a device;
receiving a request associated with a network based interface;
updating, via a display driver, the request based on the information associated with the display to obtain a query including at least a portion of the information; and
in response to receiving an image via the network based interface based on the query, displaying the image via the display, wherein the image is based on diagonally down-sampled subpixels of pixels of a block of 3 or more pixel-by-3 or more pixel blocks, wherein the pixels comprise 3 or more subpixels.

25. The non-transitory computer-readable medium of claim 24, wherein the updating the request comprises updating the request based on a dimension associated with the display.

26. A method, comprising:
dividing, by a system comprising a processor, an array of pixels into 3 or more pixel-by-3 or more pixel blocks;
receiving a query from a device including information associated with a display of the device;
diagonally down-sampling subpixels of pixels of a block of the 3 or more pixel-by-3 or more pixel blocks utilizing an interpolation filter, wherein the pixels comprise 3 or more subpixels; and
generating another array of pixels based on the information associated with the display and a result of the diagonally down-sampling subpixels of the pixels of the block.

27. The method of claim 26, wherein the utilizing the interpolation filter comprises performing a convolution operation based on the interpolation filter and a Kronecker product associated with a 3×3 matrix.

28. The method of claim 1, wherein the alternately down sampling the selected subpixels further comprises alternately sampling subpixels of the pixels according to a down-sampling factor that references a number of subpixels being sampled from the block.

29. The method of claim 28, wherein the down-sampling factor is based on a characteristic associated with a client device that is associated with the image.