# Patent

## Title
Alignment of sharp and blurred images based on blur kernel sparseness

## Abstract

The alignment of a sharp image of a subject and a blurred image of the same subject is disclosed. For example, one disclosed embodiment provides a method of determining a series of trial images. The method comprises applying a corresponding series of coordinate transforms to the sharp image, the series of coordinate transforms differing with respect to one or more of a rotational operation and a scaling operation. The method further comprises computing a series blur kernels corresponding to the series of trial images, each blur kernel mapping a trial image from the series of trial images to the blurred image. The method further includes locating a sparsest blur kernel in the series of blur kernels, and identifying one or more of the rotational operation and the scaling operation of the coordinate transform mapping the trial image corresponding to the sparsest blur kernel to the blurred image.

## Background
BACKGROUND
Image-alignment procedures are used in various image- and video processing applications, which include image stabilization, image enhancement, video summarization, panorama and satellite photo stitching, and medical imaging, as examples. Some currently used image-alignment procedures are based on feature recognition: like features in a pair of misaligned images are located, then placed into registry by subjecting one of the images to a coordinate transformation. However, alignment methods based on feature-recognition may be unreliable when one or both of the images is affected by motion blur.
SUMMARY
Accordingly, various embodiments are disclosed herein that relate to the alignment of two images, where one of the images is affected by motion blur. Thus, in one embodiment, a method to align a sharp image of a subject and a blurred image of the same subject is provided. The method comprises computing a series of trial images by applying a corresponding series of coordinate transforms to the sharp image, the series of coordinate transforms differing with respect to one or more of a rotational operation and a scaling operation. The method further comprises computing a series blur kernels corresponding to the series of trial images, each blur kernel mapping a trial image from the series of trial images to the blurred image. The method further comprises locating a sparsest blur kernel in the series of blur kernels, and identifying one or more of the rotational operation and the scaling operation of the coordinate transform mapping the trial image corresponding to the sparsest blur kernel to the blurred image.
It will be understood that the Summary above is provided to introduce in simplified form a selection of concepts that are further described in the Detailed Description. It is not meant to identify key or essential features of the claimed subject matter, the scope of which is defined by the claims that follow the Detailed Description. Further, the claimed subject matter is not limited to implementations that solve any disadvantages noted above or in any part of this disclosure.



BRIEF DESCRIPTIONS OF THE DRAWINGS
FIG. 1 represents two images of an example subject, in accordance with the present disclosure.
FIG. 2 illustrates a method to align a sharp image of a subject and a blurred image of the same subject, in accordance with an embodiment of the present disclosure.
FIG. 3 illustrates a method to align two images of a subject, in accordance with an embodiment of the present disclosure.
FIG. 4 illustrates aspects of a coarse-to-fine image alignment method, in accordance with an embodiment of the present disclosure.
FIG. 5 illustrates aspects of a method to align two images of a subject, in accordance with an embodiment of the present disclosure.
FIG. 6 illustrates aspects of a method to align two images of a subject, in accordance with an embodiment of the present disclosure.
FIG. 7 schematically represents one example embodiment of an environment in which embodiments of the present disclosure may be enacted.



DETAILED DESCRIPTION
In FIG. 1, two images of an example subject 100 are represented: first image 101 and second image 102. Subject 100 may be an any, arbitrarily chosen subject; it may be an intended subject of a photograph or video capture, for example. In some embodiments, subject 100 may be part of a larger photographic or video subject.
Accordingly, first image 101 and second image 102 may be photographic or video images. These images may have been acquired digitally, via a digital camera, for example, or otherwise acquired on photographic film, analog videotape, etc., and digitized via subsequent processing. In any case, first image 101 and second image 102 may each comprise a matrix of pixels, each pixel comprising color and/or brightness information.
It will be observed that the subject of first image 101 is misaligned (i.e., out of registry) with the subject of second image 102. In the illustrated example, misalignment is due to a translation of the subject, a rotation of the subject, and a scaling difference between the two images. It will further be observed that the subject in the first image and the subject in the second image may placed into registry by changing the alignment of the first image with respect to the second image. Such whole-image alignment may comprise translating the first image with respect to the second image, rotating the first image with respect to the second image, and/or modifying a scale of the first image or the second image.
In FIG. 1, the subjects of the two images differ with respect to sharpness as well as alignment. Thus, subject 100 is sharp in first image 101, but blurred in second image 102. It will be understood that the blurring of subject 100 is rendered in a stylized manner; viz., the doubled, dashed borders in second image 102 are intended to represent blurring. Likewise, the clean borders in first image 101 are intended to represent sharpness.
A subject in an image may be blurred as a result of motion blur. Motion blur may occur when the subject and a device that acquires the image (a camera, a video recorder, etc.) are in motion relative to each other while the image is being acquired. Such relative motion may comprise a motion of the subject, a motion of the image-acquiring device, or both.
The effect of motion blur on an image of a subject may be described mathematically. In one approach, a blur kernel k maps a sharp image I to a blurred image B,

B=I{circumflex over (x)}k,  (1)

where {circumflex over (x)} is the convolution operator. It has been demonstrated that the blur kernel mapping the sharp image to the blurred image can be estimated accurately when both the sharp image and the blurred image are known independently [L. Yuan, J. Sun, L. Quan, and H.-Y. Shum. ACM Trans. Graph. (SIG-GRAPH), 26(3):1-10, 2007]. In particular, the blur kernel can be estimated by assigning the elements {circumflex over (k)}i in kernel {circumflex over (k)} to minimize the quantity

∥B−I{circumflex over (x)}∥2+λ∥∥2,  (2)

subject to the constraints





  
    
      
        
                    ≥
          0
        
        ,
        
          
            
              ∑
              i
            
            ⁢
                      
          =
          1
        
        ,
        
          
            and
            ⁢
            
              
            
            ⁢
            λ
          
          =
          3.
        
      
    
    
      
        (
        3
        )
      
    
  




The kernel  that minimizes eq 2 according to the constraints of eq 3 is then selected as an estimate of blur kernel k.

The elements of some blur kernels occurring in photographic and video applications may follow a predictable distribution function p(ki), as follows:

p(ki)∝w1e−ki/β1+(1−w1)e−k1/β2,  (4)

where ki is the ith element in blur kernel k, and w1, β1, and β2 are parameters. Best-fit parameter values obtained for a series of sharp and blurred image pairs, e.g.,

w1=0.6, β1=0.01, β2=0.03,  (5)

show that blur kernels occurring in some photographic and video applications may be sparse in a mathematical sense.

The relationships outlined above relate sharp and blurred images that are in perfect registry with each other, but sharp and blurred images that are not in registry may be aligned according to a method that relies on blur-kernel sparseness. For example, assuming that a blur kernel linking sharp and blurred images will be maximally sparse only when the images are in proper alignment, it may be possible to align the images by iteratively adjusting one or more alignment parameters so as to maximize the blur-kernel sparseness.
To enact such a method, a metric for blur-kernel sparseness may be employed. Assuming that the probabilistic distribution of blur kernel k is given by




  
    
      
        
          
            P
            ⁡
            
              (
              k
              )
            
          
          =
          
            
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              i
            
            ⁢
            
              
            
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                  k
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        ,
      
    
    
      
        (
        6
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a sparseness E(k) of the blur kernel may be computed according to





  
    
      
        
          
            E
            ⁡
            
              (
              k
              )
            
          
          =
          
            
              -
              
                1
                Z
              
            
            ⁢
            
              
                ∑
                i
              
              ⁢
              
                ln
                ⁢
                
                  
                
                ⁢
                
                  p
                  ⁡
                  
                    (
                    
                      k
                      i
                    
                    )
                  
                
              
            
          
        
        ,
      
    
    
      
        (
        7
        )
      
    
  




where Z is the kernel size. Another measure of blur-kernel sparseness is the entropy, computed according to

S(k)=−∫xp(x)ln p(x).  (8)

In a simple, proof-of-concept experiment, a sharp and blurred image pair were acquired, and a series of misaligned images were generated by scaling the acquired sharp image by various scaling factors and rotating it through various angles. Next, a series of blur kernels mapping the acquired and misaligned images to the blurred image were computed according to the procedure set forth in eqs 2-3. Finally, they computed E(k) for each blur kernel in the series according to eqs 6-7. The inventors found that the value of E(k) computed in this manner is greatest for the unscaled, unrotated (i.e., properly aligned) sharp image.
As noted hereinabove, misalignment in an image pair may include a translational offset between the two images, in addition to rotational and scaling differences. However, when a sharp image and a blurred image are translated relative to each other, the translation is reflected linearly in the blur kernel relating them; as sparseness is an isotropic property, the sparseness of the blur kernel relating the image pair is insensitive, therefore, to the state of translational alignment of the images.
One consequence of the above analysis is that methods to correct rotational and scaling misalignment based on blur-kernel sparseness may continue to work even if the images are translationally misaligned. Another consequence is that a different method must be used to align the images translationally, if translational misalignment is present. The latter issue is further considered hereinafter, in the context of an overall image-alignment method.
FIG. 2 illustrates an example method 200 to align a sharp image of a subject and a blurred image of the same subject. The method begins at 202, where a series of trial images is computed by applying a corresponding series of coordinate transforms to the sharp image, the series of coordinate transforms differing with respect to one or more of a rotational operation and a scaling operation.
In one embodiment, the series of coordinate transforms may span a two-dimensional parameter space, the two-dimensional parameter space including a rotational parameter and a scaling parameter. This embodiment is relevant when misalignment between sharp and blurred images is modeled as a similarity transform. Neglecting the translational part of the similarity transform discussed above, each coordinate transform in the series of coordinate transforms may be of the form




  
    
      
        
          
            (
            
              
                
                  
                    x
                    ′
                  
                
              
              
                
                  
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            )
          
          =
          
            
              s
              ⁡
              
                [
                
                  
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
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                          -
                          sin
                        
                        ⁢
                        
                          
                        
                        ⁢
                        θ
                      
                    
                  
                  
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        θ
                      
                    
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
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              (
              
                
                  
                    x
                  
                
                
                  
                    y
                  
                
              
              )
            
          
        
        ,
      
    
    
      
        (
        9
        )
      
    
  




where s is a scaling parameter and θ is a rotational parameter. A coordinate transform of this form may be applied to every pixel (x, y) in the sharp image to generate a corresponding trial image. Further, the series of coordinate transforms may be constructed by varying the scaling parameter s and/or the rotational parameter θ within the series. In particular, these parameters may be varied within predetermined ranges, e.g.,

smin≦s≦smax, θmin≦θ≦θmax.  (10)

In another embodiment, the series of coordinate transforms may span a four-dimensional parameter space, the four-dimensional parameter space including two rotational parameters and two scaling parameters. This embodiment is relevant when misalignment between sharp and blurred images is modeled as an affine transformation. Neglecting the translational part of the affine transformation discussed above, each coordinate transform in the series of coordinate transforms may be of the form




  
    
      
        
          
            (
            
              
                
                  
                    x
                    ′
                  
                
              
              
                
                  
                    y
                    ′
                  
                
              
            
            )
          
          =
          
            A
            ⁡
            
              (
              
                
                  
                    x
                  
                
                
                  
                    y
                  
                
              
              )
            
          
        
        ,
      
    
    
      
        (
        11
        )
      
    
  




where transformation matrix A involves two rotations and an anisotropic scaling. Based on singular-value decomposition, transformation matrix A may be decomposed as

A=R(θ)R(−φ)DR(φ),  (12)

where R(θ) and R(φ) are rotations by θ and φ, respectively, and D is the diagonal matrix diag(s1, s2). Thus, four parameters—s1, s2, θ, and φ—may be varied within predetermined ranges to construct the series of coordinate transforms. Following the previous embodiment, coordinate transforms in the form of eq 11 may be applied to every pixel (x, y) in the sharp image to generate the series of trial images.

Method 200 continues to 204, where a series blur kernels is computed, the series of blur kernels corresponding to the series of trial images, with each blur kernel mapping a trial image from the series of trial images to the blurred image. In one embodiment, the series of blur kernels may be computed according to the method set forth in eqs 2-3.
Method 200 continues to 206, where the sparsest blur kernel in the series of blur kernels is located. In one embodiment, locating the sparsest blur kernel in the series of blur kernels comprises evaluating a blur-kernel sparseness E(k) according to eq 7. In another embodiment, locating the sparsest blur kernel in the series of blur kernels comprises evaluating an entropy S(k) of a blur-kernel distribution function according to eq 8.
Method 200 continues to 208, where a rotational operation of the coordinate transform mapping the trial image corresponding to the sparsest blur kernel to the blurred image is identified, and to 210, where a scaling operation of the coordinate transform mapping the trial image corresponding to the sparsest blur kernel to the blurred image is identified. In one embodiment, steps 208-210 may comprise identifying the scaling and/or rotational parameters s and θ from the coordinate transform corresponding to the sparsest blur kernel as optimal rotational and/or scaling parameters sopt and θopt.
In some embodiments, image-alignment based on method 200 may further comprise applying one or more of the rotational operation and the scaling operation to the sharp image—in one example, replacing the sharp image by the trial image scaled by sopt and rotated by θopt. Likewise, one or more of the rotational operation and the scaling operation may be applied to the blurred image, but in reverse.
In method 200, the sharp image is selected as an operand image, and the blurred image is selected as a target image. The operand image is the image to which the series of coordinate transforms are applied to yield a corresponding series of trial images; the target image is the image to which the series of trial images are related via the corresponding series of blur kernels. It will be understood, however, that the converse of this approach is also contemplated. Thus, the blurred image may be selected as the operand image, and the sharp image may be selected as the target image. In the latter embodiment, the series of blur kernels relates the series of trial images to the sharp image.
Additional methods disclosed herein include aspects of the above approaches and provide additional features as well. One such example is shown in FIG. 3, where a method 300 to align a first and second image of a subject is illustrated. The method includes, at 302, locating a sparsest blur kernel in a parameter space, the sparsest blur kernel relating a select sharp image to a select blurred image, each of the select sharp image and the select blurred image chosen from the first and second images or computationally derived from the first image or the second image.
The term ‘locating a sparsest blur kernel in a parameter space’ presupposes a dependency of the blur kernel on one or more parameters. In some embodiments, the blur kernel may depend on coordinate-transform parameters s and θ, because the series of coordinate transforms maps the first or second image to a corresponding series of trial images. The parameter dependency is carried through to the computed blur kernels, which relate the series of trial images to the second or first image.
Method 300 continues to 304, where an aligning operation is applied to one or more of the first image and the second image, where the aligning operation is based on a location of the sparsest blur kernel in the parameter space.
In one series of embodiments based on method 300, image-alignment may further comprise applying a series of coordinate transforms (e.g., equation 9) to the first image to yield a corresponding series of trial images; computing a corresponding series of blur kernels relating the series of trial images to the second image, wherein the series of blur kernels includes the sparsest blur kernel; and identifying within the series of coordinate transforms a select coordinate transform corresponding to the sparsest blur kernel. As in method 200, the aligning operation referred to above may include a rotational operation and/or a scaling operation, and may be based at least partly on the select coordinate transform.
In one embodiment, the second image may be the select blurred image; applying the select coordinate transform to the first image would then result in the select sharp image. In this embodiment, the series of trial images includes the select sharp image. In another embodiment, the second image may be the select sharp image; applying the select coordinate transform to the first image would then result in the select blurred image. In this embodiment, the series of trial images includes the select blurred image.
In the embodiments described above, computationally deriving the select sharp image and/or the select blurred image from the first image or the second image comprises applying the select coordinate transform to the first image or to the second image. However, it may further comprise, in these and other embodiments, a downsampling procedure. Thus, one or more of the first image and the second image may be a fine image, and one or more of the select sharp image and the select blurred image may be computationally derived from the fine image by downsampling.
FIG. 4 illustrates method 400, which includes aspects of a coarse-to-fine, image alignment procedure in one, non-limiting embodiment. The illustrated method employs scale-space representations for both image space and parameter space. At 402, two pyramids,

{Bl}l=1L and {Il}l=1L  (13)

are constructed based on full-resolution images B and I, respectively. In the expressions above, integer L represents the height of the pyramid. At 404, a family of blur kernels is computed based on Bl and Il using an initial search range (viz., eq 10) and initial search increments Δs and Δθ. At 406, optimal scaling and rotational parameters sopt and θopt are computed over the current range and pyramid level, according to one of the methods described hereinabove. Then, at 408, the pyramid level is incremented. At 412, the search increments are set to 1⁄2 Δs and 1⁄2 Δθ, and the search ranges are set to

sopt−2Δs≦s≦sopt+2Δs, θopt−2Δθ≦θ≦θopt+2Δθ.  (14)

At 414, it is determined whether the highest level of the pyramid has been reached. If the highest level of the pyramid has been reached, then the method returns; otherwise, execution resumes at 406.

FIG. 5 illustrates method 500, which includes some operational details of method 300, step 302 in one, non-limiting embodiment. At 502, a rotational parameter θ is set to θmin, and a scaling parameter s is set to smin. At 504, a maximum sparseness value, Emax is reset to an initial value. At 506, a rotational operation and a scaling operation based on s and θ are applied to sharp image I to obtain trial image Î. At 508, a blur kernel k is estimated that relates the trial image Î to blurred image B. At 510, a sparseness E(k) of blur kernel k is computed according to eq 7.
At 512, it is determined whether or not E(k) exceeds Emax. If E(k) exceeds Emax, then at 514, Emax is set to E(k); an optimal value of the rotational parameter, θopt, is set to θ; and an optimal value of the scaling parameter, sopt, is set to s. Then, at 516, θ is increased by an increment. At 518, it is determined whether or not θ exceeds a maximum rotational value θmax. If θ does not exceed θmax, then at 520, the value of θ is reset to its minimum value, and at 522, the value of s is increased by an increment.
At 524, it is determined whether s exceeds a maximum scaling value smax. If s does not exceed smax, then execution of the method continues at 505. Otherwise, the method returns.
FIG. 6 illustrates method 600, which includes some operational details of method 300, step 304 in one, non-limiting embodiment. At 602, sharp image I is rotated by an amount θopt. Then, at 604, rotated, sharp image I is scaled by an amount sopt. In some examples, θopt and sopt may be the optimal values of the rotational and scaling parameters determined in one of the above methods.
At 606, a translational offset of rotated, scaled, sharp image I with respect to blurred image B is determined. In one embodiment, a method based on feature recognition may be used to determine an optimal translational offset topt by which I is offset with respect to B. Then, at 608, rotated, scaled, sharp image I is translated by topt to align it with B.
The example control and estimation routines disclosed herein may be used with various system configurations. These routines may represent one or more different processing strategies such as event-driven, interrupt-driven, multi-tasking, multi-threading, and the like. As such, the disclosed process steps (operations, functions, and/or acts) may represent code to be programmed into, or stored on a, device-readable storage medium. The term “device-readable storage medium” includes, but is not limited to, storage media on a computing device, such as volatile or non-volatile memory, disk drives, etc., as well as portable media, such as DVDs, CDs, flash memory drives, portable hard drives, floppy disks, etc.
FIG. 7 schematically represents one example embodiment of an environment in which the above process steps may be enacted. First, FIG. 7 shows computing device 702, which includes processor 704 and memory 706. Computing device 702 may be virtually any device having at least a processor and a memory, wherein the processor is operatively coupled to the memory and configured to execute instructions (i.e., code) stored therein. Computing device 702 may be a server system, a desktop or laptop computer, a handheld computer, a cell phone, or a gaming console, for example. Computing device 702 also may be an on-board controller for a digital camera, video camera, or other such digital image capture device.
In FIG. 7, computing device 702 is operatively coupled to image-output device 708, which may include virtually any device configured to display an image. Image-output device 708 may be a printer, a monitor, a touch screen, a television, or a video headset, for example, and may be separate from or incorporated into a body of the computing device 702. Computing device 702 is further coupled to image capture device 710, which may include any device configured to capture a still image or a video image of a subject, and which may be separate from or integrated with a body of the computing device 702. In various embodiments, image capture device 710 may be a digital camera, a scanner, a video camera, or a cell phone, for example. Other embodiments may include a plurality of image-capture devices and/or image-output devices substantially the same or at least partly different from each other.
FIG. 7 shows first image 101 and second image 102 resident on image-capture device 710, and it shows derived image 712 displayed on image-output device. Derived image 712 may be an image derived from aligning first image 101 and second image 102 according to one or more of the methods disclosed herein.
It will be understood that some of the process steps described and/or illustrated herein may in some embodiments be omitted without departing from the scope of this disclosure. Likewise, the indicated sequence of the process steps may not always be required to achieve the intended results, but is provided for ease of illustration and description. One or more of the illustrated actions, functions, or operations may be performed repeatedly, depending on the particular strategy being used.
Finally, it will be understood that the systems and methods described herein are exemplary in nature, and that these specific embodiments or examples are not to be considered in a limiting sense, because numerous variations are contemplated. Accordingly, the present disclosure includes all novel and non-obvious combinations and sub-combinations of the various systems and methods disclosed herein, as well as any and all equivalents thereof.

## Claims

1. A method to align a first image and a second image of a subject, the method comprising:
for each of a series of blur kernels that vary according to alignment parameters, evaluating blur-kernel sparseness to locate a sparsest blur kernel, the sparsest blur kernel relating a select sharp image to a select blurred image, each of the select sharp image and select blurred image chosen from or computationally derived from the first or second images; and
applying an aligning operation to the first or second image, the aligning operation being dependent on the alignment parameters corresponding to the sparsest blur kernel.

2. The method of claim 1, wherein the aligning operation includes a rotational operation.

3. The method of claim 1, wherein the aligning operation includes a scaling operation.

4. The method of claim 1, wherein one or more of the first image and the second image is a fine image, and one or more of the select sharp image and the select blurred image is computationally derived from the fine image by downsampling.

5. The method of claim 1, further comprising:
applying a series of coordinate transforms based on the alignment parameters to the first image to yield a corresponding series of trial images;
computing a corresponding series of blur kernels relating the series of trial images to the second image; and
identifying within the series of coordinate transforms a select coordinate transform corresponding to the sparsest blur kernel.

6. The method of claim 5, wherein the aligning operation is at least partly based on the select coordinate transform.

7. The method of claim 5, wherein the second image is the select blurred image, wherein applying the select coordinate transform to the first image results in the select sharp image, and wherein and the series of trial images includes the select sharp image.

8. The method of claim 5, wherein the second image is the select sharp image, wherein applying the select coordinate transform to the first image results in the select blurred image, and wherein the series of trial images includes the select blurred image.

9. The method of claim 5, further comprising applying a translational alignment procedure to one or more of the first image and the second image.

10. A method to align a sharp image of a subject and a blurred image of the subject, the method comprising:
computing a series of trial images by applying a corresponding series of coordinate transforms to the sharp image, the series of coordinate transforms differing with respect to one or more of a rotational operation and a scaling operation;
computing a series blur kernels corresponding to the series of trial images, each blur kernel mapping a trial image from the series of trial images to the blurred image;
for each blur kernel in the series of blur kernels, evaluating blur-kernel sparseness to locate a sparsest blur kernel; and
identifying one or more of the rotational operation and the scaling operation of the coordinate transform mapping the trial image corresponding to the sparsest blur kernel to the blurred image.

11. The method of claim 10, wherein evaluating blur-kernel sparseness comprises evaluating a sparseness E(k) related to a blur-kernel size Z and to a blur-kernel distribution function p(ki) according to




  
    
      
        
          
            E
            ⁡
            
              (
              k
              )
            
          
          =
          
            
              -
              
                1
                Z
              
            
            ⁢
            
              
                ∑
                i
              
              ⁢
              
                ln
                ⁢
                
                  
                
                ⁢
                
                  p
                  ⁡
                  
                    (
                    
                      k
                      i
                    
                    )
                  
                
              
            
          
        
        ,
      
    
    
      
        (
        7
        )
      
    
  




where ki is an ith element in blur kernel k.

12. The method of claim 10, wherein evaluating blur-kernel sparseness comprises evaluating an entropy S(k) of a blur-kernel distribution function p(x) according to S(k)=−∫xp(x)ln p(x).

13. The method of claim 10, further comprising applying the one or more of the rotational operation and the scaling operation to the sharp image.

14. The method of claim 10, further comprising applying, in reverse, the one or more of the rotational operation and the scaling operation to the blurred image.

15. A device-readable storage medium excluding signals per-se, the device-readable storage medium comprising instructions executable by a computing device to:
compute a series of trial images by applying a series of coordinate transforms to an operand image, the series of coordinate transforms differing with respect to one or more of a rotational operation and a scaling operation;
compute a series blur kernels corresponding to the series of trial images, each blur kernel relating a trial image from the series of trial images to a target image;
for each blur kernel in the series of blur kernels, evaluating blur-kernel sparseness to locate a sparsest blur kernel;
identify one or more of the rotational operation and the scaling operation of the coordinate transform relating the trial image corresponding to the sparsest blur kernel to the target image.

16. The storage medium of claim 15, wherein the target image is a blurred image.

17. The storage medium of claim 15, wherein the target image is a sharp image.

18. The storage medium device of claim 15, wherein the series of coordinate transforms spans a two-dimensional parameter space, the two-dimensional parameter space including a rotational parameter and a scaling parameter.

19. The storage medium of claim 15, wherein the series of coordinate transforms spans a four-dimensional parameter space, the four-dimensional parameter space including two rotational parameters and two scaling parameters.

20. The storage medium of claim 19, wherein the coordinate transform is based on an affine transformation.