# Patent

## Title
Image restoration method, image restoration apparatus, and image-pickup apparatus

## Abstract

A method includes performing an approximate partially coherent imaging operation for elements of a first basis generated from a model image having no noise and no blur, and generating based upon the first basis a second basis that is blurred, the approximate partially coherent imaging operation being expressed by a convolution integral on an eigenfunction corresponding to a maximum eigenvalue of a Kernel matrix and each element of the first basis, generating an intermediate image in which each pixel value of the observed image that has been denoised is replaced with its square root, and obtaining a restored image by approximating each of a plurality of patches that are set to entirely cover the intermediate image, using a linear combination of elements of the first basis and linear combination coefficients obtained when each patch is approximated by a linear combination of elements of the second basis.

## Background
BACKGROUND OF THE INVENTION
1. Field of the Invention
The present invention relates to a restoration (reconstruction) of an image.
2. Description of the Related Art
An observed image obtained by a bright field microscope or a virtual slide including an image sensor (referred to as a “digital microscope” hereinafter) is known as an observed image obtained through a photoelectric conversion of an optical image formed by partially coherent imaging. Conventionally, no methods have been proposed for removing from the observed image a noise caused by the image sensor (which may be simply referred to as a “noise” hereinafter), or a blur caused by an optical system (which may be simply referred to as a “blur” hereinafter). It is a conventionally proposed method to restore an original image by removing the noise and blur from a deteriorated image obtained by photoelectrically converting an optical image formed by incoherent imaging.
For example, Yifei Lou, Andrea L. Bertozzi, and Stefano Soatto, “Direct Sparse Deblurring, “Journal of Mathematical Imaging and Vision,” 2011, vol. 39, p. 1-12 (“literature 1”) proposes a method for removing a noise and a blur from a deteriorated image using a set of image patches extracted from a model image and a set in which each patch is blurred by a convolution integral of a point-image intensity distribution of an optical system. Michael Elad and Michal Aharon, “Image Denoising Via Sparse and Redundant Representations Over Learned Dictionaries,” IEEE Transactions of Image Processing, 2006, vol. 15, p. 3736-3745 (“literature 2”) proposes a method for removing a noise caused by an image sensor from a deteriorated image using a set of image patches generated by machine learning from a model image.
However, the image restoration methods disclosed in the literatures 1 and 2 address incoherent imaging and cannot sufficiently deblur the observed image obtained from the optical image formed by partially coherent imaging.
SUMMARY OF THE INVENTION
The present invention provides an image restoration method, an image restoration apparatus, and an image-pickup apparatus, which can eliminate a blur caused by an optical system from an observed image obtained by a photoelectric conversion of an optical image formed by partially coherent imaging.
An image restoration method according to the present invention is configured to restore an original image from an observed image obtained through a photoelectric conversion of an optical image formed by partially coherent imaging. The image restoration method includes the steps of performing an approximate partially coherent imaging operation for elements of a first basis generated from a model image that is assumed to have no noise and no blur, and generating based upon the first basis a second basis that is blurred, the approximate partially coherent imaging operation being expressed by a convolution integral on an eigenfunction corresponding to a maximum eigenvalue of a Kernel matrix and each element in the first basis, generating an intermediate image in which each pixel value of the observed image that has been denoised is replaced with its square root, and obtaining a restored image by approximating each of a plurality of patches that are set to entirely cover the intermediate image, using a linear combination between the first basis and a linear combination coefficient obtained when each patch is approximated by a linear combination of the second basis.
Further features of the present invention will become apparent from the following description of exemplary embodiments with reference to the attached drawings.



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a block diagram of an illustrative configuration of a digital microscope according to this embodiment.
FIG. 2 is a block diagram of an illustrative configuration of a computer illustrated in FIG. 1.
FIG. 3 is a view for explaining S206 illustrated in FIG. 2.
FIG. 4 is a flowchart for explaining image processing according to first and second embodiments.
FIG. 5 is a flowchart for explaining an image processing method configured to generate a blurred image in which a noise caused by an image sensor is removed from an observed image according to the first embodiment.
FIG. 6 is a flowchart for explaining processing of removing blurs from a blurred image according to the first embodiment.
FIGS. 7A-7E are views for explaining effects of the image processing method according to the first embodiment.
FIG. 8 is a flowchart for explaining an image processing method configured to generate a blurred image in which a noise caused by an image sensor is removed from an observed image according to the second embodiment.
FIG. 9 is a flowchart for explaining processing of removing blurs from a blurred image according to the second embodiment.
FIGS. 10A-10E are views for explaining effects of the image processing method according to the second embodiment.



DESCRIPTION OF THE EMBODIMENTS
FIG. 1 is a block diagram of an illustrative configuration of a digital microscope (image-pickup apparatus) according to this embodiment. The digital microscope includes a digital microscope body 1001, an A/D converter 1008, and a computer 101.
The digital microscope body 1001 includes an illumination light source 1002, an illumination optical system 1003, a sample (specimen) stage 1004, an imaging optical system 1005, and an image sensor 1007.
The illumination light source 1002 emits light, and serves as a set of point light sources. The illumination optical system 1003 illuminates a sample (specimen) SP, such as a präparat, using the light from the illumination light source 1002 (partially coherent illumination). The sample SP is an object as an original image to be restored. The sample stage 1004 is mounted with and moves the sample SP in three directions. The imaging optical system 1005 forms an image of the sample SP on the image plane of the image sensor 1007.
This embodiment thus forms the optical image of the object through partially coherent imaging, and the optical image contains a blur caused by an optical system of the partially coherent imaging (which may be referred to as a “partially coherent imaging optical system” hereinafter). The partially coherent imaging optical system includes both the illumination optical system 1003 and the imaging optical system 1005.
The image sensor 1007 is an image-pickup element (photoelectric conversion element) configured to photoelectrically convert the optical image of the object formed by the imaging optical system 1005 into an analogue electric signal. The image sensor 1007 can use a charged coupled device (“CCD”), a complementary metal oxide semiconductor (“CMOS”), etc. The optical image contains a noise caused by the image sensor 1007.
The A/D converter 1008 converts the analogue electric signal from the image sensor 1007 into a digital signal.
The computer 101 provides various types of image processing for the digital signal output from the A/D converter 1008, in addition to storing, displaying, and communicating information. In particular, according to this embodiment, the computer 101 executes an image restoration method.
In the image restoration method, the computer 101 restores an original image by performing image processing of removing the noises caused by the image sensor and the blurs caused by the partially coherent imaging optical system, from the digital signal representing the observed image of the object. In other words, the computer 101 serves as an image restoration apparatus configured to restore the original image by performing image processing for the input digital electric signal. The image restoration apparatus may not be configured as part of the image-pickup apparatus, but may be configured as an independent apparatus, such as a PC.
The illumination light emitted from the illumination optical system 1002 passes the illumination optical system 1003, illuminates the sample SP on the sample stage 1004, then passes the imaging optical system 1005, and forms an image on an imaging plane of the image sensor 1007. An epi-illumination may be used for the illumination method, which illuminates the sample SP from the top, and forms an image using the reflected light. The analogue electric signal output from the image sensor 1007 is converted into the digital electric signal by the A/D converter 1008, and input into the computer 101.
FIG. 2 is a block diagram of an illustrative configuration of the computer 101. The computer 101 includes bus wiring 102, a controller 103, a display unit 104, a storage unit 105, an input unit 106, and a medium interface 107.
The bus wiring 102 connects the controller 103, the display unit 104, the storage unit 105, the input unit 106, and the medium interface 107 with one another. The controller 103 is, for example, a CPU, a GPU, a DSP (processor), or a microcomputer, and includes a cashe memory (not illustrated) for temporary storage.
The display unit 104 is a display device, such as a CRT display and a liquid crystal display. The storage unit 105 is, for example, a memory and a hard disk drive, configured to store various types of information.
The input unit 106 is, for example, a keyboard and a pointing device (such as a mouse). The medium interface 107 is, for example, a CD-ROM drive and a USB interface, configured to connect the recording medium 108 to the bus wiring 102. The recording medium 108 includes a CD-ROM and a USB memory.
The information stored in the storage unit 105 contains effective light source information 105a, NA information 105b, λ information 105c, model image 105d, a basis 105e, an observed image 105f, a blurred image 105g, an intermediate image 105h, a patch 105i, a linear combination coefficient 105j, an approximated image 105k, a restored image 105l, a random noise 105m, and an image processing program 105n. 
The effective light source information 105a is information relating to a light intensity distribution formed on a pupil plane 1006 in the imaging optical system 1005 when there is no sample.
The NA information 105b is information relating to a numerical aperture (“NA”) on the image side of the imaging optical system 1005.
The λ information 105c is information relating to a wavelength (λ) of the illumination light source 1002.
The model image 105d is a base image used to generate the basis 105e, and a sharp image that is assumed to have no noises and no blurs. The model image 105d is, for example, an image obtained by a scanning electron microscope (“SEM”).
The storage unit 105 can store, in part of the model image 105d, a blurred model image, which is generated by a partially coherent imaging operation from the model image 105d. 
The basis 105e is a set of image patches of N pixels×N pixels generated from random sampling or machine learning (dictionary learning) from the image, such as the model image 105d. The random sampling is user processing of extracting a predetermined number of image patches of N pixels×N pixels from the image at random locations. The machine learning is processing of generating a user designated number of image patches of N pixels×N pixels using a machine learning algorithm K-SVD method from an image, etc. For example, the basis 105e may use a set of 256 elements of 8 pixels×8 pixels.
A basis {b} (first basis) generated from the model image 105d is a basis having no noises and no blurs similar to the model image 105d, and reflects the characteristic of the model image. A basis {b′} (second basis) is generated by performing the following approximate partially coherent imaging operation for the basis {b} and is a blurred basis having no noises. Elements b1, b2, . . . , bn, of the basis {b} correspond to elements b′1, b′2, . . . , b′n, of the basis {b′}. A basis {b′′} (third basis) may be generated from the blurred model image.
The observed image 105f is an image obtained from the digital microscope body 1001 via the A/D converter 1008.
The blurred image 105g is an image that is made by removing the noise caused by the image sensor 1007 from the observed image 105f. 
The intermediate image 105h is an image made by converting pixel values in the observed image 105f or the blurred image 105g into values suitable for image processing. For example, one illustrative conversion is replacing each pixel value of the blurred image 105g with its square root.
The patches 105i are a plurality of small segments that are set so as to cover the intermediate image 105h, and each patch is an image piece having a size of N pixels×N pixels.
The linear combination coefficient 105j is a coefficient obtained when the patches 105i are approximated with a linear combination of elements of the basis 105e. The linear combination coefficient 105j is a vector having the same dimension as that of elements of the basis 105e. 
The approximated image 105k is an image generated from the approximated patches that is expressed by approximating the patches 105i using the basis. The size of the approximated image 105k is the same as the size of the observed image, and the initial values of any pixels in the approximated image 105k are zero. The blurred image 105g is a set of all patches of the approximate image 105k made by approximating the patches 105i with the linear combination of the second basis that is blurred but has no noises. In addition, the restored image 105l is a set of all patches of the approximated image 105k made by approximating the patches 105i with the linear combination of elements of the first basis that has no noises and no blurs.
The restored image 105l is an image generated from the approximated image 105k. 
The random noise 105m is an artificially generated noise in accordance with a condition designated by a user. The artificially generated noises do not perfectly accord with the noises caused by the image sensor, and thus a plurality of types of linear combination coefficients 105j are averaged obtained when a plurality of types of noises are generated by the random noise 105m. When the random noise 105m generates a plurality of types of noises, the storage unit 105 stores an end condition that defines the number of types (or the number of generations or the number of iterations).
The image processing program 105n is a program that enables the controller 103 to execute image processing of removing the noise caused by the image sensor 1007 and the blur caused by the partially coherent imaging optical system, from the observed image 105f or the blurred image 105g. 
FIG. 3 is a flowchart for explaining image processing performed for the observed image 105f or the blurred image 105g. “S” stands for the step, and this rule is true of another flowchart. The flowchart illustrated in FIG. 3 can be implemented as an image processing program 105n for executing the controller 103 in the computer 101 to execute each step, and this is true of another flowchart. The flowchart illustrated in FIG. 3 is a fundamental flowchart of embodiments, which will be described later.
Initially, the recording medium 108 in which the image processing program 105n is recorded is connected to the medium interface 107, and the image processing program 105n is installed in the storage unit 105 via the controller 103. A start-up command of the image processing program 105n is input into the input unit 106 by the user, and the controller 103, in response, refers to the memory unit 105 and runs the image processing program 105n (S201).
In addition, the image processing parameter input by the user to the input unit 106 is stored in the storage unit 105 by the controller 103 (S201). The “image processing parameter” contains the effective light source information 105a, the NA information 105b, the λ information 105c, the model image 105d, the basis 105e, and the observed image 105f. 
Next, the controller 103 refers to the storage unit 105, generates the first basis from the model image 105d through the random sampling or machine learning (S202), and stores the first basis in the storage unit 105. Depending upon the object of the image processing, the second basis may be generated by performing the partially coherent imaging operation for each element of the first basis or the blurred model image may be generated by performing the partially coherent imaging operation for the model image 105d and the third basis may be generated from the blurred model image.
Next, the controller 103 refers to the storage unit 105, and generates the intermediate image 105h from the observed image 105f or the blurred image 105g (S203), and stores the intermediate image 105h in the storage unit 105. For example, in the processing of removing the blur caused by the partially coherent imaging optical system from the blurred image 105g, the intermediate image 105h is generated by replacing each pixel value of the blurred image 105g with its square root.
S204 to S207 are processing of generating the approximated image 105k from the intermediate image 105h by the controller 103, and will be referred to as “core processing” hereinafter.
Initially, the controller 103 refers to the storage unit 105, extracts the patches 105i from the intermediate image 105h (S204), and stores them in the storage unit 105. The extracted patches may or may not have an overlapping part but the patches 105i need to comprehend the whole intermediate image or closely covers the intermediate image. In addition, the extracting rule of the patches 105i needs to be uniformly applied to the entire intermediate image, and cannot be changed in the middle of the extraction. For example, unless the patch projects from the intermediate image 105h, the patch 105i may be extracted so that it overlaps the adjacent patch 105i by one pixel at its end.
Next, the controller 103 refers to the storage unit 105, approximates one patch 105i with the linear combination of elements of the second basis, calculates the linear combination coefficient 105j (S205), and stores the linear combination coefficient 105j in the storage unit 105. The linear combination coefficient 105j is determined so as to satisfy the following expressions:




  
    
      
        
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Herein, A and Ã denote linear combination coefficients 105j of the basis 105e, ∥Ã∥0 denotes the number of non-zero elements in the linear combination coefficient Ã, x denotes a brightness value distribution of one of patches 105i, and x′ denotes a brightness value distribution of the patch x that is approximated with a linear combination of elements of the basis 105e and the linear combination coefficient Ã. ∥x−x∥22 denotes a square sum of a difference between a brightness distribution of the patch x and the linear combination approximation using the basis 105e, and ε denotes the approximation precision of the linear combination designated by the user. In other words, under the approximation precision designated by the user, the linear combination coefficients 105j having minimum non-zero elements are calculated.
Next, the controller 103 refers to the storage unit 105, and reconstructs the approximated image 105k by approximating one patch 105i extracted from the intermediate image 105h using the linear combination of the linear combination coefficient 105j and elements of the basis (S206). Thereafter, the controller 103 returns the approximated patch 105i to the original position and stores the result. When the basis used in S206 is the second basis, the blurred image is finally generated, and when the second basis is replaced with the first basis, the restored image is finally generated.
FIG. 4 is a view for explaining processing of returning the patches 105i to the extracted original locations. The patch 105i extracted from a certain location (i, j) of the intermediate image 105h is approximated by the linear combination of the linear combination coefficient 105j and elements of the basis, and the pixel value of the approximated patch 105i is added to the location (i, j) in the approximated image. Even when overlapping location occurs between adjacent patches, the pixel value of the location is expressed by additions. The reconstruction of the approximated image 105k means processing of generating an image in the course of generating the finally approximated image 105k. 
Next, the controller 103 refers to the storage unit 105, performs S205 and S206 for all patches 105i, and generates the finally approximated image 105k (S207).
The above processing is the core processing.
Next, the controller 103 refers to the storage unit 105, and executes weighted averaging between the observed image 105f and the approximated image 105k for each pixel, and generates the blurred image 105g in which the noise caused by the image sensor 1007 has been removed or the restored image 105l(S208). When the intermediate image of S203 is the observed image, the blurred image is generated. When the intermediate image is the blurred image, the restored image is generated. The blurred image 105g or the restored image 105l is stored in the storage unit 105. The weighted averaging has both an effect of removing overlapping between the patches (an effect of a division) and an effect of removing the noise. Since the restored image has no noises, only a division may be performed instead of weighted averaging. The controller 103 may display the restored image 105l on the display unit 104.
As described above, the image processing of this embodiment can remove the noise caused by the image sensor 1007 from the observed image 105f obtained by the digital microscope body 1001 or remove the blur caused by the partially coherent imaging optical system from the blurred image 105g. 
A detailed description will be given of the image processing method in the following embodiments.
First Embodiment
This embodiment performs for the observed image 105f image processing of removing the noise caused by the image sensor and blur caused by the partially coherent imaging optical system, and obtains the restored image having no noises and no blurs.
FIG. 5 is a flowchart for explaining a method for generating a blurred image 105g made by removing the noise caused by the image sensor from the observed image 105f. 
Initially, the controller 103 refers to the storage unit 105, performs a partially coherent imaging operation for the model image 105d, generates a blurred model image (S301), and stores the blurred model image in the storage unit 105. Alternatively, a user may store, as the blurred model image in the storage unit 105, an image captured under the same optical condition as the target digital microscope.
Next, the controller 103 refers to the storage unit 105, generates a blurred basis {b′′} (third basis) from the blurred model image (S302), similar to the processing of S202, and stores the blurred basis {b′′} in the storage unit 105. In addition, each element of the blurred basis {b′′} is normalized by a division using an L2 norm. The L2 norm is a scalar amount expressed as follows:




  
    
      
        
          
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Herein, bi′′ denotes an i-th element of the blurred basis {b′′}, ∥bi′′∥2 denotes the L2 norm of bi′′, and bij′′ denotes a j-th pixel value.
Next, the controller 103 refers to the storage unit 105, extracts the patches 105i from the observed image 105f similar to the processing of S204, and stores the patches 105i in the storage unit 105 (S303).
Next, the controller 103 refers to the storage unit 105, approximates one of the patches with the blurred basis {b′′}, calculates a linear combination coefficient α similar to S205 (S304), and stores the linear combination coefficient α in the storage unit 105.
Next, the controller 103 refers to the storage unit 105, reconstructs the approximated image 105k using the linear combination coefficient α and the blurred basis {b′′} (S305), similar to the processing of S206, and stores the approximated image 105k in the storage unit 105.
Next, the controller 103 refers to the storage unit 105, performs S304 and S305 for all patches 105i, generates the finally approximated image 105k (S306), and stores the pixel values of the finally approximated image 105k in the storage unit 105.
Next, the controller 103 refers to the storage unit 105, executes the weighted averaging between the observed image 105f and the approximate image 105k for each pixel, generates the blurred image 105g by removing the noise caused by the image sensor from the observed image 105f (S307), and stores the blurred image 105g in the storage unit 105.




  
    
      
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Herein, X denotes a pixel value of one pixel in the blurred image 105g. Y denotes a pixel value of one corresponding pixel in the observed image 105f. X′ denotes a pixel value of one corresponding pixel in the approximated image 105k, p is a weighting parameter designated by the user based upon the magnitude of the noise caused by the image sensor, “cnt” denotes an overlapping number of the patch 105i in one corresponding pixel of the approximated image 105k. 
The processing can be expedited by distributing the processing of S304 among a plurality of operating units. The method for generating the blurred image 105g by removing the noise caused by the image sensor from the observed image 105f has been thus described.
FIG. 6 is a flowchart for explaining a method for generating a restored image 105l by removing the blur caused by the partially coherent imaging optical system from the blurred image 105g obtained by S307.
Initially, the controller 103 refers to the storage unit 105, generates the basis {b} similar to the processing of S202 from the model image 105d (S401), and stores the basis {b} in the storage unit 105. Each element of the basis {b} is normalized through a division using its L2 norm.
Next, the controller 103 refers to the storage unit 105, performs an approximate partially coherent imaging operation for each element of the basis {b}, generates a blurred basis {b′} (S402), and stores it in the storage unit 105. In this approximate partially coherent imaging operation, the controller 103 executes an imaging operation expressed by a convolution integral on each element of and the basis {b} and an eigenfunction corresponding to the maximum eigenvalue of the Kernel matrix expressed by the following expression which has a Fourier transform of a shifted pupil function as an element.




  
    
      
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Herein, K denotes the Kernel matrix, Si denotes an intensity at an i-th point of the effective light source, and (xi, yi) is an i-th coordinate on the image plane. ψi is given as follows:

ψi=FT[P(f−fi,g−gi)]  Expression 5

Herein, FT denotes a Fourier transform, P denotes a pupil function, (fi, gi) denotes a coordinate on the pupil plane of the i-th point in the effective light source, and (f, g) denotes a coordinate of the pupil plane.
In addition, an eigenvalue of the Kernel matrix and the corresponding eigenfunction are given as follows:

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U=[Φ1,Φ2 . . . ]

S=diag[λ1,λ2 . . . ]  Expression 6

Herein, USV† denotes a matrix obtained by performing a singular value decomposition for the Kernel matrix, λi denotes an i-th eigenvalue and positive real number, Φi denotes an eigenfunction as a vector amount corresponding to the i-th eigenvalue, and “diag” denotes a diagonal matrix having λi as a diagonal component. At this time, elements of the basis {b′} are calculated based upon the elements of the basis {b} as follows:




  
    
      
        
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Herein, bj denotes the j-th element of the basis {b}, and bj′ denotes the corresponding j-th element of the basis {b′}. Other symbols are similar to those in Expression 6.
In the partially coherent imaging operation, it is known that Expression 7 can be precisely approximated as follows only using the eigenfunction corresponding to the maximum eigenvalue. The symbols that are used are similar to those of Expression 7.




  
    
      
        
          
            
              
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Next, both sides of Expression 8 are square-rooted and the basis {b′} is newly defined as follows:




  
    
      
        
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Herein, the employed symbols are similar to those of Expression 7. Thereby, the elements of the basis {b} and the corresponding elements of the basis {b′} are approximated with a linear relationship. The imaging operation that is the partially coherent imaging operation expressed as in Expression 9, by a convolution integral of the eigenfunction corresponding to the maximum eigenvalue of the Kernel matrix is herein referred to as an approximate partially coherent imaging operation. Conventionally, it has not yet been proposed to generate the basis {b′} from the basis {b} using the approximate partially coherent imaging operation.
Next, the controller 103 refers to the storage unit 105, generates the intermediate image 105h by replacing each pixel value of the blurred image 105g with its square root (S403), and stores it in the storage unit 105.




  
    
      
        
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        ⁢
        
          
        
        ⁢
        10
      
    
  




Herein, Iij denotes an ij-th pixel value of the intermediate image 105h, and Jij denotes an ij-th pixel value of the blurred image 105g. According to the approximate partially coherent imaging operation, as described for Expression 9, this operation is performed because a square root of the blurred image 105g has an approximately linear relationship with a non-blurred image. S403 has not yet been conventionally proposed.
Next, the controller 103 refers to the storage unit 105, extracts the patches 105i from the entire intermediate image similar to the processing of S204 (S404), and stores the patches 105i in the storage unit 105.
Next, the controller 103 refers to the storage unit 105, approximates one of the patches 105i with the linear combination of elements of the basis {b′} similar to the processing of S205, calculates the linear combination coefficient β (S405), and stores the combination coefficient β in the storage unit 105.
Next, the controller 103 refers to the storage unit 105, calculates the patches 105i by removing the blur using the linear combination of the linear combination coefficient β and elements of the basis {b}, and reconstructs the approximated image 105k by returning the patches 105i to the original locations similar to the processing of S206 (S406). The controller stores the approximated image 105k in the storage unit 105.
S406 replaces the basis {b′} (which is a set of elements b′1, b′2, . . . , b′n) with the basis {b} (which is a set of elements b1, b2, . . . , bn). In other words, when the linear combination coefficient β is expressed by β=a1, a2, . . . , an, the patches are expressed as a1·b′1+a2·b′2+ . . . , an·b′n in S405 and as a1·b1+a2·b2+ . . . , an·bn in S406.
Next, the controller 103 performs processing of S405 and S406 for all patches 105i (S407), and stores the finally obtained pixel values of the approximated image 105k in the storage unit 105.
Next, the controller 103 refers to the storage unit 105, and generates the pixel values of the finally restored image 105l by dividing the pixel value at the overlapping location of the patches 105i in the approximated image 105k by the number of overlapping patches (S408). The controller 103 displays the generated restored image on the display unit 104 or stores it in the storage unit 105.
The processing can be expedited when the processing of S405 is distributed among a plurality of operating units. The method of generating the restored image 105l by removing the blur caused by the partially coherent imaging optical system from the blurred image 105g is thus described.
Thus, this embodiment obtains the restored image 105l by approximating each patch using the linear combination of elements of the basis {b} and the linear combination coefficient β obtained when each of the plurality of patches set to cover the entire intermediate image is approximated with the linear combination of the second basis.
FIGS. 7A-7E are views for explaining the effect of the image restoration method according to the first embodiment. These figures illustrate the result when the blurred image 105g is generated by removing the noise from the observed image 105f by a partially coherent imaging simulator, and then the restored image 105l is generated by removing the blur from the blurred image 105g. In addition, in order to compare this embodiment with the prior art, the result of generating the restored image 105l that is made by denoising and deblurring the observed image 105f by the method disclosed in the literature 1 is also illustrated.
As a simulation condition, assume that a sampling pitch on the object is 0.2 μm, the object side NA of the imaging optical system 1005 is 0.7, the illumination optical system 1003 provides an annular illumination in which an outer σ is 0.7 and an inner σ is 0.3, and a wavelength of the illumination light source 1002 is 0.55 μm. The basis includes 256 elements each of which has 8 pixels×8 pixels, and is generated by a machine learning algorithm. The image size is uniformly 100 pixels×100 pixels. In the linear combination coefficient α, there are about 10 to 15 non-zero elements. This is true of the linear combination coefficient β.
FIG. 7A illustrates an original image corresponding to an object.
FIG. 7B illustrates a virtual object image in which a deterioration corresponding to a blur caused by the partially coherent imaging optical system is added to the original image by the partially coherent imaging simulator, and a white Gaussian noise of a standard deviation of 5% corresponding to the noise caused by the image sensor is further added.
FIG. 7C illustrates a blurred image that is made by denoising the observed image in accordance with the flow of FIG. 5.
FIG. 7D illustrates a restored image that is made by deblurring the blurred image in accordance with the flow of FIG. 6.
FIG. 7E illustrates a restored image that is made by denoising and deblurring the observed image by the method disclosed in the literature 1.
Table 1 summarizes a quantitative evaluation result of the image quality. In order to compare the first embodiment with the prior art, the image restoration result made by the method disclosed in the literature 1 is included. The evaluation index uses the PSNR. This is calculated from a difference of a pixel value between the original image and the comparative image, and indicates that the larger the value is the closer to the original image the comparative image is.











TABLE 1







Denoised 

Restored image




image

made by the



Observed
(blurred
Restored
method of



Image
image)
image
literature 1



















PSNR [dB]
12.17
13.07
19.85
13.71









It is understood from FIG. 7 and Table 1 that this embodiment can generate a higher-quality restored image.
Second Embodiment
This embodiment also performs for the observed image 105f image processing of removing a noise caused by the image sensor and a blur caused by the partially coherent imaging optical system, and obtains the restored image having no noise and no blur.
FIG. 8 is a flowchart for explaining a method for generating a blurred image 105g by removing a noise caused by the image sensor from the observed image 105f. 
Initially, the controller 103 refers to the storage unit 105, generates the basis {b} from the model image 105d similar to the processing of S202 (S501), and stores it in the storage unit 105. Each element of the basis {b} is normalized through a division using its L2 norm.
Next, the controller 103 generates the basis {b′} from the basis {b} similar to the processing of S402 (S502), and stores it in the storage unit 105.
Next, the controller 103 stores the pixel value generated for each pixel as the artificial random noise n in the random noise 105m of the storage unit 105 (S503). For example, the following normal random number may be used for random noise:

η=N(μ,ρ)  Expression 11

Herein, N(μ,ρ) denotes the normal random number having an average value μ and a standard deviation ρ.
Next, the controller 103 refers to the storage unit 105, generates an intermediate image A by subtracting the random noise η from the observed image 105f for each pixel (S504), and stores the intermediate image A in the storage unit 105.
Next, the controller 103 refers to the storage unit 105, generates an intermediate image B that is made by replacing each pixel value of the intermediate image A with its square root (S505), and stores the intermediate image B in the storage unit 105.
Next, the controller 103 refers to the storage unit 105, and extracts the patches 105i from the entire intermediate image B similar to the processing of S204 (S506), and stores it in the storage unit 105.
Next, the controller 103 refers to the storage unit 105, calculates a linear combination coefficient γ using one of the patches 105i extracted from the whole intermediate image B and the basis {b′} similar to the processing of S205 (S507), and stores it in the storage unit.
Next, the storage unit 103 executes S507 for all patches 105i (S508).
Next, the controller 103 refers to the storage unit 105, adds the linear combination coefficient γ obtained by the previous iteration to the linear combination coefficient γ obtained by this iteration for each patch 105i (S509), and stores it in the storage unit 105.
Next, the controller 103 ends the iteration when the end condition designated by the user is satisfied, and the flow returns to S503 when the end condition is not satisfied (S510). The end condition designated by the user may be an upper limit of the number of iterations, for example. In other words, this embodiment generates a plurality of types of artificial random noises in the step of generating the random noises (S503).
Next, the controller 103 refers to the storage unit 105, averages the linear combination coefficients γ summed up for each iteration by the number of iterations for each patch 105i (S511), and stores the result in the storage unit 105. In other words, this embodiment averages the plurality of linear combination coefficients obtained corresponding to the plurality of types of random noises.
Next, the controller 103 refers to the storage unit 105, calculates the patch denoised with the linear combination coefficient γ and the basis {b′}, and reconstructs the approximated image by returning the patch to the extracted original location similar to the processing of S206 (S512), and stores it in the storage unit 105.
Next, the controller 103 performs S512 to all patches 105i, and generates the blurred image (S513), and stores it in the storage unit 105.
Next, the controller 103 refers to the storage unit 105, generates the finally blurred image 105g by weighted averaging between the observed image and the approximated image (S514), and stores the pixel value of the blurred image 105g in the storage unit 105.
The processing can be expedited when the processing of S507 is distributed among a plurality of operating units. The method of generating the blurred image 105g by removing the noise caused by the image sensor from the observed image 105f has been thus described.
FIG. 9 is a flowchart for explaining a method for generating a restored image 105l by removing the blur caused by the partially coherent imaging optical system from the blurred image 105g obtained in S514.
The controller 103 refers to the storage unit 105, calculates the patches 105i deblurred with the linear combination of elements of the basis {b} and the averaged linear combination coefficient, and reconstructs the approximated image 105k that has been deblurred by returning the patches 105i to the original locations (S601).
This embodiment also replaces the basis {b′} (which is a set of elements b′1, b′2, . . . , b′n) with the basis {b} (which is a set of elements b1, b2, . . . , bn). In other words, when the linear combination coefficient is expressed by a1, a2, . . . , an, the patches are expressed as a1·b′1+a2·b′2+ . . . , an·b′n in S512 and as a1·b1+a2·b2+ . . . , an·bn in S601. The controller stores the approximated image 105k in the storage unit 105.
Next, the controller 103 performs S604 for all patches 105i, generates the finally approximated image (S602), and stores it in the storage unit 105.
Next, the controller 103 refers to the storage unit 105, and generates the pixel values of the finally restored image by dividing the pixel values at the overlapping locations of the patches 105i by the number of patches 105i (S603). The controller 103 displays the generated restored image 105l on the display unit 104 or stores it in the storage unit 105.
The method of generating the restored image 105l by removing the blur caused by the partially coherent imaging optical system from the blurred image 105g has been thus described.
FIGS. 10A-10E are views for explaining the effect of the image restoration method according to the second embodiment. These figures illustrate the result when the blurred image 105g is generated by removing the noise from the observed image 105f by a partially coherent imaging simulator, and then the restored image 105l is generated by removing the blur from the blurred image 105g. In addition, in order to compare this embodiment with the prior art, the result of generating the restored image 105l that is made by denoising and deblurring the observed image by the method disclosed in the literature 1 is also illustrated. The simulation condition is the same as that of the first embodiment. In the linear combination coefficient γ, there are about 10 to 15 non-zero elements.
FIG. 10A illustrates an original image corresponding to an object, similar to the original image of the first embodiment.
FIG. 10B illustrates a virtual object image, similar to the observed image of the first embodiment. In other words, a deterioration corresponding to a blur caused by the partially coherent imaging optical system is added to the original image by the partially coherent imaging simulator, and a white Gaussian noise of a standard deviation of 5% corresponding to the noise caused by the image sensor is further added.
FIG. 10C illustrates a blurred image that is made by denoising the observed image in accordance with the flow of FIG. 8.
FIG. 10D illustrates a restored image that is made by deblurring the blurred image in accordance with the flow of FIG. 9.
FIG. 10E illustrates a restored image that is made by denoising and deblurring the observed image by the method disclosed in the literature 1, similar to the restored image of the first embodiment.
Table 2 summarizes a quantitative evaluation result of the image quality. In order to compare the second embodiment with the prior art, the image restoration result made by the method disclosed in the literature 1 is included. The evaluation index uses the PSNR, similar to the first embodiment.











TABLE 2







Denoised 

Restored image




image

made by the



Observed
(blurred
Restored
method of



Image
image)
image
literature 1



















PSNR [dB]
12.17
16.62
19.87
13.71









It is understood from FIG. 10 and Table 2 that this embodiment can generate a higher-quality restored image.
The present invention is applicable to an application of restoring an image obtained by a digital microscope, suitable, for example, for an image obtained by a virtual slide.
While the present invention has been described with reference to exemplary embodiments, it is to be understood that the invention is not limited to the disclosed exemplary embodiments. The scope of the following claims is to be accorded the broadest interpretation so as to encompass all such modifications and equivalent structures and functions.
This application claims the benefit of Japanese Patent Application No. 2012-185318, filed Aug. 24, 2012, which is hereby incorporated by reference herein in its entirety.

## Claims

1. An image restoration method configured to restore an original image from an observed image obtained through a photoelectric conversion of an optical image formed by partially coherent imaging, the image restoration method comprising the steps of:
performing an approximate partially coherent imaging operation for elements of a first basis generated from a model image that is assumed to have no noise and no blur, and generating based upon the first basis a second basis that is blurred, the approximate partially coherent imaging operation being expressed by a convolution integral on an eigenfunction corresponding to a maximum eigenvalue of a Kernel matrix and each element of the first basis;
generating an intermediate image in which each pixel value of the observed image that has been denoised is replaced with its square root; and
obtaining a restored image by approximating each of a plurality of patches that are set to entirely cover the intermediate image, using a linear combination of elements of the first basis and linear combination coefficients obtained when each patch is approximated by a linear combination of the second basis.

2. The image restoration method according to claim 1, further comprising the steps of:
performing a partially coherent imaging operation for the model image and generating a blurred model image;
generating a third basis from the blurred model image; and
generating the observed image denoised through weighted averaging between the observed image and an approximated image obtained by approximating each of the plurality of patches set to entirely cover the observed image using a linear combination of elements of the third basis.

3. The image restoration method according to claim 1, further comprising the steps of:
generating a plurality of artificial random noises; and
generating the observed image that has been denoised by subtracting the random noises from the observed image,
wherein the linear combination coefficient is an average of a plurality of types of linear combination coefficients corresponding to a plurality of types of random noises.

4. The image restoration method according to claim 3, wherein a normal random number is used for the artificial random noises.

5. An image restoration apparatus configured to restore an original image from an observed image obtained through a photoelectric conversion of an optical image formed by partially coherent imaging, the image restoration method comprising the steps of:
a unit configured to perform an approximate partially coherent imaging operation for elements of a first basis generated from a model image that is assumed to have no noise and no blur, and to generate based upon the first basis a second basis that is blurred, the approximate partially coherent imaging operation being expressed by a convolution integral on an eigenfunction corresponding to a maximum eigenvalue of a Kernel matrix and each element of the first basis;
a unit configured to generate an intermediate image in which each pixel value of the observed image that has been denoised is replaced with its square root; and
a unit configured to obtain a restored image by approximating each of a plurality of patches that are set to entirely cover the intermediate image, using a linear combination of elements of the first basis and linear combination coefficients obtained when each patch is approximated by a linear combination of the second basis.

6. A non-transitory tangible medium storing a program that enables a computer to execute an image restoration method configured to restore an original image from an observed image obtained through a photoelectric conversion of an optical image formed by partially coherent imaging, the image restoration method comprising the steps of:
performing an approximate partially coherent imaging operation for elements of a first basis generated from a model image that is assumed to have no noise and no blur, and generating based upon the first basis a second basis that is blurred, the approximate partially coherent imaging operation being expressed by a convolution integral on an eigenfunction corresponding to a maximum eigenvalue of a Kernel matrix and each element of the first basis;
generating an intermediate image in which each pixel value of the observed image that has been denoised is replaced with its square root; and
obtaining a restored image by approximating each of a plurality of patches that are set to entirely cover the intermediate image, using a linear combination of elements of the first basis and linear combination coefficients obtained when each patch is approximated by a linear combination of the second basis.

7. An image-pickup apparatus comprising:
an imaging optical system configured to form an optical image of an object by partially coherent imaging;
an image sensor configured to photoelectrically convert the optical image formed by the imaging optical system; and
a computer configured to restore an original image as the object from an observed image obtained from the image sensor,
wherein the computer
performs an approximate partially coherent imaging operation for elements of a first basis generated from a model image that is assumed to have no noise and no blur, and generates based upon the first basis a second basis that is blurred, the approximate partially coherent imaging operation being expressed by a convolution integral on an eigenfunction corresponding to a maximum eigenvalue of a Kernel matrix and each element of the first basis;
generates an intermediate image in which each pixel value of the observed image that has been denoised is replaced with its square root; and
obtains a restored image by approximating each of a plurality of patches that are set to entirely cover the intermediate image, using a linear combination of elements of the first basis and linear combination coefficients obtained when each patch is approximated by a linear combination of the second basis.