Abstract:
In one embodiment, an embedded device is provided which comprises a region of interest defining mechanism to define a region of interest (ROI) within an image. A transformation mechanism of the embedded device applies a nonlinear magnification or pinching transformation to the region of interest such that magnification or pinching within the region of interest varies from a greater amount at a focal point of the region of interest to a lesser amount at an outer border of the region of interest.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     Priority is claimed to U.S. Provisional Application No. 60/614,214, filed Sep. 28, 2004. 
    
    
     COPYRIGHT NOTICE 
     This patent document contains information subject to copyright protection. The copyright owner has no objection to the facsimile reproduction by anyone of the patent document or the patent, as it appears in the US Patent and Trademark Office files or records, but otherwise reserves all copyright rights whatsoever. 
     BACKGROUND OF THE DISCLOSURE 
     The disclosure relates to digital image manipulation in general, and more particularly, to digital image magnification and pinching. 
     Digital image manipulation describes many different types of modifications and transformations that may be performed on digital images. Examples of image manipulation operations include rotation, magnification, pinching, warping, edge detection, and filtering. 
     In some applications, operations such as magnification and pinching may help a user to see or appreciate fine details in an image. Rotation may help a user to understand an image from a certain perspective, or may orient an image for a specific use. In other applications, digital image manipulation may be performed for the sake of amusement, for example, pinching or magnifying a portion of an image to change a facial expression in a photograph. Digital image manipulation techniques are also used in industry, in applications including pattern recognition, feature extraction (e.g. in video surveillance and human motion analysis), image restoration, image enhancement, warping/morphing for computer animated sequences, and biomedical image processing. 
     A number of digital image manipulation techniques are commercially available in the form of photograph editing software. Embedded devices, such as digital cameras and mobile telephones, also have digital image manipulation functionality. 
     BRIEF SUMMARY OF THE DISCLOSURE 
     According to one embodiment or aspect of the disclosure, an embedded device is provided which comprises a region of interest defining mechanism to define a region of interest (ROI) within an image. A transformation mechanism of the embedded device applies a nonlinear magnification or pinching transformation to the region of interest such that magnification or pinching within the region of interest varies from a greater amount at a focal point of the region of interest to a lesser amount at an outer border of the region of interest. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Embodiments will be described with reference to the following drawing figures, in which like numerals represent like items throughout the figures, and in which: 
         FIG. 1  is a block diagram of an exemplary embedded device capable of performing transformations on an image; 
         FIG. 2  is a schematic illustration of an image with an identified region of interest for transformation; 
         FIG. 3  is an original size 520×390 pixel image before transformation using the illustrated method; 
         FIGS. 4-16  illustrate the image of  FIG. 3  as transformed according to the illustrated embodiments using various parameters for the transformations; 
         FIGS. 17-22  illustrate the image of  FIG. 3  as transformed by prior art image transformation methods; 
         FIG. 23  is a block diagram of an exemplary embedded device with an integer microprocessor capable of performing transformations on images; 
         FIG. 24  is a block diagram of an exemplary embedded device with a floating-point microprocessor capable of performing transformations on images; 
         FIG. 25  is a schematic flow diagram illustrating the tasks involved in an implementation of the transformation methods; 
         FIG. 26  is an illustration of a mobile telephone with a digital camera, illustrating the use of the transformation methods on a portable device; 
         FIG. 27  is a facial image of original size 520×390 pixels before using transformation methods according to the illustrated embodiments; and 
         FIGS. 28 and 29  illustrate the image of  FIG. 27  as transformed by the transformation methods, using various parameters. 
     
    
    
     DETAILED DESCRIPTION 
       FIG. 1  is a block diagram of an exemplary embedded device  10 , which, in the illustrated embodiment, comprises a wireless mobile communication device. The illustrated embedded device  10  comprises a system bus  14 , a device memory  16  (which is a main memory in the illustrated device  10 ) connected to and accessible by other portions of the embedded device  10  through system bus  14 , and hardware entities  18  connected to the system bus  14 . At least some of the hardware entities  18  perform actions involving access to and use of main memory  16 . The hardware entities  18  may include microprocessors, ASICs, and other hardware. 
     A graphics entity  20  is connected to the system bus  14 . The graphics entity  20  may comprise a core or portion of a larger integrated system (e.g., a system on a chip (SoC)), or it may comprise a graphics chip, such as a graphics accelerator. In the illustrated embodiment, the graphics entity  20  comprises a graphics pipeline (not shown), a graphics clock  23 , a buffer  22 , and a bus interface  19  to interface graphics entity  20  with system bus  14 . 
     Buffer  22  holds data used in per-pixel processing by graphics entity  20 . Buffer  22  provides local storage of pixel-related data, such as pixel information from buffers (not shown) within main memory  16 . 
     In the illustrated embodiment, graphics entity  20  is capable of performing localized image transformations on portions of images. To that end, graphics entity  20  includes a region of interest defining mechanism  24  to display and allow a user to select a region of interest within an image to be transformed and a transformation device  26  to perform the image transformation. As shown, the region of interest defining mechanism  24  is coupled to the user interface  28  of the embedded device  10 . The image transformations that may be performed by embedded device  10  will be described in greater detail below. The image on which the embedded device  10  operates may be stored in the main memory  16  of the embedded device  10 , the buffer  22  of the embedded device, or on another machine-readable medium interoperable with the embedded device. Additionally, although the graphics entity  20  performs the transformation functions in the illustrated embodiment, in other embodiments, those functions may be performed by the other hardware  18 . 
       FIG. 2  is a schematic illustration of an image  50 . The image  50  has a width W and a height H. In the illustrated embodiment, the width W and height H are expressed in units of pixels, although other measurement units may be used. The height H of the image  50  extends along the y-axis  52  in  FIG. 2 , and the width W of the image extends along the x-axis  54 . In  FIG. 2 , the width coordinates of the image  50  extend from  0  to W- 1  and the height coordinates extend from  0  to H- 1 , as shown. 
     Image  50  may originally be created in a number of ways, including digital photography, film photography followed by digitization, digitization from a non-photographic source, and pure digital illustration/rendering. Particular implementations of the image transformation methods presented here on specific types of images and specific platforms or computing systems will be described in greater detail below. 
     Transformation methods illustrated herein provide for localized transformation of an image. As shown in  FIG. 2 , the transformation may be localized using a defined region of interest  56 , such as, for example, a circular region of radius R centered at (x o ,y o ). More specifically, the transformation may be localized by limiting it to the area within the region of interest  56 . The center coordinates (x o ,y o ) of the circular region  56  may be arbitrarily selected, and the entire circle need not be located within the bounds of the image. Although the region of interest  56  is illustrated as a circle, it need not be a circle, and may vary in shape and dimensions. Regions of interest of other shapes will be described in more detail below. 
     Most image transformations can be described as sets of mathematical transformation functions represented by sets of mathematical equations; these equations are descriptive of the operations being performed on the image regardless of the particular platform on which the transformations are implemented. The mathematical equations describing one exemplary set of transformation functions for the illustrated embodiment are given below as Equations (1) and (2). For each pixel in image  50 : 
                     x   out     =     {             x   o     +       (       x   in     -     x   o       )     ·     a       [     1   -           (       x   in     -     x   o       )     2     +       (       y   in     -     y   o       )     2         R   2         ]     k               for               (       x   in     -     x   o       )     2     +       (       y   in     -     y   o       )     2       ≤     R   2                 x   in         otherwise                               (   1   )                 y   out     =     {             y   o     +       (       y   in     -     y   o       )     ·     a       [     1   -           (       x   in     -     x   o       )     2     +       (       y   in     -     y   o       )     2         R   2         ]     k               for               (       x   in     -     x   o       )     2     +       (       y   in     -     y   o       )     2       ≤     R   2                 y   in         otherwise                               (   2   )               
In Equations (1) and (2), (x in , y in ) is the input pixel location, (x out , y out ) is the output pixel location, and the parameters a and k control the type of distortion (i.e., magnification or pinching) and the level of magnification or pinching. The parameter a can take a value between zero and infinity; the parameter k can take a value between negative infinity and infinity. (The effect of varying the parameters a and k will be described in greater detail below with respect to certain examples.) As Equations (1) and (2) state, pixels within the region of interest  56 , which is circular in this embodiment, are transformed, while for all other pixels, the output is the same as the input.
 
     The parameter a, as given in Equations (1) and (2), has effects on both the magnitude and type of distortion. While Equations (1) and (2) may be directly applied in some circumstances, it is useful to separate the magnitude effects of the parameter a from its effects on the type of distortion. This can be done by restricting the permissible values of parameter a to values between one and infinity and introducing a separate binary parameter m that determines whether the distortion is magnification (m=0) or pinching (m=1). Equations (3) and (4) illustrate the use of the binary parameter m: 
                     x   out     =     {             x   o     +       (       x   in     -     x   o       )     ·       a         (     -   1     )     m     ·     [     1   -           (       x   in     -     x   o       )     2     +       (       y   in     -     y   o       )     2         R   2         ]         k             for               (       x   in     -     x   o       )     2     +       (       y   in     -     y   o       )     2       ≤     R   2                 x   in         otherwise                               (   3   )                 y   out     =     {             y   o     +       (       y   in     -     y   o       )     ·     a         (     -   1     )     m     ·       [     1   -           (       x   in     -     x   o       )     2     +       (       y   in     -     y   o       )     2         R   2         ]     k                 for               (       x   in     -     x   o       )     2     +       (       y   in     -     y   o       )     2       ≤     R   2                 y   in         otherwise                               (   4   )               
Equations (3) and (4) are identical in effect to Equations (1) and (2), taking into account the mathematical identity:
 
     
       
         
           
             
               
                 
                   
                     a 
                     
                       - 
                       
                         
                           [ 
                           
                             1 
                             - 
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         x 
                                         in 
                                       
                                       - 
                                       
                                         x 
                                         o 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 + 
                                 
                                   
                                     ( 
                                     
                                       
                                         y 
                                         in 
                                       
                                       - 
                                       
                                         y 
                                         o 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                               
                               
                                 R 
                                 2 
                               
                             
                           
                           ] 
                         
                         k 
                       
                     
                   
                   = 
                   
                     
                       ( 
                       
                         1 
                         a 
                       
                       ) 
                     
                     
                       
                         [ 
                         
                           1 
                           - 
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       x 
                                       in 
                                     
                                     - 
                                     
                                       x 
                                       o 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               + 
                               
                                 
                                   ( 
                                   
                                     
                                       y 
                                       in 
                                     
                                     - 
                                     
                                       y 
                                       o 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                             
                             
                               R 
                               2 
                             
                           
                         
                         ] 
                       
                       k 
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     If a varies in the range 1≦a≦∞, 
               1   a     =     a     -   1             
varies in the range
 
             0   &lt;     1   a     &lt;   1.         
Thus, if a is restricted to the range 1≦a≦∞ and a negative exponent is used by setting m=1 in Equations (3) and (4), it is equivalent to varying a in the range 0&lt;a&lt;1 in the original transformation functions. Alternatively, setting m=0 to get a positive exponent in Equations (3) and (4) is equivalent to varying a in the range 1≦a≦∞ in the original transformation functions. By adjusting the value of m, the new transformation functions cover the same range of a as the original transformation functions.
 
     Equations (1)-(4) perform the transformation, whatever its parameters, in both the horizontal and vertical directions. In an alternate embodiment, the transformation may be applied in only one direction. In that case, an exemplary set of transformation functions for one dimensional transformation along the horizontal are: 
                     x   out     =     {             x   o     +       (       x   in     -     x   o       )     ·     a         (     -   1     )     m     ·       [     1   -         (       x   in     -     x   o       )     2       d   2         ]     k                   for                    x   in     -     x   o            ≤   d               x   in         otherwise                               (   6   )                 y   out     =     y   in             (   7   )               
and an exemplary set of transformation functions for the one dimensional transformation along the vertical are:
 x out =x in    (8) 
                     y   out     =     {             y   o     +       (       y   in     -     y   o       )     ·     a         (     -   1     )     m     ·       [     1   -         (       y   in     -     y   o       )     2       d   2         ]     k                 for                  y   in     -     y   o            ≤   d               y   in         otherwise                               (   9   )               
in which d is half the width or height of the region of interest. The effect of transformation Equations (3) and (4) and the values of parameters a, k, and m are better understood in view of the following two examples.
 
     EXAMPLE 1 
     a=2, k=1, m=0 
     When the three parameters in Equations (3) and (4) are set as indicated above, Equations (3) and (4) reduce to: 
     
       
         
           
             
               
                 
                   
                     x 
                     out 
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               x 
                               o 
                             
                             + 
                             
                               
                                 ( 
                                 
                                   
                                     x 
                                     in 
                                   
                                   - 
                                   
                                     x 
                                     o 
                                   
                                 
                                 ) 
                               
                               · 
                               
                                 2 
                                 
                                   [ 
                                   
                                     1 
                                     - 
                                     
                                       
                                         
                                           
                                             ( 
                                             
                                               
                                                 x 
                                                 in 
                                               
                                               - 
                                               
                                                 x 
                                                 o 
                                               
                                             
                                             ) 
                                           
                                           2 
                                         
                                         + 
                                         
                                           
                                             ( 
                                             
                                               
                                                 y 
                                                 in 
                                               
                                               - 
                                               
                                                 y 
                                                 o 
                                               
                                             
                                             ) 
                                           
                                           2 
                                         
                                       
                                       
                                         R 
                                         2 
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                         
                         
                           for 
                         
                         
                           
                             
                               
                                 
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                                 2 
                               
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                                       y 
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                                   ) 
                                 
                                 2 
                               
                             
                             ≤ 
                             
                               R 
                               2 
                             
                           
                         
                       
                       
                         
                           
                             x 
                             in 
                           
                         
                         
                           otherwise 
                         
                         
                           
                               
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
             
               
                 
                   
                     y 
                     out 
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 y 
                                 o 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     y 
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                             · 
                             
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                                             ( 
                                             
                                               
                                                 x 
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                                           2 
                                         
                                       
                                       
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                           for 
                         
                         
                           
                             
                               
                                 
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                                 2 
                               
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                                       y 
                                       in 
                                     
                                     - 
                                     
                                       y 
                                       o 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                             
                             ≤ 
                             
                               R 
                               2 
                             
                           
                         
                       
                       
                         
                           
                             y 
                             in 
                           
                         
                         
                           otherwise 
                         
                         
                           
                               
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     Equations (10) and (11) produce a magnified image with a maximum magnification power of two. At the center of the region of interest  56 , where (x in , y in )=(x o ,y o ), the exponential term is equal to two; therefore, the center is magnified by a factor of two. However, at the edge of the region of interest  56 , where (x in −x o ) 2 +(y in −y o ) 2 =R 2 , the exponential term equals one; therefore, pixels along the edge are unmagnified. The overall effect of Equations (10) and (11) is to provide a magnification power of two at the center of the region of interest  56  which gradually decreases as the distance from the center of the region of interest  56  increases. 
       FIG. 3  is an image in RGB format with an original image size of 520×390 pixels.  FIG. 4  is the transformed image of  FIG. 3 , illustrating the application of Equations (10) and (11) using the parameters of Example 1 with a magnification radius of 100 pixels. 
     EXAMPLE 2 
     a=2, k=1, m=1 
     When the three parameters in Equations (3) and (4) are set as indicated above, Equations (3) and (4) reduce to: 
     
       
         
           
             
               
                 
                   
                     x 
                     out 
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               x 
                               o 
                             
                             + 
                             
                               
                                 ( 
                                 
                                   
                                     x 
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                                     x 
                                     o 
                                   
                                 
                                 ) 
                               
                               · 
                               
                                 2 
                                 
                                   - 
                                   
                                     [ 
                                     
                                       1 
                                       - 
                                       
                                         
                                           
                                             
                                               ( 
                                               
                                                 
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                                                   o 
                                                 
                                               
                                               ) 
                                             
                                             2 
                                           
                                           + 
                                           
                                             
                                               ( 
                                               
                                                 
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                                                   in 
                                                 
                                                 - 
                                                 
                                                   y 
                                                   o 
                                                 
                                               
                                               ) 
                                             
                                             2 
                                           
                                         
                                         
                                           R 
                                           2 
                                         
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                           
                         
                         
                           for 
                         
                         
                           
                             
                               
                                 
                                   ( 
                                   
                                     
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                                 2 
                               
                             
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                               2 
                             
                           
                         
                       
                       
                         
                           
                             x 
                             in 
                           
                         
                         
                           otherwise 
                         
                         
                           
                               
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
             
               
                 
                   
                     y 
                     out 
                   
                   = 
                   
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                               y 
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                               · 
                               
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                                             2 
                                           
                                         
                                         
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                                           2 
                                         
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                           
                         
                         
                           for 
                         
                         
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       x 
                                       in 
                                     
                                     - 
                                     
                                       x 
                                       o 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               + 
                               
                                 
                                   ( 
                                   
                                     
                                       y 
                                       in 
                                     
                                     - 
                                     
                                       y 
                                       o 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                             
                             ≤ 
                             
                               R 
                               2 
                             
                           
                         
                       
                       
                         
                           
                             x 
                             in 
                           
                         
                         
                           otherwise 
                         
                         
                           
                               
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     Equations (12) and (13) produce a locally pinched image with a maximum pinching factor of two. At the center of the region of interest  56 , the exponential term is equal to one half; therefore, the center is pinched by a factor of two. At the edge of the region of interest  56 , the exponential term is equal to one; therefore, pixels at the edge of the region of interest  56  are unpinched. The overall effect of Equations (12) and (13) is to provide a pinching power of two at the center of the region of interest  56  which gradually decreases as the distance from the center of the region of interest  56  increases.  FIG. 12  illustrates an image transformed with these paramaters. 
     Table 1 below presents the results of several additional examples, illustrating the use and selection of the parameters a, k, and m for Equations (3) and (4). All of the examples presented below used nearest-neighbor pixel duplication, although other methods, such as interpolation, could be used to fill in pixels in the magnified images. The image size and the radius and location of the region of interest in the examples presented below are the same as those in Examples 1 and 2. In Table 1, certain examples are duplicative of others, but are presented nonetheless for ease of reference. 
     
       
         
               
             
               
               
               
               
               
               
             
               
               
               
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 Examples 
               
             
          
           
               
                   
                 Example No. 
                 a 
                 k 
                 m 
                 Result Figure 
               
               
                   
                   
               
             
          
           
               
                   
                 1 
                 2 
                 1 
                 0 
                 FIG. 4 
               
               
                   
                 2 
                 2 
                 1 
                 1 
                 FIG. 12 
               
               
                   
                 3 
                 1.5 
                 1 
                 0 
                 FIG. 5 
               
               
                   
                 4 
                 3 
                 1 
                 0 
                 FIG. 6 
               
               
                   
                 5 
                 2 
                 1.5 
                 0 
                 FIG. 7 
               
               
                   
                 6 
                 2 
                 2 
                 0 
                 FIG. 8 
               
               
                   
                 7 
                 2 
                 3 
                 0 
                 FIG. 9 
               
               
                   
                 8 
                 1 
                 1 
                 1 
                 FIG. 10 
               
               
                   
                   
                   
                   
                   
                 (same as the untrans- 
               
               
                   
                   
                   
                   
                   
                 formed image of FIG. 3) 
               
               
                   
                 9 
                 1.5 
                 1 
                 1 
                 FIG. 11 
               
               
                   
                 10  
                 2 
                 1 
                 1 
                 FIG. 12 
               
               
                   
                 (same as 
               
               
                   
                 Example 2) 
               
               
                   
                 11  
                 3 
                 1 
                 1 
                 FIG. 13 
               
               
                   
                 12  
                 2 
                 1.5 
                 1 
                 FIG. 14 
               
               
                   
                 13  
                 2 
                 2 
                 1 
                 FIG. 15 
               
               
                   
                 14  
                 2 
                 3 
                 1 
                 FIG. 16 
               
               
                   
                   
               
             
          
         
       
     
     In general, the examples presented above show that as the value of the paramater k increases with the values of a and m held constant, the transition between the point of greatest magnification or pinching and the points of least magnification or pinching becomes smoother and more gradual. Thus, the parameter k can be interpreted as determining the size and the degree of distortion of the transition region between the most and least distorted areas of the image. 
     The examples presented above also show that as the value of parameter a inscreases with the values of k and m held constant, the maximum power of magnification or pinching increases. 
     Table 1 shows the effect of varying the parameters a, k, and m on the transformed image. However, there are two cases in which the output image is the same as the input image. The first case is when a=1, k=1, and m=0. The second case is when a=1, k=1, and m=1. 
     In addition to the examples presented above, certain comparative examples were prepared using the image editing program ADOBE PHOTOSHOP® and its SPHERIZE and PINCH operations. Six cases could be approximated using the conventional software. These are presented in Table 2. 
     
       
         
               
             
               
               
               
               
             
           
               
                 TABLE 2 
               
             
             
               
                   
               
               
                 Comparative Examples 
               
             
          
           
               
                 Comparative 
                 PHOTOSHOP ® 
                   
                   
               
               
                 Example No. 
                 Operations 
                 Figure 
                 Compare to Example 
               
               
                   
               
               
                 C1 
                 SPHERIZE 100%, 
                 FIG. 17 
                 Example 1 (a = 2, 
               
               
                   
                 then 75% 
                   
                 k = 1, m = 0) 
               
               
                 C2 
                 SPHERIZE 100%, 
                 FIG. 18 
                 Example 5 (a = 2, 
               
               
                   
                 then 38% 
                   
                 k = 1.5, m = 0) 
               
               
                 C3 
                 SPHERIZE 90% 
                 FIG. 19 
                 Example 3 (a = 1.5, 
               
               
                   
                   
                   
                 k = 1, m = 0) 
               
               
                 C4 
                 PINCH 90% 
                 FIG. 20 
                 Example 10 (a = 2, 
               
               
                   
                   
                   
                 k = 1, m = 1) 
               
               
                 C5 
                 PINCH 70% 
                 FIG. 21 
                 Example 12 (a = 2, 
               
               
                   
                   
                   
                 k = 1.5, m = 1) 
               
               
                 C6 
                 PINCH 50% 
                 FIG. 22 
                 Example 9 (a = 1.5, 
               
               
                   
                   
                   
                 k = 1, m = 1) 
               
               
                   
               
             
          
         
       
     
     Two out of the six comparative examples, Examples C1 and C2, required two PHOTOSHOP® operations to produce a comparable effect. (Although comparable, the effect created by the PHOTOSHOP® software was not identical, as can be seen from the figures.) Thus, one advantage of these transformation methods is that fewer transformation operations may be required to produce a desired effect. These transformation methods also appear to provide slightly more magnification and pinching at the center of the transformation region. 
     In the examples above, all of which used Equations (3) and (4), the area on which the transformation is performed is circular. However, the area of the transformation need not be circular, and may be chosen depending on the application, provided that appropriate equations are used for the transformation. For example, Equations (14) and (15) below provide for a transformation in an elliptical area. In Equations (14) and (15), two additional parameters, b and c, describe the major and minor axes of the ellipse, i.e., its width and height. (However, the parameters b and c do not themselves equal the major and minor axes of the ellipse. The major axis is equal to 2bR and the minor axis is equal to 2cR.) 
                     x   out     =     {             x   o     +       (       x   in     -     x   o       )     ·     a         (     -   1     )     m     -       [     1   -           b   ⁡     (       x   in     -     x   o       )       2     +       c   ⁡     (       y   in     -     y   o       )       2         R   2         ]     k                 for               b   ⁡     (       x   in     -     x   o       )       2     +       c   ⁡     (       y   in     -     y   o       )       2       ≤     R   2                 x   in         otherwise                               (   14   )                 y   out     =     {             y   o     +       (       y   in     -     y   o       )     ·     a         (     -   1     )     m     -       [     1   -           b   ⁡     (       x   in     -     x   o       )       2     +       c   ⁡     (       y   in     -     y   o       )       2         R   2         ]     k                 for               b   ⁡     (       x   in     -     x   o       )       2     +       c   ⁡     (       y   in     -     y   o       )       2       ≤     R   2                 y   in         otherwise                               (   15   )               
In embodiments in which the area of transformation or region of interest is not a geometric shape with an easily located center, an arbitrary focal point may be chosen. Even where the region of interest  56  has an easily located geometric center, a different (not co-located) focal point may be chosen.
 
     The illustrated transformation methods may be implemented to run on a computing system of limited capabilities, such as an integer microprocessor. Integer microprocessors are commonly used on mobile devices, such as mobile telephones, mobile telephones with digital cameras, and other portable computing devices. While integer microprocessors typically include a floating-point (i.e., decimal) mathematics emulator, it can be more time consuming and computationally expensive to use the emulator. The transformations may be implemented using integer arithmetic. 
     When implementing transformation equations such as Equations (3) and (4) on an integer microprocessor, two considerations arise: the calculation of the power functions in those equations using only integer arithmetic, and the ordering of operations so as to avoid integer overflow (i.e., the condition in which a calculated number exceeds the largest integer that the microprocessor can handle). 
       FIG. 23  is a block diagram of an exemplary embedded device  60  that is adapted to perform the transformations described above using integer arithmetic. The embedded device  60  includes a main memory  16  connected to a system bus  14 , a graphics entity  66  connected by an interface  19  to the system bus  14 , and a integer microprocessor  61  connected to the system bus  14 . Embedded device  60  also includes a transformation operations facilitator  62  connected to the microprocessor. An integer operations facilitator  64  is included within the transformation operations facilitator  62 . 
     The transformation operations facilitator  62  calculates the power functions of Equations (3) and (4) and performs the other transformation operations in a manner compatible with the microprocessor  61 . The integer operations facilitator  64  ensures that all of the necessary calculations are performed using integer arithmetic with an order of calculation that avoids integer overflow in the integer microprocessor  61 . (The functions of both components  62 ,  64  and the calculations that are performed will be described below in more detail.) An advantage of an embedded device such as device  60  is that no floating-point emulator is used, which makes the transformations more efficient on the integer microprocessor  61 . The transformation operations facilitator  62  and the integer operations facilitator  64  may be implemented in hardware, in software, in some combination of hardware and software, or in any other way compatible with the microprocessor  61 . 
     Although illustrated in  FIG. 23 , the graphics entity  66  need not be included in embedded device  60 . 
     In order to calculate the power functions in Equations (3) and (4), in the illustrated embodiment, a Taylor series expansion of the function is used. For an arbitrary power function, the Taylor series expansion is given by Equation (16): 
                     a   n     =     1   +       (     ln   ⁢           ⁢   a     )     ⁢   n     +           (     ln   ⁢           ⁢   a     )     2       2   !       ⁢     n   2       +           (     ln   ⁢           ⁢   a     )     3       3   !       ⁢     n   3       +   …   ⁢           +           (     ln   ⁢           ⁢   a     )     k       k   !       ⁢     n   k       +   …             (   16   )               
As in any use of a Taylor series, the approximation becomes more accurate as more terms are added. However, the more terms of a Taylor series that are used, the more computationally expensive the process becomes. Additionally, successive terms of a Taylor series add ever more diminishing amounts of accuracy to the final result. Therefore, the number of Taylor series terms that are used to calculate the power function will depend on the accuracy desired as well as the computing power available. In one implementation, which will be described below in greater detail, the first four terms of the Taylor series were found to provide sufficient accuracy without requiring undue computing power. Using the first four terms of the series with a=2, Equation (16) above reduces to Equation (17):
 
     
       
         
           
             
               
                 
                   
                     2 
                     n 
                   
                   ≅ 
                   
                     1 
                     + 
                     
                       
                         ( 
                         
                           ln 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                         
                         ) 
                       
                       ⁢ 
                       n 
                     
                     + 
                     
                       
                         
                           
                             ( 
                             
                               ln 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                             
                             ) 
                           
                           2 
                         
                         
                           2 
                           ! 
                         
                       
                       ⁢ 
                       
                         n 
                         2 
                       
                     
                     + 
                     
                       
                         
                           
                             ( 
                             
                               ln 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                             
                             ) 
                           
                           3 
                         
                         
                           3 
                           ! 
                         
                       
                       ⁢ 
                       
                         n 
                         3 
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
     Although Equation (17) does not contain strictly integer terms, the non-integer terms can be converted to integers for the purpose of performing the calculations. For example, the natural logarithm of 2 can be multiplied by 2 23  (i.e., shifted 23 bits to the left) to result in the integer 5767168. The results of the calculations can subsequently be shifted back (i.e., divided by 2 23 ) to remove the effect of the multiplier. In general, large factors of 2 are used to preserve accuracy by preserving a number of significant digits; smaller factors may be used if less accuracy is desired. Additionally, although any large integer factor can be used when converting floating-point numbers to integers, factors of 2 are used in the illustrated embodiment so that relatively slow multiplication operations can be replaced by relatively fast bit-shifting operations. 
     A sample of the implementation code for a 32-bit microprocessor using the four-term Taylor series expansion of Equation (17) and a 2 23  integer conversion multiplier for the magnification operation is as follows for the case in which a=2, k=1, and m =0: 
     
       
         
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
           
               
                   
                   
               
             
             
               
                   
                 int32 r, xo, yo, xin, xout, yin, yout, rSq, k1, k2, xy, factor; 
               
             
          
           
               
                   
                 rSq = r * r; 
               
               
                   
                 k1 = 5767168 / r; 
               
               
                   
                 k2 = 2048 / r; 
               
               
                   
                 xy = (xin − xo) * (xin − xo) + (yin − yo) * (yin − yo); 
               
               
                   
                 factor = 8388608 + (5767168 − (xy * k1) / r) + 
               
             
          
           
               
                   
                 (2048 − (xy * k2) / r) * (1024 − ((xy * k2) &gt;&gt; 1) / r) 
               
             
          
           
               
                   
                 + 
               
             
          
           
               
                   
                 (128 − (xy * 128) / rSq) * (64 − (xy * 64) / rSq) * 
               
               
                   
                 (64 − (xy * 64) / rSq); 
               
             
          
           
               
                   
                 xout = xo + ((factor * (xin − xo)) &gt;&gt; 23); 
               
               
                   
                 yout = yo + ((factor * (yin − yo)) &gt;&gt; 23); 
               
               
                   
                   
               
             
          
         
       
     
     In the above code snippet, 8388608 is 1×2 23 , and the operations are ordered so as to avoid integer overflow on the 32-bit microprocessor. The value of the Taylor series is calculated as a multiplicative factor, is multiplied by the difference between the location of the input pixel and the center of the transformation region, and is added to the location of the center of the transformation region. A shifting operation at the end removes the effect of the 2 23  multiplier. These operations are performed on each input pixel in the region of interest. 
     In general, the difference between the magnification and pinching transformations lies in the sign (i.e., addition versus subtraction) of certain operations. The code for the pinching operation for the case in which a=2, k=1, and m=1 is as follows: 
     
       
         
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
           
               
                   
                   
               
             
             
               
                   
                 int32 r, xo, yo, xin, xout, yin, yout, rSq, k1, k2, xy, factor; 
               
             
          
           
               
                   
                 rSq = r * r; 
               
               
                   
                 k1 = 5767168 / r; 
               
               
                   
                 k2 = 2048 / r; 
               
               
                   
                 xy = (xin − xo) * (xin − xo) + (yin − yo) * (yin − yo); 
               
               
                   
                 factor = 8388608 − (5767168 − (xy * k1) / r) + 
               
             
          
           
               
                   
                 (2048 − (xy * k2) / r) * (1024 − ((xy * k2 ) &gt;&gt; 1) / r) 
               
             
          
           
               
                   
                 − 
               
             
          
           
               
                   
                 (128 − (xy * 128) / rSq) * (64 − (xy * 64) / rSq) * 
               
               
                   
                 (64 − (xy * 64) / rSq); 
               
             
          
           
               
                   
                 xout = xo + ((factor * (xin − xo)) &gt;&gt; 23); 
               
               
                   
                 yout = yo + ((factor * (yin − yo)) &gt;&gt; 23); 
               
               
                   
                   
               
             
          
         
       
     
     The above code snippets were found to provide real-time results on an ARM926EJ-S 32-bit integer microprocessor. Although this described embodiment is coded in C and implemented on a 32-bit microprocessor, other embodiments may be coded in any programming language, including C, C++, Java, J++, and assembler, may be implemented on microprocessors of any capabilities, including 64-bit microprocessors and 128-bit microprocessors, and may use any values of the parameters a, k, and m. The implementations need not use integer-only arithmetic and need not be ordered so as to avoid integer overflow. If these methods are implemented on an integer microprocessor, they may be provided as image processing functions on a mobile telephone with a digital camera or other portable electronic devices. It should also be understood that these methods may be implemented in software, hardware or any combination of software and hardware on a microprocessor, an ASIC, or any other platform with sufficient computing capability to implement them. 
       FIG. 24  a block diagram of an exemplary embedded device  70  that is adapted to perform the transformations described above using floating-point arithmetic. The components of embedded device  70  are generally similar to those of embedded device  60 , and thus, the description above will suffice with respect to the similar components. Unlike embedded device  60 , embedded device  70  includes a floating-point microprocessor  72 . Embedded device  70  also includes a transformation operations facilitator  74  coupled to the floating-point microprocessor  72 , but the transformation operations facilitator  74  has no integer operations facilitator. Calculations are performed in embedded device  70  using floating-point numbers, omitting, for example, the tasks of converting the terms of Equations (3) and (4) to integers. Although an integer-only implementation of the illustrated transformation methods would function correctly if performed on embedded device  70 , it is advantageous to make use of the floating-point capabilities of microprocessor  72 . 
       FIG. 25  is a more general flow diagram illustrating a method  100  for applying localized magnification or pinching to an image. Method  100  may be implemented on any platform capable of performing the necessary calculations. 
     Method  100  begins with input image processing at S 102  and control passes to S 104 . In S 104 , the region of interest in the input image is selected. The region of interest is typically defined by a geometric shape (such as the circles and ellipses described above), although an arbitrary geometric region may be used if the transform calculations are modified appropriately. In S 104 , the user would select the center and radius or other dimensions of the region of interest. Once the region of interest is selected, method  100  continues with S 106 , in which a pixel of the input image is selected. Following S 106 , method  100  continues with S 108 , a decision task in which it is determined whether or not the selected pixel is in the region of interest. If the selected pixel is in the region of interest (S 108 :YES), that pixel is transformed at S 114  by performing one or more of the operations described above and a resulting output pixel of an output image is generated. If the selected pixel is not in the region of interest (S 108 :NO), control of method  100  is transferred to S 110 , in which it is determined whether there are other pixels remaining in the input image. If there are other pixels remaining in the image (S 110 :YES), control of method  100  returns to S 106 . If there are no other pixels remaining in the image (S 110 :NO), control passes to S 112 . In S 112 , any interpolation or replication of missing pixels in the output image necessary to create a complete transformed output image may be performed. (In the simplest cases, any necessary pixel replication may be performed by nearest neighbor duplication.) Any other tasks required to create a whole, viewable image may also be performed at S 112 , including the writing of header information for the output image file. Once S 112  is complete, method  100  terminates and returns at S 116 . 
     In some of the foregoing description, it has been assumed that the image to be transformed is in the RGB (red-green-blue) format, in which each image pixel has a value for the red content of that pixel, a value for the green content, and a value for the blue content. However, the illustrated transformation methods can be used directly on other image formats without first converting to RGB. This is advantageous because although RGB-format images are relatively easy to transform, they are more difficult to compress, and generally consume more storage space. 
     Two other common image formats are YCbCr and YCrCb. Whereas in an RGB image, data is stored in terms of the red, green, and blue color values for each pixel, the YCbCr and YCrCb formats store image data by recording the luminance (Y) and chrominance (Cb, Cr) values for each pixel. The YCbCr and YCrCb formats are popular because they are used in the common JPEG picture file format. 
     The ability to operate on RGB, YCbCr, and YCrCb images is advantageous if image transforms are implemented on a portable device such as a digital camera, because all three formats may be used in a digital camera. This is because of the way digital images are created and processed. 
     For example, most digital camera image sensors are composed of individual sensor cells that are sensitive to only one of red, green, or blue light, not to light of all three colors. Therefore, individual cells are typically deployed in a pattern, called a Bayer pattern, in which cells sensitive to green are dispersed among and alternated with cells sensitive to red and blue. In consumer products, green cells usually predominate because the human visual system is more sensitive to green, and the inclusion of more green cells tends to increase the perceived image quality. In one typical Bayer pattern, an array of 16 sensor cells may include 8 green cells, 4 red cells, and 4 blue cells arranged roughly in a checkerboard pattern. When an image is taken by a digital device that uses single-color cells in a Bayer pattern, the raw image is typically interpolated such that each pixel has a red value, a green value, and a blue value and stored, at least in an intermediate stage of processing, as an RGB image. The image may be further converted to YCbCr or YCrCb for compression and storage. 
     Although images in YCbCr and YCrCb formats may be directly processed by applying the transformations described above, there are some circumstances in which additional tasks may be performed, for example, with subsampled YCbCr and YCrCb images. In a subsampled image, some chrominance values are discarded or subsampled in order to reduce the size of the file. For example, in the common H2V1 YCbCr 4:2:2 format, pixel columns are subsampled, but pixel rows are unaffected. In this subsampling scheme, if the columns are numbered starting from zero, only even columns have the Cb component and only odd columns have the Cr component. Another subsampled format is the YCbCr 4:2:0 format, in which each 2×2 pixel array shares a single Cb value and a single Cr value. YCrCb format is generally the same as YCbCr, except that the order of Cb and Cr components is reversed. 
     The transformation methods described above may be directly applied to subsampled YCbCr and YCrCb formats, although doing so may not result in an end image with correctly alternating Cb and Cr components. To overcome this issue, a temporary unsubsampled image (YCbCr 4:4:4 or YCrCb 4:4:4) may be created from the subsampled image by considering pairs of adjacent pixels and duplicating the appropriate Cb and Cr values so that each pixel has a Cb and a Cr value. For storage purposes after transformation, the extra Cb and Cr values may be discarded. Tests performed by the inventor showed no visually perceptible differences between the processed result of an RGB image and the processed result of that same image in YCbCr and YCrCb fornats. 
       FIG. 26  shows an embodiment of a mobile telephone  200  with a digital camera  202 . The mobile telephone  200  and its digital camera  202  include the region of interest defining mechanism  24  and the transform device  26  of  FIG. 1 , or other mechanisms for performing image transformations as described herein. In typical use, a user would take a digital picture using the digital camera  202  of the mobile telephone  200 , and would then use the processing capabilities of the mobile telephone  200  to perform a transformation. As shown in  FIG. 26 , a digital image  204  is displayed on the display screen  206  of the mobile telephone  200 . (Typically, the display screen  206  is a relatively small liquid crystal display driven by graphics entity  20 , although other types of display screens  206  may be used.) As shown, the image  204  has been transformed by local magnification of a region of interest  208 . An overlay or pull-down menu  214  temporarily overlaid on the image  204  may provide instructions for changes in the type and magnitude of transformation. For example, the user may be instructed to use the arrow keys  210  of the mobile telephone  204  to move the region of interest  208 . (If the region of interest  208  is moved, the transformation would be repeated, centered about a new focal point, by performing a method such as method  100  again.) The user may also be instructed that some combination of number/letter keys  212  can be used to change the magnification/pinch level, switch between magnification and pinch, or use both on the same image  204 . (In which case, a method such as method  100  would be repeated with new parameters.) Depending on the implementation, the user may or may not be able to directly modify the values of the parameters a, k, and m; in some embodiments, the user may simply modify settings such as “magnification factor,” the values for which are mapped to particular parameter values. 
     Depending on the implementation, the parameters of the transformation may be hard-coded or pre-set into the device, such that the transformation always results in, for example, magnification about the same predetermined point with the same radius of transformation. This may be useful in image analysis applications with a number of similar images. 
     An advantage of the implementation shown in  FIG. 26  is that the user is presented with detail while preserving the context of the image as a whole. Whereas in a traditional linear transformation magnification scheme, the user would typically see only a portion of the image on screen and would scroll to change the visible portion, thus losing the view of the entire image, localized magnification keeps the entire image  204  visible while a desired region  208  is magnified. This may increase user efficiency by lessening the amount of time a user spends changing the magnification of the image and scrolling to see the entire image. 
     Transformations may also be applied to images to create artistic effects. In addition, the illustrated transformations may be implemented on portable devices such as mobile telephone  200  for these purposes. For example,  FIGS. 27-29  show the effect of these transformation methods on a facial image.  FIG. 27  is an original, unmodified facial image.  FIG. 28  illustrates the image of  FIG. 27  after magnifying a circular region of radius 60 pixels localized around the mouth using parameters a=2, k=3, and m=0.  FIG. 29  illustrates the image of  FIG. 27  after pinching a circular region of radius 70 pixels localized around the nose using parameters a=2, k=1, and m=1. Combinations of transformations performed on the same image may produce additional effects. 
     Each element described hereinabove may be implemented with a hardware processor together with computer memory executing software, or with specialized hardware for carrying out the same functionality. Any data handled in such processing or created as a result of such processing can be stored in any type of memory available to the artisan. By way of example, such data may be stored in a temporary memory, such as in a random access memory (RAM). In addition, or in the alternative, such data may be stored in longer-term storage devices, for example, magnetic disks, rewritable optical disks, and so on. For purposes of the disclosure herein, a computer-readable media may comprise any form of data storage mechanism, including such different memory technologies as well as hardware or circuit representations of such structures and of such data. 
     While certain illustrated embodiments are disclosed, the words which have been used herein are words of description rather than words of limitation. Changes may be made, for example, within the purview of the appended claims.