Patent Publication Number: US-9904046-B2

Title: Image pickup apparatus, image pickup system, and image processing method

Description:
TECHNICAL FIELD 
     The present invention relates to an image pickup apparatus as a microscope including an image sensor. 
     BACKGROUND ART 
     As for image data of a pathological specimen (virtual slides), not only horizontal information of the specimen but also its information in the optical axis direction of an imaging optical system (hereinafter the “optical axis direction” for short) is an important material for pathological diagnosis. Accordingly, a conventional method for a microscope changes the focus position in the optical axis direction of the imaging optical system, thereby acquiring a plurality of images (Z-stack images) of the specimen. 
     In order to observe in detail the specimen structure in the optical axis direction, space intervals at which the focus position is changed when acquiring Z-stack images, need to be small enough. Hereinafter, unless otherwise specified, Z-stack images are acquired with different focus positions at equal space intervals, and the interval is called a “focusing step”. The sampling theorem is known as a standard to determine the focus step. When this is applied to the acquisition of Z-stack images, the inverse of the focus step needs to be twice or more the maximum value of the spatial frequency in the optical axis direction that a three-dimensional optical image has (the Nyquist condition). Hereinafter, only the focus step (the sampling interval in the optical axis direction) is subject to the Nyquist condition, and for horizontal directions, which are perpendicular to the optical axis direction, the Nyquist condition is disregarded. The spatial frequency of the optical image is a frequency range in which the spectrum has a non-zero value obtained by performing a discrete Fourier transform for the intensity distribution data of the optical image. If the focusing step is determined according to the Nyquist condition, the focusing step will be a relatively small value on the order of the wavelength of the illumination light, so that the data volume of Z-stack images is enormous. This method results in an increase in the cost of hardware related to the acquisition, processing, and storing of images and an increase in processing time. 
     Subjected to the resolving power defined by the optical system of the microscope, acquired images of a specimen are degraded relative to the actual specimen. In order to restore these degraded images, PLT1 restores an image by image processing without considering the sparseness of a specimen, but the resolution of acquired Z-stack images and the resolution of restored images are the same. As a time period required to acquire Z-stack images is shortened with a larger focusing step, the resolution in the optical axis direction of the specimen degrades accordingly. Further, if the Nyquist condition is not satisfied, aliasing (fold distortion) occurs, resulting in the occurrence of a false pattern in the structure in the optical axis direction in Z-stack images. One solution for this problem is the interpolation that increases the resolution in the optical axis direction of the Z-stack images, but according to the sampling theorem, the correctness of interpolation is not ensured when the Nyquist condition is not satisfied. 
     Accordingly, there has been growing interest in a novel signal processing technique referred to as compressed sensing or compressive sensing in these years. The compressive sensing is a technique which accurately reconstructs information about an object subject to observation from data sampled without the Nyquist condition being satisfied. 
     For example, NPLT1 discloses a method of reconstructing three-dimensional shape information of a specimen from one image by applying the compressive sensing to a hologram. PLT2 discloses a method which, with an improvement in the optical element or image pickup element, generates an image from which the amount of information obtained is not essentially reduced even when the sampling interval is increased, to reconstruct an image higher in resolution than an acquired image (a super-resolution process). PLT3 discloses a method of optimizing an objective function including a noise suppression term and a sparse regularization term in a tomographic image acquiring apparatus such as an MRI (Magnetic Resonance Imaging) for the image reconstruction. This method uses a regularization with the sparseness of a solution as prior information, and provides a highly accurately reconstruction with a reduced number of acquired data if the three-dimensional information of a specimen is sparse (the number of non-zero elements is small). 
     NPLT2 describes imaging by a microscope, and NPLT3 describes the accuracy of reconstruction in the compressive sensing. NPLT4 describes a TwIST algorithm, and NPLT5 describes a weak-object optical transfer function. 
     CITATION LIST 
     Patent Literature 
     
         
         [PTL 1] U.S. Patent Application, Publication No. 2010/0074486 
         [PTL 2] U.S. Pat. No. 7,532,772 
         [PTL 3] U.S. Patent Application, Publication No. 2011/0293158 
       
    
     Non-Patent Literature 
     
         
         [NPLT 1] D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, S. Lim, “Compressive Hoography,” Optics Express, USA, Optical Society of America, 2009, Vol. 17, No. 15, p. 13040-13049 
         [NPLT 2] Y. Sung, C. J. R. Sheppard, “Three-dimensional imaging by partially coherent light under nonparaxial condition,” Journal of the Optical Society of America A, USA, Optical Society of America, 2011, Vol. 28, No. 4, p. 554-559 
         [NPLT 3] D. L. Donoho, M. Elad, V. N. Temlyakov, “Stable recovery of sparse overcomplete representations in the presence of noise,” IEEE Transactions on Information Theory, USA, IEEE, 2006, Vol. 52, Issue 1, p. 6-18 
         [NPLT 4] J. M. Bioucas-Dias, M. A. T. Figueiredo, “A New TwIST: Two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Transactions on Image Processing, USA, IEEE, 2007, Vol. 16, Issue 12, p. 2992-3004 
         [NPLT 5] C. J. R. Sheppard, X. Q. Mao, “Three-dimensional imaging in a microscope,” Journal of the Optical Society of America A, USA, Optical Society of America, 1989, Vol. 6, No. 9, p. 1260-1269 
       
    
     SUMMARY OF INVENTION 
     Technical Problem 
     It is difficult for the method of PLT1 to reduce the number of Z-stack images while maintaining the resolution in the optical axis direction of the specimen. It is in principle unable to obtain a resolution higher than that of the obtained image when the focusing step does not satisfy the Nyquist condition. The method of PLT2 uses the compressive sensing, but does not discuss an image reconstruction method for the Z-stack images. It needs a complicated optical element or image sensor, causing an increase of costs of the apparatus. The methods of PLT3 and NPLT1 are silent about a method of improving the image pickup apparatus so as to improve the precision of the reconstruction. 
     The present invention precisely provides a three-dimensional reconstruction of a specimen with the smaller number of samples in the image pickup apparatus. 
     Solution to Problem 
     An image pickup apparatus according to the present invention includes an illumination optical system configured to illuminate a specimen, an imaging optical system configured to form an optical image of the specimen, a light modulator configured to generate at least one of a transmittance distribution and a phase distribution which are asymmetric with respect to an optical axis on a pupil plane of at least one of the illumination optical system and the imaging optical system, an image sensor configured to photoelectrically convert the optical image of the specimen formed by the imaging optical system, and a driver configured to change a relative position along an optical axis direction of the imaging optical system between a focal plane of the imaging optical system and at least one of the specimen and the image sensor. The driver change the relative position in acquiring a plurality of images of the specimen. 
     Further features and aspects of the present invention will become apparent from the following description of exemplary embodiments with reference to the attached drawings. 
     Advantageous Effects of Invention 
     The present invention precisely provides a three-dimensional reconstruction of a specimen with the smaller number of samples in the image pickup apparatus. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         FIG. 1  is a block diagram illustrating the configuration of an image pickup system according to this embodiment; 
         FIG. 2  is a flow chart for explaining operations of the image pickup system illustrated in  FIG. 1 ; (Example 1) 
         FIGS. 3A-3B  are views illustrating the structure and refractive index distribution of a specimen; (Examples 1, 2) 
         FIGS. 4A-4C  are views illustrating the effective light sources of Numerical Example 1 and a conventional example, and the pupil function of an imaging optical system; (Example 1) 
         FIGS. 5A-5C  are Z-stack images of Numerical Example 1; (Example 1) 
         FIGS. 6A-6B  are views illustrating the specimen reconstructed in Numerical Example 1; (Example 1) 
         FIGS. 7A-7C  are views illustrating the effective light source of Numerical Example 2 and the pupil function of an imaging optical system; (Example 2) 
         FIGS. 8A-8C  are Z-stack images of Numerical Example 2; (Example 2) 
         FIGS. 9A-9B  are views illustrating the specimen reconstructed in Numerical Example 2; (Example 2) 
         FIGS. 10A-10B  are views illustrating the structure and light emission intensity distribution of a specimen; (Example 3) 
         FIG. 11  is a view illustrating the pupil function of an imaging optical system of Numerical Example 3; (Example 3) 
         FIGS. 12A-12C  are Z-stack images of Numerical Example 3; (Example 3) and 
         FIGS. 13A-13B  are views illustrating the specimen reconstructed in Numerical Example 3. (Example 3) 
     
    
    
     DESCRIPTION OF EMBODIMENTS 
       FIG. 1  is a block diagram illustrating the configuration of an image pickup system according to this embodiment. The present invention can be embodied as an image pickup apparatus, and may also be embodied as an image pickup system that is a combination of an image pickup apparatus and a computer (and further a display unit). Such an image pickup apparatus or image pickup system is suitable for a virtual slide generating system or a digital microscope and very useful for applications such as the pathological diagnosis. 
     As illustrated in  FIG. 1 , the image pickup system of this embodiment includes an image pickup apparatus  10  and a computer (PC)  503 . A display unit  504  and an input unit  505  are connected to the PC  503 . A storage unit  501  and an operating unit  502  are connected to the image pickup apparatus  10  or the PC  503 . The system configuration of  FIG. 1  is merely illustrative, and the image pickup apparatus  10  and the PC  503  may be integrally combined, so that the image pickup apparatus includes an operating unit. Any or all of the storage unit  501 , the operating unit  502 , the display unit  504 , and the input unit  505  may be integrally combined with the image pickup apparatus  10  or the PC  503 . 
     As illustrated in  FIG. 1 , the image pickup apparatus  10  includes an illumination optical system  100 , a specimen stage  201 , an imaging optical system  300 , an image sensor  401 , and the like. The image pickup apparatus  10  may include a bright field microscope and a fluorescence microscope. 
     The illumination optical system  100  includes a light source  101 , an illumination light modulator  102 , and an optical system  103  and illuminates a specimen. 
     When a specimen is bright field observed, for example, a halogen lamp or an LED (Light Emitting Diode) is used as the light source  101 . When a specimen is fluorescence observed, instead of an LED, a xenon lamp or a laser light source may be used. 
     The illumination light modulator  102  adjusts at least one of the transmittance distribution (intensity distribution of transmitted light) and phase distribution on the pupil plane of the illumination optical system  100 . The illumination light modulator  102  may adjust at least one of the transmittance distribution (intensity distribution of transmitted light) and phase distribution on the pupil plane of the illumination optical system  100  each time a driver changes a relative position. The illumination light modulator  102  fixes an effective light source, or the intensity distribution of illumination light on a specimen, at a desired distribution or changes it freely, and, for example, a variable diaphragm, a light shield having an opening, a spatial light modulator, or the like placed close to the pupil plane of the illumination optical system can be used as the illumination light modulator  102 . 
     The effective light source is an image, of emitted light from the illumination light modulator  102 , formed on the pupil plane of the imaging optical system in the bright field microscope when there is no specimen, and it is commonly known that the distribution of the effective light source affects the resolving power and the depth of focus. The illumination light modulator  102  is optically conjugate to the pupil plane of the imaging optical system. Although the illumination light modulator  102  of  FIG. 1  is shown as a light transmitting element, it may be a reflective element such as a DMD (digital micro-mirror device). 
     The emitted light from the illumination optical system  100  is led to a specimen  202  mounted on the specimen stage  201 . The specimen stage  201  may cooperate with a mechanism (not shown) to put out the specimen  202  after observed and a mechanism (not shown) to take a specimen  202  to be observed next out of a specimen holder such as a cassette and to feed it in so as to automatically observe sequentially a plurality of specimens. However, the specimen  202  does not necessarily need to be automatically replaced, but may be replaced manually. Further, the specimen stage  201  may include a driving unit to displace the specimen  202  minutely in the optical axis direction of the imaging optical system  300 . By operating this driving unit to acquire a plurality of images, Z-stack images can be obtained. The specimen  202  may be a preparation in which an object subject to observation such as a piece of tissue is mounted on a slide glass and covered with a light transmitting cover glass to be fixed. 
     The imaging optical system  300  includes a light modulator  301  and optical elements (lenses)  302  and forms an optical image of a specimen. An element similar to that of the illumination light modulator  102  is used for the light modulator  301 , and the light modulator  301  fixes the intensity distribution, phase distribution, or both of the emitted light at a desired distribution or changes it freely.  FIG. 1  is a schematic diagram for where a bright field microscope is used, but not being limited to this, a fluorescence microscope may be used, or an imaging optical system of a scheme such as a phase contrast method or a differential interference method can be used. 
     The image sensor  401  photoelectrically converts the optical image of the specimen  202  formed on the image plane by the imaging optical system  300  and transmits the resulting signal. There may be provided a driving unit to displace the image sensor  401  minutely in the optical axis direction of the imaging optical system  300 . By operating this driving unit to acquire a plurality of images, Z-stack images can be obtained. The image sensor  401  is connected to the PC  503  or the storage unit  501  and the operating unit  502  in such a way that it can transmit signals. 
     Thus, the image pickup apparatus  10  includes a driver to change the relative position in the optical axis direction between the focal plane of the imaging optical system  300  and at least one of the specimen  202  and the image sensor  401  and changes the relative position by the driver to acquire a plurality of images of the specimen  202 . 
     When the three-dimensional information of the specimen  202  is not reconstructed immediately after the image acquisition, the image data is transmitted from the image sensor  401  to the PC  503  or the storage unit  501  and stored there. The three-dimensional information is, for example, data representing three-dimensional distributions of physical quantities such as the refractive index, extinction coefficient, and fluorescent intensity of the specimen  202 . When the reconstruction follows the image acquisition immediately, the image data is transmitted to the PC  503  or the operating unit  502 , and the three-dimensional information of the specimen  202  is reconstructed. In accordance with a user&#39;s instruction via the input unit  505  or information stored in the PC  503 , one or both of the displaying process on the display unit  504  and the transmitting process to the PC  503  or the storage unit  501  for storage are performed for the reconstructed data. 
     Note that all the modules except for the image pickup apparatus  10  in  FIG. 1  do not necessarily need to be directly connected to the image pickup apparatus  10 . For example, the image pickup apparatus  10  may be connected to a remote server via a LAN (Local Area Network) or a cloud computing service. In this case, the entities of the modules except for the image pickup apparatus  10  exist on the remote server. Advantageously, the reconstruction can be provided even if the image pickup apparatus  10  and peripheral devices such as the operating unit  502  cannot be integrated because of the restrictions in placement and costs. In addition, the operating unit of the latest performance can be always used, and data can be shared between remote sites. 
     A description will now be given of the reconstruction of the three-dimensional information of the specimen  202  by the PC  503  or the operating unit  502 . The operating unit  502  performs a compressive sensing reconstruction algorithm for Z-stack images to reconstruct information of the specimen that has a greater number of data elements than the number of all pixels of the Z-stack images. 
     Although it is now contemplated that Z-stack images are acquired by displacing the image sensor  401  by a micro distance in the optical axis direction of the imaging optical system.  300 , the method may displace the specimen stage  201  by the micro distance. In defining the coordinate system of the image plane, one coordinate axis (here Z axis) in the three-dimensional orthogonal coordinate system is set parallel to the optical axis direction of the imaging optical system  300 , Z=0 is set to the position conjugate to the surface of the specimen, and the positive direction is set to the direction from the image plane to the specimen. 
     A description will now be given of the formulation of imaging by a microscope in the reconstruction process. In a bright field microscope, strictly speaking, a relationship between the specimen and the optical image is non-linear, due to a partially coherent imaging system. However, if the specimen is close to transparent, so that diffracted light fluxes other than the 0-th order are relatively faint as compared with the 0-th order diffracted light (straightforward traveling light), then the linearity can be approximated by ignoring the interferences among the diffracted light fluxes other than the 0-th order. Based upon this assumption, a two-dimensional optical image I(X, Y, Z) on an XY plane of which the Z coordinate is Z is expressed by expressions (1) and (2).
 
 I ( X,Y,Z )= I   0 +Re[α∫∫∫ C   1 ( f,g,h )FT[ T ( x,y,z )]exp { i 2π( fX+gY+hZ )} df dg dh]   (1)
 
 T ( x,y,z )=1+ i π( n   0   2   −n ( x,y,z ) 2 )  (2)
 
     Herein, n 0  is a background refractive index, n(x, y, z) is a complex refractive index at a coordinate (x, y, z) inside the specimen, FT is a three-dimensional Fourier transform, (f, g, h) is a three-dimensional coordinate in a frequency space, (X, Y, Z) is a three-dimensional coordinate near the image plane, and Re is an operation outputting a real part of a complex number. The background refractive index is the refractive index of light transmitting material uniformly filling the space between structures. In the case of a pathological specimen, the refractive index of a light transmitting material uniformly filled in the space between the cover glass and the slide glass, such as an encapsulant or an intercellular substance inside a piece of tissue. The real part of the complex refractive index denotes a refractive index, the imaginary part denotes an extinction coefficient, and T is commonly called a scattering potential. C 1 (f, g, h) is a three-dimensional function in the frequency space, referred to as a weak-object optical transfer function, and is uniquely determined from the effective light source and the pupil function of the imaging optical system  300 . The effective light source is an intensity distribution defined in a two-dimensional coordinate plane of f, g, and the pupil function is a complex transmittance distribution defined in the two-dimensional coordinate plane of f, g. The absolute value of the complex value of each point of the pupil function denotes amplitude transmittance, and the argument thereof denotes the relative variation amount of phase in transmitted light. I 0  is a constant representing background light having a uniform intensity over the entire surface, and α is a constant determined from the product of the sampling interval of (f, g, h) and the complex conjugate of the value at the origin of the three-dimensional Fourier spectrum of T. NPLT2 describes the expression (1) in more detail. 
     In order to enable numerical calculation, (f, g, h) is set to discrete coordinates spaced at regular intervals, the triple integral in the expression (1) is rewritten as a sum, and FT is defined as a three-dimensional discrete Fourier transform. As a result, the expression (1) is rewritten using the product of matrices and a vector as illustrated in expressions (3) to (7). 
     
       
         
           
             
               
                 
                   
                       
                   
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                                         ⁢ 
                                         
                                           X 
                                           Nx 
                                         
                                       
                                       + 
                                       
                                         
                                           g 
                                           Ny 
                                         
                                         ⁢ 
                                         
                                           Y 
                                           Ny 
                                         
                                       
                                       + 
                                       
                                         
                                           h 
                                           k 
                                         
                                         ⁢ 
                                         
                                           Z 
                                           j 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 } 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     In the three coordinate systems (x, y, z), (X, Y, Z), (f, g, h), assume that N x ×N y ×N z  is the number of coordinate points used to calculate the expression (1). For example, using N x  coordinate points {f 1 , f 2 , . . . , f Nx } on the f axis, the matrices of the expressions (5) to (7) are formed. Further, assume that {Z 1 , Z 2 , . . . , Z N } are individual Z coordinate points used to acquire N Z-stack images and Z in the expression (1) is set to one of them. 
     I in the expression (3) is an (N x ×N y ×N)×1 vector (an M×1 vector obtained by combining a plurality of images and subtracting a constant) containing the luminance values of all the pixels of Z-stack images determined by the expression (1) for these Z&#39;s. “n” is an (N x ×N y ×N z )×1 vector (an N×1 vector representing the three-dimensional refractive index distribution of the specimen  202 ), and • is an operator for taking the element-wise product of matrices (the Hadamard product). ΦB is an (N x ×N y ×N)×(N x ×N y ×N z ) complex matrix (an M×N matrix). T is an (N x ×N y ×N z )×1 vector (an N×1 vector representing three-dimensional information of the specimen  202 ) defined by the expression (4). 1 is an (N x ×N y ×N z )×1 vector having all elements of 1. 
     As understood from the expression (3), there is a linear relationship between the scattering potential T that is three-dimensional information of the specimen and the Z-stack images I. The linear relationship between observed data and the reconstruction object and the smaller number of elements of observed data than the number of elements of the reconstruction object can provide an accurate reconstruction of specimen information only when compressive sensing is applied. The number of elements refers to, but is not limited to, the number of numerical values in the data or the dimension number of the vector, such as the number of pixels of an image or the number of values representing the refractive index distribution of a specimen. Of course, the reconstruction can be provided where the number of elements of observed data is the same as or larger than the number of elements in the reconstruction object. 
     The fluorescence microscope is known as incoherent imaging, and a two-dimensional optical image I(X, Y, Z) on an XY plane with the Z coordinate of Z is expressed by expression (8).
 
 I ( X,Y,Z )=∫∫∫OTF( f,g,h )FT[ O ( x,y,z )]exp{ i 2π( fX+gY+hZ )} df dg dh   (8)
 
     OTF is an optical transfer function of the imaging optical system  300 , and O denotes a fluorescence intensity distribution of the specimen. 
     Similar to the bright field microscope, the right side of the expression (8) can be written as the product of matrices and a vector as illustrated in expressions (9) to (13), and on the basis of this linearity, compressive sensing can be applied. 
     
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     I 
                     = 
                     
                       
                         Φ 
                         F 
                       
                       ⁢ 
                       O 
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     O 
                     = 
                     
                       ( 
                       
                         
                           
                             
                               O 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     x 
                                     1 
                                   
                                   , 
                                   
                                     y 
                                     1 
                                   
                                   , 
                                   
                                     z 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         
                           
                             ⋮ 
                           
                         
                         
                           
                             
                               O 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     x 
                                     Nx 
                                   
                                   , 
                                   
                                     y 
                                     Ny 
                                   
                                   , 
                                   
                                     z 
                                     Nz 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
             
               
                 
                   
                     Φ 
                     F 
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             
                               Φ 
                               11 
                             
                           
                           
                             … 
                           
                           
                             
                               Φ 
                               
                                 1 
                                 ⁢ 
                                 Nz 
                               
                             
                           
                         
                         
                           
                             ⋮ 
                           
                           
                             ⋱ 
                           
                           
                             ⋮ 
                           
                         
                         
                           
                             
                               Φ 
                               
                                 N 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                               
                             
                           
                           
                             … 
                           
                           
                             
                               Φ 
                               NNz 
                             
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               OTF 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     f 
                                     1 
                                   
                                   , 
                                   
                                     g 
                                     1 
                                   
                                   , 
                                   
                                     h 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             
                                 
                             
                           
                           
                             0 
                           
                         
                         
                           
                             
                                 
                             
                           
                           
                             ⋱ 
                           
                           
                             
                                 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                                 
                             
                           
                           
                             
                               OTF 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     f 
                                     Nx 
                                   
                                   , 
                                   
                                     g 
                                     Ny 
                                   
                                   , 
                                   
                                     h 
                                     Nz 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     F 
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
             
               
                 
                   F 
                   = 
                   
                     ( 
                     
                       
                         
                           
                             exp 
                             ⁢ 
                             
                               { 
                               
                                 - 
                                 
                                   ⅈ2π 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         
                                           f 
                                           1 
                                         
                                         ⁢ 
                                         
                                           x 
                                           1 
                                         
                                       
                                       + 
                                       
                                         
                                           g 
                                           1 
                                         
                                         ⁢ 
                                         
                                           y 
                                           1 
                                         
                                       
                                       + 
                                       
                                         
                                           h 
                                           1 
                                         
                                         ⁢ 
                                         
                                           z 
                                           1 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                               } 
                             
                           
                         
                         
                           … 
                         
                         
                           
                             exp 
                             ⁢ 
                             
                               { 
                               
                                 - 
                                 
                                   ⅈ2π 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         
                                           f 
                                           1 
                                         
                                         ⁢ 
                                         
                                           x 
                                           Nx 
                                         
                                       
                                       + 
                                       
                                         
                                           g 
                                           1 
                                         
                                         ⁢ 
                                         
                                           y 
                                           Ny 
                                         
                                       
                                       + 
                                       
                                         
                                           h 
                                           1 
                                         
                                         ⁢ 
                                         
                                           z 
                                           Nz 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                               } 
                             
                           
                         
                       
                       
                         
                           ⋮ 
                         
                         
                           ⋱ 
                         
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             exp 
                             ⁢ 
                             
                               { 
                               
                                 - 
                                 
                                   ⅈ2π 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         
                                           f 
                                           Nx 
                                         
                                         ⁢ 
                                         
                                           x 
                                           1 
                                         
                                       
                                       + 
                                       
                                         
                                           g 
                                           Ny 
                                         
                                         ⁢ 
                                         
                                           y 
                                           1 
                                         
                                       
                                       + 
                                       
                                         
                                           h 
                                           Nz 
                                         
                                         ⁢ 
                                         
                                           z 
                                           1 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                               } 
                             
                           
                         
                         
                           … 
                         
                         
                           
                             exp 
                             ⁢ 
                             
                               { 
                               
                                 - 
                                 
                                   ⅈ2π 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         
                                           f 
                                           Nx 
                                         
                                         ⁢ 
                                         
                                           x 
                                           Nx 
                                         
                                       
                                       + 
                                       
                                         
                                           g 
                                           Ny 
                                         
                                         ⁢ 
                                         
                                           y 
                                           Ny 
                                         
                                       
                                       + 
                                       
                                         
                                           h 
                                           Nz 
                                         
                                         ⁢ 
                                         
                                           z 
                                           Nz 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                               } 
                             
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
             
               
                 
                   
                     Φ 
                     jk 
                   
                   = 
                   
                     
                       1 
                       
                         
                           N 
                           x 
                         
                         ⁢ 
                         
                           N 
                           y 
                         
                         ⁢ 
                         
                           N 
                           z 
                         
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               exp 
                               ⁢ 
                               
                                 { 
                                 
                                   ⅈ2π 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         
                                           f 
                                           1 
                                         
                                         ⁢ 
                                         
                                           X 
                                           1 
                                         
                                       
                                       + 
                                       
                                         
                                           g 
                                           1 
                                         
                                         ⁢ 
                                         
                                           Y 
                                           1 
                                         
                                       
                                       + 
                                       
                                         
                                           h 
                                           k 
                                         
                                         ⁢ 
                                         
                                           Z 
                                           j 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 } 
                               
                             
                           
                           
                             … 
                           
                           
                             
                               exp 
                               ⁢ 
                               
                                 { 
                                 
                                   ⅈ2π 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         
                                           f 
                                           Nx 
                                         
                                         ⁢ 
                                         
                                           X 
                                           1 
                                         
                                       
                                       + 
                                       
                                         
                                           g 
                                           Ny 
                                         
                                         ⁢ 
                                         
                                           Y 
                                           1 
                                         
                                       
                                       + 
                                       
                                         
                                           h 
                                           k 
                                         
                                         ⁢ 
                                         
                                           Z 
                                           j 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 } 
                               
                             
                           
                         
                         
                           
                             ⋮ 
                           
                           
                             
                                 
                             
                           
                           
                             ⋮ 
                           
                         
                         
                           
                             
                               exp 
                               ⁢ 
                               
                                 { 
                                 
                                   ⅈ2π 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         
                                           f 
                                           1 
                                         
                                         ⁢ 
                                         
                                           X 
                                           Nx 
                                         
                                       
                                       + 
                                       
                                         
                                           g 
                                           1 
                                         
                                         ⁢ 
                                         
                                           Y 
                                           Ny 
                                         
                                       
                                       + 
                                       
                                         
                                           h 
                                           k 
                                         
                                         ⁢ 
                                         
                                           Z 
                                           j 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 } 
                               
                             
                           
                           
                             … 
                           
                           
                             
                               exp 
                               ⁢ 
                               
                                 { 
                                 
                                   ⅈ2π 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         
                                           f 
                                           Nx 
                                         
                                         ⁢ 
                                         
                                           X 
                                           Nx 
                                         
                                       
                                       + 
                                       
                                         
                                           g 
                                           Ny 
                                         
                                         ⁢ 
                                         
                                           Y 
                                           Ny 
                                         
                                       
                                       + 
                                       
                                         
                                           h 
                                           k 
                                         
                                         ⁢ 
                                         
                                           Z 
                                           j 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 } 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     A description will now be given of a method of reconstructing a three-dimensional distribution of the scattering potential T or the fluorescence intensity distribution O based on the above imaging expression. 
     In the compressive sensing, observed data is acquired via an encoder as needed, and a reconstruction algorithm follows which estimates a sparse solution based on the observed data. This method provides highly accurately reconstructed data having a larger number of elements than the observed data. This reconstruction algorithm will be hereinafter referred to as a “compressive sensing reconstruction algorithm” or simply a “reconstruction algorithm.” 
     For a successful reconstruction, data as a reconstruction object is desirably sparse. The bright field microscope provides encoding by giving an appropriate transmittance or phase modulation to one or both of the illumination light modulator  102  and the light modulator  301 . For example, a highly accurate reconstruction is provided by generating at least one of a transmittance distribution and a phase distribution which are asymmetric with respect to the optical axis on the pupil plane of at least one of the illumination light modulator  102  and the light modulator  301 . 
     In general, the purpose of encoding in the compressive sensing is to obtain observed data that theoretically guarantees the reconstruction accuracy, and it is a conventional method to provide a modulation based on a Gaussian random number. However, for images captured via an optical system, the reconstruction of three-dimensional information of the specimen does not become sufficiently accurate only by providing a modulation based on a Gaussian random number to the amplitude or phase of transmitted light on the pupil plane of the imaging optical system. As evident from the embodiments, the reconstruction accuracy can be improved by providing an asymmetric transmittance or phase distribution as discussed above. 
     A description will now be given of the reason why a transmittance or phase distribution asymmetric with respect to the optical axis improves the reconstruction accuracy. Assume that φ i  is an i′-th column vector of Φ B  or Φ F , it is generally known that the reconstruction accuracy in the compressive sensing is determined by the coherence μ defined by expression (14). 
     
       
         
           
             
               
                 
                   μ 
                   = 
                   
                     
                       max 
                       
                         
                           i 
                           &lt; 
                           j 
                         
                         ∈ 
                         
                           { 
                           
                             1 
                             , 
                             
                                 
                             
                             ⁢ 
                             … 
                             ⁢ 
                             
                                 
                             
                             , 
                             
                               N 
                               c 
                             
                           
                           } 
                         
                       
                     
                     ⁢ 
                     
                       
                          
                         
                           〈 
                           
                             
                               ϕ 
                               i 
                             
                             , 
                             
                               ϕ 
                               j 
                             
                           
                           〉 
                         
                          
                       
                       
                         
                           
                              
                             
                               ϕ 
                               i 
                             
                              
                           
                           2 
                         
                         ⁢ 
                         
                           
                              
                             
                               ϕ 
                               j 
                             
                              
                           
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     Herein, N C  is the number of columns of Φ B  or Φ F , &lt; &gt; is the inner product, and the denominator on the right side denotes an L2 norm product of the two column vectors. The coherence μ means the maximum value of the degree of correlation between columns of a matrix representing the observation process. The coherence μ has a positive real value, and it is said that the closer to zero this value is, the higher the reconstruction accuracy becomes in the compressive sensing. The above asymmetric transmittance distribution consequently reduces the coherence of Φ B  or Φ F  and thus can improve the reconstruction accuracy in the compressive sensing. NPLT3 describes the coherence in more detail. The coherence defined by the expression (14) is irrelevant to the optical coherence. 
     This reconstruction algorithm numerically solves a minimization problem expressed in expression (15) based on matrix Φ describing the linear relationship between observed data and specimen information, defined by the expression (5) or (11), and on observed data I. 
     
       
         
           
             
               
                 
                   
                     θ 
                     ^ 
                   
                   = 
                   
                     
                       
                         
                           arg 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           min 
                         
                         θ 
                       
                       ⁢ 
                       
                         
                            
                           
                             I 
                             - 
                             
                               
                                 Φ 
                                 X 
                               
                               ⁢ 
                               θ 
                             
                           
                            
                         
                         2 
                       
                     
                     + 
                     
                       λΨ 
                       ⁡ 
                       
                         ( 
                         θ 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     Herein, argmin is an operation that outputs a value of a variable at which a function on the right side has a minimum value, and θ on the left side is an estimated value of the solution that minimizes the function on the right side. The first term on the right side denotes an L2 norm of an estimated residue; Φ X  is Φ B  or Φ F  mentioned above; and θ is T or O mentioned above. The second term on the right side is referred to as a regularization term, and generally uses a function that has a characteristic where the sparser θ is, the smaller value it takes on for the compressive sensing. λ is a constant called a regularization parameter, but its value may be changed in the iterative reconstruction process described later. This regularization term is a critical factor for reconstructing sparse θ highly accurately. Although the L1 norm of θ is typical as Ψ, the TV (Total Variation) norm defined by expression (16) or the function defined by expressions (17) to (19), for example, can also provide the similar effects. 
     
       
         
           
             
               
                 
                   
                     Ψ 
                     ⁡ 
                     
                       ( 
                       θ 
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       n 
                     
                     ⁢ 
                     
                       
                         
                           
                             ( 
                             
                               
                                 
                                   ∂ 
                                   θ 
                                 
                                 
                                   ∂ 
                                   x 
                                 
                               
                               ⁢ 
                               
                                 ❘ 
                                 n 
                               
                             
                             ) 
                           
                           2 
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 
                                   ∂ 
                                   θ 
                                 
                                 
                                   ∂ 
                                   y 
                                 
                               
                               ⁢ 
                               
                                 ❘ 
                                 n 
                               
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
             
               
                 
                   
                     Ψ 
                     ⁡ 
                     
                       ( 
                       θ 
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       n 
                     
                     ⁢ 
                     
                       log 
                       ⁡ 
                       
                         ( 
                         
                           
                             
                                
                               
                                 θ 
                                 n 
                               
                                
                             
                             σ 
                           
                           + 
                           1 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
             
               
                 
                   
                     Ψ 
                     ⁡ 
                     
                       ( 
                       θ 
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       n 
                     
                     ⁢ 
                     
                       ( 
                       
                         1 
                         - 
                         
                           exp 
                           ⁢ 
                           
                             { 
                             
                               - 
                               
                                 
                                    
                                   
                                     θ 
                                     n 
                                   
                                    
                                 
                                 σ 
                               
                             
                             } 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
             
               
                 
                   
                     Ψ 
                     ⁡ 
                     
                       ( 
                       θ 
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       n 
                     
                     ⁢ 
                     
                       
                          
                         
                           θ 
                           n 
                         
                          
                       
                       
                         
                            
                           
                             θ 
                             n 
                           
                            
                         
                         + 
                         σ 
                       
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
     Herein, the subscript n of θ indicates a value at a coordinate of an n-th sampling point among N x ×N y ×N z  number in the specimen. σ is a constant and preset to an appropriate value for the object. 
     The TwIST algorithm disclosed in NPLT3, for example, may be used for an algorithm to solve the minimization problem expressed in expression (15). NPLT4 describes the TwIST algorithm in detail. However, the applicable algorithm is not limited, and may be a general algorithm having the purpose of estimating a sparse solution from data obtained through linear observation. Further, the object function is not limited to the expression (15), and an object function having a plurality of regularization terms for example, may be used. 
     First Embodiment 
     Referring now to  FIG. 2 , a description will be given of an operation of the acquisition of images of the specimen  202  and the reconstruction of three-dimensional information in the image pickup system illustrated in  FIG. 1 . In the first embodiment, the image pickup apparatus  10  is a bright field microscope in which the optical characteristic of the illumination light modulator  102  is suitably adjusted to the compressive sensing.  FIG. 2  is a flowchart of a processing procedure according to the first embodiment, and “S” stands for the “step.” The flowchart illustrated in  FIG. 2  can be implemented as a program that enables a computer to realize the function of each step. 
     In S 201 , the specimen  202  is mounted on the specimen stage  201 . For example, an automatic transporter in association with the specimen stage  201  picks up the specimen  202  out of a specimen holder such as a cassette and mounts it onto the specimen stage  201 . This mounting may be performed not automatically but manually by a user. 
     In S 202 , the relative position between the focal plane of the imaging optical system  300  and the specimen or the image sensor is adjusted. For example, when the specimen stage  201  includes an accompanying driving unit, the specimen stage  201  is displaced by a micro distance as needed so that the position of the specimen  202  in the optical axis direction of the imaging optical system  300  takes on a set value. Instead, a driving unit attached to the image sensor  401  may minutely displace the image sensor  401  in the optical axis direction of the imaging optical system  300 . Alternatively, the relative position of the focal plane to the specimen may be changed by driving a lens of the imaging optical system  300 . 
     In S 203 , images are acquired. More specifically, while the illumination optical system  100  guides light emitted from the light source  101  to the specimen  202 , the image sensor  401  acquires images of the specimen  202  via the imaging optical system  300 . 
     When the transmittance distribution of the illumination light modulator  102  placed near the pupil plane of the illumination optical system  100  is variable and the current state of the illumination light modulator  102  is different from the capturing setting, the illumination light modulator  102  is adjusted to a specified state before S 203 . 
     The transmittance distribution of the illumination light modulator  102  may be, for example, asymmetric with respect to the optical axis and the aperture area may be smaller than that of the usual capturing condition in which no compressive sensing is performed. When the image pickup in which compressive sensing is performed follows the usual image pickup in which no compressive sensing is performed, the process proceeds to S 203  after the optical characteristic of the illumination light modulator  102  is changed to a state suitable for the compressive sensing. 
     In the compressive sensing, when the light modulator  301  is placed near the pupil plane of the imaging optical system  300 , the optical characteristic thereof may be changed at the same time as that of the illumination light modulator  102 . 
     After S 203 , if all Z-stack images in the specified condition are not acquired (NO at S 204 ), then S 202  and S 203  are repeated until all Z-stack images in the specified condition are completely acquired. 
     Image data acquired in all given conditions (YES at S 204 ) may be temporarily stored in the storage unit  501  or another storage device (not sown) or may be immediately transmitted from the image sensor  401  to the PC  503  or the operating unit  502 . 
     In reconstructing the three-dimensional information of the specimen  202  (YES at S 205 ), in S 206  the acquired Z-stack images are transmitted to the PC  503  or the operating unit  502 . If reconstruction is not to be performed (NO at S 205 ), in S 208  the Z-stack images are transmitted to the storage unit  501  or the PC  503  to be stored. 
     In S 207 , the PC  503  or the operating unit  502  performs the reconstruction processing of the three-dimensional information of the specimen  202  based on the acquired image data and information about image pickup conditions. This process may be executed over a plurality of distributed operating units in parallel or may be executed on an operating unit (not shown) connected via a network. 
     According to this embodiment, the pupil plane of at least one of the illumination optical system and the imaging optical system has at least one of the transmittance distribution (intensity distribution of transmitted light) and the phase distribution which are asymmetric with respect to the optical axis so as to perform the compressive sensing reconstruction algorithm for a plurality of images. Thus, the necessary number of images in S 202  and S 203  becomes less than that in not applying this embodiment. 
     In S 209  subsequent to S 208 , the reconstructed three-dimensional information of the specimen  202  is displayed on the display unit  504  in accordance with an instruction from a user or the advance setting. For this display, the reconstructed three-dimensional information is converted into volume data that is a set of color or gray-scaled images. A cross-section of the volume data may be displayed, or a three-dimensional view of the volume data may be displayed by using a rendering function of the PC  503  after it is partially made transparent or separated into a displayed region and a non-displayed region using threshold processing. 
     In the above procedure, there may be a time lag between capturing and reconstruction processing. For example, the following use case can be expected. Z-stack images acquired by capturing for the compressive sensing for many specimens are stored in the storage unit  501  or the like. Thereafter, the user reconstructs only data of a selected specimen as needed. This method suppresses the data capacity to be stored for a long time, enables a provision of high resolution data of all the specimens, reducing the cost related to data storage. 
     Next follows a numerical example of the first embodiment. 
     Numerical Example 1 
     Assume that a wavelength of illumination light irradiated onto the specimen from the illumination optical system  100  is 550 nm, a numerical aperture on the specimen side of the imaging optical system  300  is 0.7, and a magnification is 1.0 times for description convenience. Also, assume that the specimen is a group of particles randomly positioned as illustrated in  FIG. 3A .  FIG. 3A  illustrates a ground truth three-dimensional shape of the specimen. Assume that a particle has a radius of 0.25 μm, a refractive index is 1.1, a background refractive index near the particles is 1.0, and a ratio of volume occupied by all the particles to the space is 2.0%.  FIG. 3B  illustrates a refractive index distribution on the section of y=0 illustrated in  FIG. 3A . 
     A depth of focus D on the specimen side of the imaging optical system  300  is expressed by expression (20), and the value of D is 0.96 μm under the above conditions. 
     
       
         
           
             
               
                 
                   D 
                   = 
                   
                     λ 
                     
                       2 
                       ⁢ 
                       
                         ( 
                         
                           1 
                           - 
                           
                             
                               1 
                               - 
                               
                                 NA 
                                 2 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
     Herein, λ is a wavelength of the illumination light, NA denotes a numerical aperture on the specimen side of the imaging optical system  300 . The depth of focus on the image side is obtained by setting the numerical aperture on the image side to NA in the expression (20). The width along the spatial frequency coordinate h direction of a region where the weak-object optical transfer function has a non-zero value coincides with the inverse of D. The reason for this is described in detail in, e.g., NPLT5. 
     According to the sampling theorem, it is impossible to accurately reconstruct information of the specimen in the optical axis direction with a resolution equal to or higher than that of the acquisition unless Z-stack images are acquired at intervals smaller than this depth of focus Din the optical axis direction. In the numerical example 1, a variation amount of the relative position of the specimen is larger than the depth of focus D. The image acquisition interval in the Z axis direction or the focusing step being 0.96 μm or less is referred to as the Nyquist condition. 
     When the image sensor  401  is driven, the depth of focus D′ on the image side of the imaging optical system  300  is expressed by expression (21), and in the numerical example 1 the variation amount of the relative position of the image sensor is higher than the depth of focus D′. Herein, NA′ denotes a numerical aperture on the image side of the imaging optical system. 
     
       
         
           
             
               
                 
                   
                     D 
                     ′ 
                   
                   = 
                   
                     λ 
                     
                       2 
                       ⁢ 
                       
                         ( 
                         
                           1 
                           - 
                           
                             
                               1 
                               - 
                               
                                 NA 
                                 ′2 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     Assume that Z coordinate points at which Z-stack images are acquired be three points of Z=±1.1 μm, Z=0.0 μm. In other words, three sections through the points of Z=±1.1 μm, Z=0.0 μm and orthogonal to the Z axis are acquired with the image sensor  401  from the three-dimensional light intensity distribution of the specimen  202  formed near the image plane. Although this embodiment discloses the image processing method of acquiring Z-stack images with the effective light source and the pupil function are fixed, the effective light source, the pupil function, or both may be changed each time each of Z-stack images is acquired. 
       FIG. 4A  illustrates the effective light source distribution on the pupil plane of the illumination optical system  100  in the numerical example 1.  FIG. 4B  illustrates the effective light source distribution on the pupil plane of the illumination optical system  100  in a conventional example.  FIG. 4C  illustrates the transmittance distribution on the pupil plane of the imaging optical system  300  in the numerical example 1. The effective light source, with the white portion illustrated in  FIG. 4A  emitting light with uniform intensity, has a transmittance distribution with a monopole at a position offset from the optical axis. The area of the monopole is equal to or narrower than a circle having a diameter d, and the value of d is 1% to 20% of the radius of the pupil. Assume that the pupil function of the imaging optical system  300  provides the white portion illustrated in  FIG. 4C  that transmits light with uniform intensity. The coordinate axes f, g of  FIGS. 4A-4C  represent normalized values obtained by dividing the spatial frequencies along X and Y directions by NA/λ. Although the effective light source illustrated in  FIGS. 4A-4C  is close to coherent illumination because its light emitting portion is relatively small, the light emitting portion of the effective light source can be made larger. 
     The coherence μ of matrix Φ B  with the effective light source of  FIG. 4A  has a value of 0.0048 for the real part of Φ B  and a value of 0.0032 for the imaginary part of Φ B . The coherence μ of matrix Φ B  with the effective light source of  FIG. 4B  has a value of 8.4003 for the real part of Φ B  and a value of 1.1680 for the imaginary part of Φ B . Thus, it can be inferred that the reconstruction accuracy is higher with the effective light source of  FIG. 4A . 
       FIGS. 5A-5C  illustrate three Z-stack images acquired from the three-dimensional light intensity distribution calculated based on the expression (3) under the optical system conditions illustrated in  FIGS. 4A and 4C .  FIG. 5A  illustrates an image of Z=−1.1 μm.  FIG. 5B  illustrates an image of Z=0.0 μm.  FIG. 5C  illustrates an image of Z=+1.1 μm. Hereinafter, the effective light source refers to the intensity distribution of transmitted light emitted from the illumination light modulator  102 . Similarly, the pupil function means a transmittance distribution of the light modulator  301 . 
     A solution of the expression (15) is calculated based on the expression (3) using the TwIST algorithm from these three Z-stack images. The operating unit  502  that performs the reconstruction processing may be integrally combined with the image pickup apparatus  10  or connected to it via a network. 
       FIG. 6A  illustrates a binarized result using 30% of a maximum value as a threshold, the three-dimensional refractive index distribution reconstructed with the regularization term of the expression (15) as an L1 norm and the regularization parameter as 5E-5. This indicates that the three-dimensional shape information of the specimen  202  can be reconstructed from Z-stack images that do not satisfy the Nyquist condition.  FIG. 6B  illustrates a reconstructed refractive index distribution on the section through y=0. For the quantitative evaluation of this, an RMSE (Root Mean Square Error) given by the expression (21) is defined. 
     
       
         
           
             
               
                 
                   RMSE 
                   = 
                   
                     
                       
                         1 
                         
                           
                             N 
                             x 
                           
                           ⁢ 
                           
                             N 
                             y 
                           
                           ⁢ 
                           
                             N 
                             z 
                           
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             x 
                             , 
                             y 
                             , 
                             z 
                           
                         
                         ⁢ 
                         
                           
                              
                             
                               
                                 
                                   n 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     x 
                                     , 
                                     y 
                                     , 
                                     z 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   n 
                                   1 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     x 
                                     , 
                                     y 
                                     , 
                                     z 
                                   
                                   ) 
                                 
                               
                             
                              
                           
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
     Herein, n 1  is a ground truth three-dimensional refractive index distribution of the specimen  202 , and n 2  is a reconstructed refractive index distribution. A value of the RMSE for the reconstructed refractive index distribution in  FIGS. 6A-6B  is 1.09E-2. 
     When the conventional effective light source of the illumination optical system illustrated in  FIG. 4B  and the pupil function of the imaging optical system  300  illustrated in  FIG. 4C  are used, a value of the RMSE is 1.23E-2. Thus, the three-dimensional refractive index distribution of the specimen  202  can be reconstructed within a certain error range from Z-stack images that do not satisfy the Nyquist condition, and the error is reduced by encoding using the effective light source asymmetric with respect to the optical axis. 
     Second Embodiment 
     Referring now to  FIG. 2 , a description will be given of an operation of the acquisition of images of the specimen  202  and the reconstruction of three-dimensional information in the image pickup system illustrated in  FIG. 1 . In the second embodiment, the image pickup apparatus  10  is a bright field microscope in which the optical characteristic of the light modulator  301  is suitably adjusted to the compressive sensing. A description of the same operation as that of the first embodiment will be omitted. 
     After S 201  and S 202  in  FIG. 2 , while the illumination optical system  100  guides light emitted from the light source  101  to the specimen  202 , the image sensor  401  acquires images of the specimen  202  via the imaging optical system  300  (corresponding to S 203  in  FIG. 2 ). Herein, if an optical characteristic (the transmittance or phase modulation amount) of the light modulator  301  placed near the pupil plane of the imaging optical system  300  is variable and the current state of the light modulator  301  is different from the capturing setting, the light modulator  301  is adjusted to a specified state before S 203 . At this time, the transmittance or phase distribution of the light modulator  301  may be asymmetric with respect to the optical axis. When the image pickup in which compressive sensing is performed follows the usual image pickup in which no compressive sensing is performed, the process proceeds to S 203  after the optical characteristic of the light modulator  301  is changed to a state suitable for the compressive sensing. In the compressive sensing, the optical characteristic of the illumination light modulator  102  may be placed near the pupil plane of the illumination optical system  100  and changed at the same time as that of the light modulator  301 . The operation after this is the same as that in the first embodiment. 
     Next follows a numerical example of the second embodiment. 
     Numerical Example 2 
     The wavelength, the numerical aperture, the magnification, the Nyquist condition, the specimen, and Z coordinate points at which Z-stack images are acquired, are the same as those of the first embodiment. In the effective light source, the white portion illustrated in  FIG. 7A  emits light with uniform intensity. Assume that the pupil function of the imaging optical system  300  has the transmittance distribution illustrated in  FIG. 7B  and the phase distribution in radians illustrated in  FIG. 7C . Each point of the phase distribution of  FIG. 7C  is generated independently by normal random numbers with a standard deviation π. Such a pupil of the imaging optical system can be realized by using, for example, a liquid crystal spatial light modulator or the like. The coordinate axes f, g of  FIGS. 7A-7C  represent normalized values obtained by dividing the spatial frequencies along X and Y directions by NA/λ. 
     The coherence μ of matrix Φ B  with the effective light source and the pupil function of  FIGS. 7A-7C  has a value of 0.0083 for the real part of Φ B  and a value of 0.0080 for the imaginary part of Φ B . The coherence μ of matrix Φ B  with the effective light source and the pupil function of  FIGS. 7A-7B  and the phase distribution of  FIG. 7C  replaced with a uniform distribution has a value of 0.1056 for the real part of Φ B  and a value of 0.1016 for the imaginary part of Φ B . Thus, it can be inferred that the reconstruction accuracy is improved by the phase distribution as well as the transmittance distribution of the pupil plane having asymmetry. 
       FIGS. 8A-8C  illustrate three Z-stack images acquired from the three-dimensional light intensity distribution calculated based on the expression (3) under these conditions.  FIG. 8A  illustrates an image of Z=−1.1 μm.  FIG. 8B  illustrates an image of Z=0.0 μm.  FIG. 8C  illustrates an image of Z=+1.1 μm. 
     A solution of the expression (15) is calculated based on the expression (3) using the TwIST algorithm from these three Z-stack images. The expression (15) is the same as that of the first embodiment.  FIG. 9A  illustrates a result of binarizing the reconstructed three-dimensional refractive index distribution using 30% of its maximum value as a threshold. This indicates that the three-dimensional shape information of the specimen  202  can be reconstructed from Z-stack images that do not satisfy the Nyquist condition.  FIG. 9B  illustrates the reconstructed refractive index distribution on the section through y=0. A value of the RMSE for the reconstructed refractive index distribution of  FIGS. 9A-9B  is 1.13E-2. A value of the RMSE for the refractive index distribution reconstructed with the phase distribution of  FIG. 7C  being replaced with a uniform distribution is 1.29E-2. As understood from  FIGS. 9A-9B , the three-dimensional refractive index distribution of the specimen  202  can be reconstructed within a certain error range from Z-stack images not satisfying the Nyquist condition, and further the error is reduced by using the pupil function asymmetric with respect to the optical axis. 
     Third Embodiment 
     A description will now be given of an operation of the acquisition of images of the specimen  202  and the reconstruction of three-dimensional information when the image pickup apparatus  10  illustrated in  FIG. 1  is a fluorescence microscope. A description of the same operation as that of the first embodiment will be omitted. 
     The specimen  202  is dyed for fluorescence observations, and a specific area where fluorescent dye exists emits light in response to the illumination light as described later. 
     In general, the fluorescence microscope does not include an illumination light modulator  102  such as a diaphragm in the illumination optical system  100 . After only the light modulator  301  is adjusted as needed, the image sensor  401  acquires images of the specimen  202  via the imaging optical system  300  (an equivalent to S 203  in  FIG. 2 ). As stated for the first embodiment, the pupil function may be changed each time each of Z-stack images is acquired. 
     Next follows a numerical example of third embodiment. 
     Numerical Example 3 
     Assume that the image pickup apparatus  10  is a fluorescence microscope, the wavelength of monochromatic light emitted by fluorescent dye is 550 nm, the numerical aperture on the specimen side of the imaging optical system  300  is 0.7, and the magnification be 1.0 times for description convenience. The Nyquist condition is the same as that of the first embodiment. Assume that the specimen is a group of particles randomly positioned as illustrated in  FIG. 10A . Assume that the radius of a particle is 0.25 μm and a ratio of volume occupied by all the particles to the space is 2.0%. Only the particles emit light with an intensity of 1 (in an arbitrary unit) and the background behind the particles does not emit light at all.  FIG. 10B  illustrates a light emission intensity distribution on the section through y=0 illustrated in  FIG. 10A . 
     Assume that Z coordinate points at which Z-stack images are acquired are three points of Z=±1.1 μm, Z=0.0 μm. In other words, three sections through the points of Z=±1.1 μm, Z=0.0 μm and orthogonal to the Z axis of the three-dimensional light intensity distribution of the specimen  202  formed near the image plane are acquired with the image sensor  401 . 
       FIGS. 12A-12C  illustrate three Z-stack images acquired from the three-dimensional light intensity distribution calculated based on the expression (9) when the pupil plane of the imaging optical system  300  has the transmittance distribution illustrated in  FIG. 11 .  FIG. 12A  is an image of Z=−1.1 μm.  FIG. 12B  is an image of Z=0.0 μm.  FIG. 12C  is an image of Z=+1.1 μm. In the fluorescence observation, the effective light source does not have an influence due to the incoherent imaging system and this embodiment uniformly illuminates the specimen using excitation light to cause light emission at the aforementioned wavelength. 
     The solution of the expression (15) is calculated based on the expression (9) using the TwIST algorithm from these three Z-stack images. The binarized result using 1% of the maximum value as the threshold, the three-dimensional light emission intensity distribution reconstructed with the regularization term of the expression (15) as a TV norm and the regularization parameter as 1E-4 is illustrated in  FIG. 13A .  FIG. 13B  illustrates the reconstructed light emission intensity distribution on the cross-section going through y=0. The value of the RMSE for the reconstructed light emission intensity distribution of  FIGS. 13A-13B  is 1.31E-2. The value of the RMSE when the pupil has a uniform transmittance distribution is 1.35E-2, and thus the transmittance distribution asymmetric with respect to the optical axis is effective in reconstruction. When comparing  FIGS. 13A-13B  and  FIGS. 10A-10B , it is understood that the three-dimensional shape information of the specimen  202  can be estimated from Z-stack images not satisfying the Nyquist condition. 
     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. 2013-007453, filed on Jan. 18, 2013 which is hereby incorporated by reference herein in its entirety. 
     INDUSTRIAL APPLICABILITY 
     The present invention is applicable to an image pickup apparatus configured to acquire three-dimensional information of a specimen such as a virtual slide generating system and a digital microscope. 
     REFERENCE SIGNS LIST 
     
         
           10  image pickup apparatus 
           100  illumination optical system 
           202  specimen 
           300  imaging optical system 
           401  image sensor