Patent Publication Number: US-2017363853-A1

Title: Reconstruction algorithm for fourier ptychographic imaging

Description:
REFERENCE TO RELATED PATENT APPLICATION(S) 
     This application claims the benefit under 35 U.S.C. §119 of the filing date of Australian Patent Application No. 2014280898, filed Dec. 23, 2014, hereby incorporated by reference in its entirety as if fully set forth herein. 
     TECHNICAL FIELD 
     The current invention relates to systems and apparatus for Fourier Ptychographic imaging. 
     BACKGROUND 
     Fourier Ptychographic Microscopy (FPM) is a kind of microscopy that forms an image of a specimen using Fourier Ptychographic imaging. This imaging method is based on capturing many lower resolution images under different lighting conditions, and combining them using an iterative computational process to generate a higher resolution image. Although the lower resolution images are real images, the higher resolution image is complex. FPM can achieve a high resolution and a wide field of view simultaneously without moving the specimen relative to the imaging optics. 
     Virtual microscopy is a technology that gives physicians the ability to navigate and observe a biological specimen at different simulated magnifications and through different two-dimensional (2D) or three-dimensional (3D) views as though they were controlling a microscope. Virtual microscopy can be achieved using a display device such as a computer monitor or tablet with access to a database of images of microscope images of the specimen. There are a number of advantages of virtual microscopy over traditional microscopy. Firstly, the specimen itself is not required at the time of viewing, thereby facilitating archiving, telemedicine and education. Virtual microscopy can also enable the processing of the specimen images to change the depth of field and to reveal pathological features that would be otherwise difficult to observe by eye, for example as part of a computer aided diagnosis system. 
     Conventional capture of images for virtual microscopy is generally performed using a high throughput slide scanner. The specimen is loaded mechanically onto a stage and moved under the microscope objective as images of different parts of the specimen are captured on a sensor. Depth and thickness information for the specimen being imaged are generally required in order to perform an efficient capture. 
     Any two adjacent images have an overlap region so that the multiple images of the same specimen can be combined into a 2D layer or a 3D volume in a computer system attached to the microscope. Mosaicing and other software algorithms are used to register both the neighbouring images at the same depth and at different depths so that there are no defects between adjoining images to give a seamless 2D or 3D view. Virtual Microscopy is different from other image mosaicing tasks in a number of important ways. Firstly, the specimen is typically moved by the stage under the optics, rather than the optics being moved to capture different parts of the subject as would take place in a panorama. The stage movement is can be controlled very accurately and the specimen may be fixed in a substrate. 
     The microscope is used in a controlled environment—for example mounted on vibration isolation platform in a laboratory with a custom illumination set up so that the optical tolerances of the system (alignment and orientation of optical components and the stage) are very tight. Therefore, the coarse alignment of the captured tiles for mosaicing can be fairly accurate, the lighting even, and the transform between the tiles well represented by a rigid transform. On the other hand, the scale of certain important features of a specimen can be of the order of several pixels and the features can be densely arranged over the captured tile images. This means that the required stitching accuracy for virtual microscopy is very high. Additionally, given that the microscope can be loaded automatically and operated in batch mode, the processing throughput requirements are also high. 
     Fourier Ptychographic Microscopy (FPM) is an alternative to the above high throughput slide scanner. FPM can produce a 2D image of a specimen with both a high resolution and wide field of view without transverse motion of the specimen under the objective lens. This is achieved by capturing many lower resolution images of the specimen under different lighting conditions, and combining the captured images using an iterative computational process. Each iteration analyses the set of captured images sequentially to converge towards a high quality higher resolution image. The captured images are combined in the Fourier domain so that there are no image seams in real space. The ability to generate an image without discrete stitching artefacts in the spatial domain in this way is a second advantage of FPM over traditional slide scanners. A third advantage is that the generated image is complex, that is to say it includes phase information. 
     On the other hand, the capture of the set of images may be slow as the illumination strength may be reduced. Also, the iterative computational process can require significant processing and storage resources in order to achieve an acceptable quality. It is desirable, therefore to develop a system for FPM that is efficient and accurate. 
     SUMMARY 
     According to one aspect of the present disclosure there is provided a method of generating an image of a substantially translucent specimen, the method comprising: 
     (a) illuminating and imaging the specimen based on light filtered by an optical element; 
     (b) acquiring a plurality of relatively lower resolution intensity images of the specimen for which content of the images corresponds to partially overlapping regions in frequency space; and 
     (c) reconstructing a relatively higher resolution image of the specimen by iteratively updating overlapping regions of the relatively higher resolution image in Fourier space with the plurality of relatively lower resolution intensity images, wherein said iterative updating processes the plurality of relatively lower resolution intensity images in a first sequence which progresses from a centre region of the relatively higher resolution image in increasing spatial frequency followed by a second sequence which progresses towards the centre region in decreasing spatial frequency. 
     The method may use a variable illuminator to control the spatial frequency associated with the relatively lower resolution intensity images according to angles of illumination between individual light sources of the variable illuminator and the specimen. Alternatively a scanning aperture to control the spatial frequency associated with the intensity images. In another implementation a spatial light modulator may be used to control the spatial frequency associated with the intensity images. 
     Preferably the first sequence starts with one said lower resolution image corresponding to a spatial frequency that is at or near to zero. Also preferably the second sequence ends with one said lower resolution image corresponding to a spatial frequency that is at or near to zero. In another example the iterative updating concludes towards the centre region such that the second sequence is the final sequence. 
     Alternatively or additionally the first sequence is selected in order of increasing maximum modulus of spatial frequency, and then in an order according to an angle of the radial spatial frequency. Desirably the order according to the angle of progression is one of an increasing or decreasing angle around an optical axis in a plane of illumination. Also the second sequence may be selected in order of decreasing maximum modulus of spatial frequency, and then in an order according to an angle of progression toward the centre region. In a further implementation, the second sequence is selected in order of decreasing transverse spatial frequency, and then in order of one of increasing or decreasing angle relative to an x-axis in a plane of illumination. In another, the order according to the angle of progression is one of an increasing or decreasing angle relative to an x-axis in a plane of illumination. 
     Advantageously the first sequence is selected in order of increasing radial spatial frequency, and then in order of one of increasing or decreasing angle of the radial spatial frequency. Preferably the second sequence is selected in order of decreasing radial spatial frequency, and then in order of one of increasing or decreasing angle of the radial spatial frequency. 
     In specific implementations the variable illuminator comprises positions of illumination on a plane perpendicular to an optical axis of imaging and configured to illuminate the specimen from a plurality of angles of illumination, wherein at least one of:
         (a) positions of illumination on the plane map to two-dimensional (2D) spatial frequencies in a Fourier reconstruction space that are approximately evenly spaced;   (b) positions of illumination on the plane map to 2D spatial frequencies in a Fourier reconstruction space such that the density is greater towards the spatial frequency corresponding to the DC term of the Fourier reconstruction;   (c) positions of illumination on the plane map to 2D spatial frequencies in a Fourier reconstruction space such that the density is greater towards the spatial frequency corresponding to the DC term of the Fourier reconstruction according to a power law;   (d) positions of illumination on the plane map to 2D spatial frequencies in a Fourier reconstruction space such that the density is greater towards the spatial frequency corresponding to the DC term of the Fourier reconstruction by the illumination angles being arranged with a substantially regular pattern in a polar coordinate system defined by a radial coordinate that depends on the magnitude of the angle relative to an optical axis and an angular coordinate corresponding to the orientation of the angle relative to the optical axis;   (e) a density of positions of illumination drops substantially to zero outside a circular region;   (f) positions of illumination on a plane perpendicular to the optical axis are spaced evenly on concentric circles such that the number of angular locations selected around each circle increases monotonically with the radius of the circle; and   (g) positions of illumination are defined one or more spiral arrangements.       

     In other implementations the illuminating and imaging comprises scanning an aperture in a plane perpendicular to an optical axis of imaging utilizing the above. 
     Other aspects are also disclosed. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       At least one embodiment of the invention will now be described with reference to the following drawings, in which: 
         FIG. 1  shows a high-level system diagram for a Fourier Ptychographic Microscopy system; 
         FIGS. 2A and 2B  show two prior art variable illuminator designs for a Fourier Ptychographic Microscopy system based on a square lattice and a hexagonal lattice, respectively; 
         FIGS. 3A and 3B  illustrate the relative geometry of a small light source (such as an LED)  330 , a specimen  380  and the optical axis  390  of the microscope  101 ; 
         FIG. 4  illustrates a variable illumination system  108  for FPM that is not flat, taking the form of a hemisphere  410 ; 
         FIG. 5  is a schematic flow diagram of a process  500  that generates a higher resolution image of a specimen by Fourier Ptychographic imaging according to the present disclosure; 
         FIG. 6  is a schematic flow diagram of a method of generating a higher resolution image  110  from the set of lower resolution captured images  104 ; 
         FIGS. 7A and 7B  illustrate an exemplary partitioning of the images that may be used at step  610  of method  600 ; 
         FIG. 8  is a schematic flow diagram of a method of generating a higher resolution partition image from set of lower resolution partition images; 
         FIG. 9  is a schematic flow diagram of a method of updating a higher resolution partition image based on a single lower resolution partition image; 
         FIGS. 10A and 10B  illustrate respectively the real space and Fourier space representations of a specimen; 
         FIGS. 11A to 11F  illustrate spatial arrangements of light sources as projected on a plane perpendicular to the optical axis and the corresponding transverse wavevectors; 
         FIGS. 12A to 12F  illustrate spatial arrangements of light sources as projected on a plane perpendicular to the optical axis and the corresponding transverse wavevectors; 
         FIGS. 13A and 13B  illustrate two alternative spatial arrangements of light sources for a variable illuminator  108 ; 
         FIGS. 14A to 14F  illustrate spatial arrangements of light sources as projected on a plane perpendicular to the optical axis and the corresponding transverse wavevectors; 
         FIGS. 15A to 15F  illustrate spatial arrangements of light sources as projected on a plane perpendicular to the optical axis and the corresponding transverse wavevectors; 
         FIG. 16A to 16F  illustrate spatial arrangements of light sources as projected on a plane perpendicular to the optical axis and the corresponding transverse wavevectors; 
         FIG. 17  shows plots of the SSIM index against the simulated number of light sources for a number of reconstruction algorithms described herein; 
         FIGS. 18A and 18B  form a schematic block diagram of a general purpose computer system upon which arrangements described can be practiced; and 
         FIGS. 19A to 19C  illustrate the order of selection of lower resolution images based on the ascending and descending square and the ascending and descending radial sequences. 
     
    
    
     DETAILED DESCRIPTION INCLUDING BEST MODE 
     Context 
       FIG. 1  shows a high-level system diagram for a microscope capture system  100  suitable for Fourier Ptychographic Microscopy (FPM). A specimen  102  is physically positioned on a stage  114  under an optical element, such as a lens  109 , and within the field of view of a microscope  101 . The microscope  102  in the illustrated implementation has a stage  114  that may be configured to move in order to correctly place the specimen in the field of view of the microscope at an appropriate depth. The stage  114  may also move as multiple images of the specimen  102  are captured by a camera  103  mounted to the microscope  101 . In a standard configuration, the stage  114  may be fixed during image capture of the specimen. 
     A variable illumination system (illuminator)  108  is positioned in association with the microscope  101  so that the specimen  102  may be illuminated by coherent or partially coherent light incident at different angles. The illuminator  108  typically includes small light emitters  112  arranged at distance from the specimen  102 , the distance being large compared to the size of the emitters and also compared to the size of the specimen  102 . With such an arrangement, the light emitters  112  act somewhat like point sources, and the light from the emitters  112  approximates plane waves at the specimen  102 . An alternate configuration may use larger light emitters and a lens to focus the light to a plane wave. The specimen  102  is typically substantially translucent such that the illuminating light can pass through the specimen  102  and be focussed by the lens  109  of the microscope  101  for detection by the camera  103 . The arrangement of the microscope  101 , the lens  109  and camera  103  represent a detector that forms an optical axis and is configured to capture or acquire images of the specimen  102  subject to the variable illumination afforded by the illuminator  108 . 
     The microscope  101  forms an image of the specimen  102  on a sensor in the camera  103  by means of an optical system. The optical system may be based on an optical element that may include an objective lens  109  with low numerical aperture (NA), or some other arrangement. The camera  103  captures one or more images  104  corresponding to each illumination configuration. Multiple images may be captured at different stage positions and/or different colours of illumination. The arrangement provides for the imaging of the specimen  102 , including the capture and provision of multiple images of the specimen  102  to the computer  105 . 
     The captured images  104 , also referred to as relatively low or lower resolution images, are intensity images that may be greyscale images or colour images, depending on the sensor and illumination. The images  104  are passed to a computer system  105  which can either start processing the images immediately or store them in temporary storage  106  for later processing. As part of the processing, the computer  105  generates a relatively high or higher resolution image  110  corresponding to one or more regions of the specimen  102 . The higher resolution image may be reproduced upon a display device  107 . As illustrated, the computer  105  may be configured to control operation of the individual light emitters  112  of the illuminator  108  via a control line  116 . Also, the computer  105  may be configured to control movement of the stage  114 , and thus the specimen  102 , via a control line  118 . A further control line  120  may be used by which the computer  105  may control the camera  103  for capture of the images  104 . 
     The transverse optical resolution of the microscope may be estimated based on the optical configuration of the microscope and is related to the point spread function of the microscope. A standard approximation to this resolution in air is given by: 
     
       
         
           
             
               
                 
                   
                     
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     where NA is the numerical aperture, and λ is the wavelength of light. A conventional slide scanner might use an air immersion objective lens with an NA of 0.7. At a wavelength of 500 nm, the estimated resolution is 0.4 μm. A typical FPM system would use a lower NA of the order of 0.08 for which the estimated resolution drops to 4 μm. 
     The numerical aperture of a lens defines a half-angle, θ H , of the maximum cone of light that can enter or exit the lens. In air, this is defined by: 
       θ H =arcsin( NA ),  (2)
 
     in terms of which the full acceptance angle of the lens can be expressed as θ F =2θ H . 
     The specimen  102  being observed may be a biological specimen such as a histology slide consisting of a tissue fixed in a substrate and stained to highlight specific features. Such specimens are substantially translucent. Such a slide may include a variety of biological features on a wide range of scales. The features in a given slide depend on the specific tissue sample and stain used to create the histology slide. The dimensions of the specimen on the slide may be of the order of 10 mm×10 mm or larger. If the transverse resolution of a virtual slide was selected as 0.4 μm, each layer would consist of at least 25,000 by 25,000 pixels. 
     Computer Implementation 
       FIGS. 18A and 18B  depict a general-purpose computer system  1800 , upon which the various arrangements to be described can be practiced. The computer system  1800  is configured to perform the functions and operations of the computer  105 , data storage  106 , and display device  107  of  FIG. 1  and thereby with the microscope  101  form apparatus for ptychographic imaging of biological specimens and the like. 
     As seen in  FIG. 18A , the computer system  1800  includes: a computer module  1801  ( 105 ); input devices such as a keyboard  1802 , a mouse pointer device  1803 , a scanner  1826 , the camera  103 , and a microphone  1880 ; and output devices including a printer  1815 , a display device  1814  ( 107 ) and loudspeakers  1817 . An external Modulator-Demodulator (Modem) transceiver device  1816  may be used by the computer module  1801  for communicating to and from a communications network  1820  via a connection  1821 . The communications network  1820  may be a wide-area network (WAN), such as the Internet, a cellular telecommunications network, or a private WAN. Where the connection  1821  is a telephone line, the modem  1816  may be a traditional “dial-up” modem. Alternatively, where the connection  1821  is a high capacity (e.g., cable) connection, the modem  1816  may be a broadband modem. A wireless modem may also be used for wireless connection to the communications network  1820 . 
     The computer module  1801  typically includes at least one processor unit  1805 , and a memory unit  1806 . For example, the memory unit  1806  may have semiconductor random access memory (RAM) and semiconductor read only memory (ROM). The computer module  1801  also includes an number of input/output (I/O) interfaces including: an audio-video interface  1807  that couples to the video display  1814 , loudspeakers  1817  and microphone  1880 ; an I/O interface  1813  that couples to the keyboard  1802 , mouse  1803 , scanner  1826 , camera  103 , the illuminator  108 , the stage  114 , and optionally a joystick or other human interface device (not illustrated); and an interface  1808  for the external modem  1816  and printer  1815 . In some implementations, the modem  1816  may be incorporated within the computer module  1801 , for example within the interface  1808 . The computer module  1801  also has a local network interface  1811 , which permits coupling of the computer system  1800  via a connection  1823  to a local-area communications network  1822 , known as a Local Area Network (LAN). As illustrated in  FIG. 18A , the local communications network  1822  may also couple to the wide network  1820  via a connection  1824 , which would typically include a so-called “firewall” device or device of similar functionality. The local network interface  1811  may comprise an Ethernet circuit card, a Bluetooth™ wireless arrangement or an IEEE 802.11 wireless arrangement; however, numerous other types of interfaces may be practiced for the interface  1811 . 
     The I/O interfaces  1808  and  1813  may afford either or both of serial and parallel connectivity, the former typically being implemented according to the Universal Serial Bus (USB) standards and having corresponding USB connectors (not illustrated). Storage devices  1809  are provided and typically include a hard disk drive (HDD)  1810 . Other storage devices such as a floppy disk drive and a magnetic tape drive (not illustrated) may also be used. An optical disk drive  1812  is typically provided to act as a non-volatile source of data. Portable memory devices, such optical disks  1825  (e.g., CD-ROM, DVD, Blu-ray Disc™), USB-RAM, portable, external hard drives, and floppy disks, for example, may be used as appropriate sources of data to the system  1800 . In the arrangement illustrated, the data storage  106  of  FIG. 1  may be implemented in whole or part by any one or more of the memory  1806 , the HDD  1810 , the disk  1825 , or the networks  1820  or  1822  when operate as storage servers or the like. 
     The components  1805  to  1813  of the computer module  1801  typically communicate via an interconnected bus  1804  and in a manner that results in a conventional mode of operation of the computer system  1800  known to those in the relevant art. For example, the processor  1805  is coupled to the system bus  1804  using a connection  1818 . Likewise, the memory  1806  and optical disk drive  1812  are coupled to the system bus  1804  by connections  1819 . Examples of computers on which the described arrangements can be practised include IBM-PC&#39;s and compatibles, Sun Sparcstations, Apple Mac™ or a like computer systems. 
     The methods of image acquisition to be described may be implemented using the computer system  1800  wherein the processes of  FIGS. 3A to 17 , may be implemented as one or more software application programs  1833  executable within the computer system  1800 . In particular, the steps of the methods of image acquisition are effected by instructions  1831  (see  FIG. 18B ) in the software  1833  that are carried out within the computer system  1800 . The software instructions  1831  may be formed as one or more code modules, each for performing one or more particular tasks. The software may also be divided into two separate parts, in which a first part and the corresponding code modules performs the image acquisition methods and a second part and the corresponding code modules manage a user interface between the first part and the user. 
     The software may be stored in a computer readable medium, including the storage devices described below, for example. The software is loaded into the computer system  1800  from the computer readable medium, and then executed by the computer system  1800 . A computer readable medium having such software or computer program recorded on the computer readable medium is a computer program product. The use of the computer program product in the computer system  1800  preferably effects an advantageous apparatus for ptychographic imaging. 
     The software  1833  is typically stored in the HDD  1810  or the memory  1806 . The software is loaded into the computer system  1800  from a computer readable medium, and executed by the computer system  1800 . Thus, for example, the software  1833  may be stored on an optically readable disk storage medium (e.g., CD-ROM)  1825  that is read by the optical disk drive  1812 . A computer readable medium having such software or computer program recorded on it is a computer program product. The use of the computer program product in the computer system  1800  preferably effects an apparatus for ptychographic imaging. 
     In some instances, the application programs  1833  may be supplied to the user encoded on one or more CD-ROMs  1825  and read via the corresponding drive  1812 , or alternatively may be read by the user from the networks  1820  or  1822 . Still further, the software can also be loaded into the computer system  1800  from other computer readable media. Computer readable storage media refers to any non-transitory tangible storage medium that provides recorded instructions and/or data to the computer system  1800  for execution and/or processing. Examples of such storage media include floppy disks, magnetic tape, CD-ROM, DVD, Blu-ray Disc™, a hard disk drive, a ROM or integrated circuit, USB memory, a magneto-optical disk, or a computer readable card such as a PCMCIA card and the like, whether or not such devices are internal or external of the computer module  1801 . Examples of transitory or non-tangible computer readable transmission media that may also participate in the provision of software, application programs, instructions and/or data to the computer module  1801  include radio or infra-red transmission channels as well as a network connection to another computer or networked device, and the Internet or Intranets including e-mail transmissions and information recorded on Websites and the like. 
     The second part of the application programs  1833  and the corresponding code modules mentioned above may be executed to implement one or more graphical user interfaces (GUIs) to be rendered or otherwise represented upon the display  1814 . Through manipulation of typically the keyboard  1802  and the mouse  1803 , a user of the computer system  1800  and the application may manipulate the interface in a functionally adaptable manner to provide controlling commands and/or input to the applications associated with the GUI(s). Other forms of functionally adaptable user interfaces may also be implemented, such as an audio interface utilizing speech prompts output via the loudspeakers  1817  and user voice commands input via the microphone  1880 . 
       FIG. 18B  is a detailed schematic block diagram of the processor  1805  and a “memory”  1834 . The memory  1834  represents a logical aggregation of all the memory modules (including the HDD  1809  and semiconductor memory  1806 ) that can be accessed by the computer module  1801  in  FIG. 18A . 
     When the computer module  1801  is initially powered up, a power-on self-test (POST) program  1850  executes. The POST program  1850  is typically stored in a ROM  1849  of the semiconductor memory  1806  of  FIG. 18A . A hardware device such as the ROM  1849  storing software is sometimes referred to as firmware. The POST program  1850  examines hardware within the computer module  1801  to ensure proper functioning and typically checks the processor  1805 , the memory  1834  ( 1809 ,  1806 ), and a basic input-output systems software (BIOS) module  1851 , also typically stored in the ROM  1849 , for correct operation. Once the POST program  1850  has run successfully, the BIOS  1851  activates the hard disk drive  1810  of  FIG. 18A . Activation of the hard disk drive  1810  causes a bootstrap loader program  1852  that is resident on the hard disk drive  1810  to execute via the processor  1805 . This loads an operating system  1853  into the RAM memory  1806 , upon which the operating system  1853  commences operation. The operating system  1853  is a system level application, executable by the processor  1805 , to fulfil various high level functions, including processor management, memory management, device management, storage management, software application interface, and generic user interface. 
     The operating system  1853  manages the memory  1834  ( 1809 ,  1806 ) to ensure that each process or application running on the computer module  1801  has sufficient memory in which to execute without colliding with memory allocated to another process. Furthermore, the different types of memory available in the system  1800  of  FIG. 18A  must be used properly so that each process can run effectively. Accordingly, the aggregated memory  1834  is not intended to illustrate how particular segments of memory are allocated (unless otherwise stated), but rather to provide a general view of the memory accessible by the computer system  1800  and how such is used. 
     As shown in  FIG. 18B , the processor  1805  includes a number of functional modules including a control unit  1839 , an arithmetic logic unit (ALU)  1840 , and a local or internal memory  1848 , sometimes called a cache memory. The cache memory  1848  typically includes a number of storage registers  1844 - 1846  in a register section. One or more internal busses  1841  functionally interconnect these functional modules. The processor  1805  typically also has one or more interfaces  1842  for communicating with external devices via the system bus  1804 , using a connection  1818 . The memory  1834  is coupled to the bus  1804  using a connection  1819 . 
     The application program  1833  includes a sequence of instructions  1831  that may include conditional branch and loop instructions. The program  1833  may also include data  1832  which is used in execution of the program  1833 . The instructions  1831  and the data  1832  are stored in memory locations  1828 ,  1829 ,  1830  and  1835 ,  1836 ,  1837 , respectively. Depending upon the relative size of the instructions  1831  and the memory locations  1828 - 1830 , a particular instruction may be stored in a single memory location as depicted by the instruction shown in the memory location  1830 . Alternately, an instruction may be segmented into a number of parts each of which is stored in a separate memory location, as depicted by the instruction segments shown in the memory locations  1828  and  1829 . 
     In general, the processor  1805  is given a set of instructions which are executed therein. The processor  1805  waits for a subsequent input, to which the processor  1805  reacts to by executing another set of instructions. Each input may be provided from one or more of a number of sources, including data generated by one or more of the input devices  1802 ,  1803 , data received from an external source across one of the networks  1820 ,  1822 , data retrieved from one of the storage devices  1806 ,  1809  or data retrieved from a storage medium  1825  inserted into the corresponding reader  1812 , all depicted in  FIG. 18A . The execution of a set of the instructions may in some cases result in output of data. Execution may also involve storing data or variables to the memory  1834 . 
     The disclosed ptychographic imaging arrangements use input variables  1854 , which are stored in the memory  1834  in corresponding memory locations  1855 ,  1856 ,  1857 . The arrangements produce output variables  1861 , which are stored in the memory  1834  in corresponding memory locations  1862 ,  1863 ,  1864 . Intermediate variables  1858  may be stored in memory locations  1859 ,  1860 ,  1866  and  1867 . 
     Referring to the processor  1805  of  FIG. 18B , the registers  1844 ,  1845 ,  1846 , the arithmetic logic unit (ALU)  1840 , and the control unit  1839  work together to perform sequences of micro-operations needed to perform “fetch, decode, and execute” cycles for every instruction in the instruction set making up the program  1833 . Each fetch, decode, and execute cycle comprises:
         (i) a fetch operation, which fetches or reads an instruction  1831  from a memory location  1828 ,  1829 ,  1830 ;   (ii) a decode operation in which the control unit  1839  determines which instruction has been fetched; and   (iii) an execute operation in which the control unit  1839  and/or the ALU  1840  execute the instruction.       

     Thereafter, a further fetch, decode, and execute cycle for the next instruction may be executed. Similarly, a store cycle may be performed by which the control unit  1839  stores or writes a value to a memory location  1832 . 
     Each step or sub-process in the processes of  FIGS. 3A to 17  is associated with one or more segments of the program  1833  and is performed by the register section  1844 ,  1845 ,  1846 , the ALU  1840 , and the control unit  1839  in the processor  1805  working together to perform the fetch, decode, and execute cycles for every instruction in the instruction set for the noted segments of the program  1833 . 
     Overview 
     The variable illumination system  108  may be formed using a set of LEDs arranged on a flat substrate, referred to as an LED matrix. The LEDs may be monochromatic or multi-wavelength, for example they may illuminate at 3 separate wavelengths corresponding to red, green and blue light, or they may illuminate at an alternative set of wavelengths appropriate to viewing specific features of the specimen. The appropriate spacing of the LEDs on the substrate depends on the microscope optics and the distance from the specimen  102  to the illumination plane, being that plane defined by the flat substrate supporting the emitters  112 . Each emitter  112 , operating as a point light source, establishes a corresponding angle of illumination  495  to the specimen  102 . Where the distance between the light source  112  and the specimen  102  is sufficiently large, the light emitted from the light source  112  approximates a plane wave. In general, the spacing of the LEDs on the substrate should be chosen so that the difference in angle of illumination arriving from a pair of neighbouring LEDs is less than the acceptance angle θ F  defined by the numerical aperture of the lens  109  according to Equation 2 above. 
     An exemplary illuminator  108  is formed of a set of LEDs forming a matrix capable of illumination at 632 nm, 532 nm and 472 nm with a spacing of approximately 4 mm. The LED matrix is placed 8 cm below the sample stage  114 , and cooperates with an optical system with NA of 0.08 and magnification of 2×, and a sensor pixel size of 5.5 μm.  FIG. 2A  illustrates an LED matrix  210  formed of a square arrangement of 121 LEDs  220 , where the LED spacing  230  is indicated.  FIG. 2B  illustrates an LED matrix  240  formed of a 2D hexagonal lattice arrangement of 115 LEDs  220 , where the LED spacing  260  is also indicated. 
     Alternative variable illumination systems to the LED matrix may be used. For example, various display technologies capable of emitting light from particular locations (pixels) could be used, such as LCD, plasma, OLED, SED, CRT or other display technology. Also, the variable illumination may be achieved by mechanically moving a light source such as an LED to a variety of locations, or even by a combination of mechanical motion, multiple sources, and display technology. 
       FIG. 3A  illustrates the relative geometry of a small light source (such as an LED)  330  ( 220 ), a specimen  380  ( 102 ), and the optical axis  390  of the microscope  101 , which is typically coincident with an optical axis of the camera  103 . A plane  310  can be constructed that is perpendicular to the optical axis  390  of the microscope  101  and includes the light source  330 . If a flat LED matrix is used as the variable illuminator  108  then the plane  310  and the LED matrix should be roughly coincident. The optical axis  390  may be considered to define a z-axis, and the x- and y-axes may be defined on the plane  310 . Ideally the x- and y-axes should be selected to coincide with the axes of the sensor in the camera  103 . The position of the light source  330  may then be defined in terms of the axis relative to a point on the specimen  335  and the corresponding point  340  projected along the optical axis  390  to the plane  310 . The point  340  may be referred to as the DC point, and the light arriving at the specimen point  335  from a light source at this position propagates along the optical axis  390 . The light source position is indicated by three offsets dx  360 , dy  370 , and dz  380 .  FIG. 3B  illustrates the geometry of  FIG. 3A  in the plane  310  transverse to the optical axis  390 . 
     The variable illumination system  108  is not constrained to be flat. The illumination system  108  may take some non-flat geometry, such as the hemisphere  410  illustrated in  FIG. 4 . The hemisphere  410  may be covered or otherwise populated by a discrete set of light sources  430  ( 220 ). It is possible to construct a plane  420  perpendicular to the optical axis  490  ( 390 ) at a distance dz  480  that may be the same as the axial distance to one of the light sources ( 380  of  FIG. 3 ), but can be at a different distance. A point  435  on the specimen  440  is projected along the optical axis  490  to the plane  420  to intersect it at an axial position  445 . The axial position  445  may be referred to as the DC point, and the light arriving at the specimen point  435  from a light source at this position propagates along the optical axis  490 . The position of each light source  450  may be projected along a line  455  joining the light source  450  and the point on the specimen  435  to a point  460  on the projected plane  420 . This point can be defined in terms of the x-, y- and z-axis in terms of three offsets dx  465 , dy  470 , and dz  475  which are a generalisation of  360 ,  370  and  380  above for a projected plane. The line  455  and the optical axis  490  subtend an angle of illumination  495  associated with the light source  450 . 
     A normalised offset vector may be formed for the offset vector of the i th  angled illumination in (dx i , dy i , dz i ) by dividing by the distance from the specimen point to the point on the plane corresponding to the illumination (i.e. from  435  to  420 , or from  335  to  330 ): 
     
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         i 
                       
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                         i 
                       
                       , 
                       
                         i 
                       
                     
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                   = 
                   
                     
                       1 
                       
                         
                           ( 
                           
                             
                               dx 
                               i 
                               2 
                             
                             + 
                             
                               dy 
                               i 
                               2 
                             
                             + 
                             
                               dz 
                               i 
                               2 
                             
                           
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                      
                     
                       
                         ( 
                         
                           
                             dx 
                             i 
                           
                           , 
                           
                             dy 
                             i 
                           
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                             i 
                           
                         
                         ) 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   3 
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     Using this approach, it is thereby possible to define the wavevector of the i th  angled illumination as the product of the normalised offset vector for this illumination and the wavenumber of illumination in vacuum, k 0 =2π/λ: 
       ( k   x   i   ,k   y   i   ,k   z   i )= k   0 (   l ,   l ,   l )  (4)
 
     The projected positions ( 460  of  FIG. 4 ) for an LED matrix with 169 LEDs is illustrated in  FIG. 14A , and the corresponding transverse (i.e. 2D) wavevectors (k x   i , k y   i ) are shown in  FIG. 14B . If the distance dz is large relative to the specimen size then the illumination approximates to plane waves at the specimen with no curvature, and the transverse wavevectors are fairly constant across the specimen. 
     It is helpful to consider aspects of the optical system in Fourier space. Two-dimensional (2D) Fourier space is a space defined by a 2D Fourier transform of the 2D real space in which the captured images are formed, or the transverse spatial properties of the specimen may be defined. The coordinates in this Fourier space are the transverse wavevectors (k x , k y ). The transverse wavevectors represent the spatial frequency of the image, with low frequencies (at or near zero) being toward the centre of the coordinate representation (e.g.  FIG. 14B ) and higher frequencies being toward the periphery of the coordinate representation. The terms transverse wavevector’ and ‘spatial frequency’ are used interchangeably in this description. The terms radial transverse wavevector and radial spatial frequency are likewise interchangeable. 
     Each lower resolution capture image is associated with a region in Fourier space defined by the optical transfer function of the optical element and also by the angle of illumination set by the variable illuminator. For the case where the optical element is an objective lens, the region in Fourier space can be approximated as a circle of radius r k  defined by the product of the wavenumber of illumination in vacuum, k 0 =2π/λ, and the numerical aperture: 
         r   k   =k   0   NA.   (5)
 
     The position of the circular region is offset according to the angle of illumination. For the i th  illumination angle, the offset is defined by the transverse components of the wavevector (k x   i , k y   i ). This is illustrated in  FIGS. 10A and 10B  which show real space and Fourier space representations of a specimen respectively. The dashed circle in  FIG. 10B  represents the region associated with a single capture image with an illumination for which the transverse wavevector is shown by the solid arrow of  FIG. 10B . The transverse wavevectors (k x   i , k y   i ) may be considered as representing the light source position on a synthetic aperture. 
     In an alternative mode of Fourier Ptychographic imaging, lower resolution capture images may be obtained using a shifted or scanning aperture (also referred to as aperture-scanning) rather than angled illumination. In this arrangement, the sample is illuminated using a single plane wave incident approximately along the optical axis. The aperture is set in the Fourier plane of the imaging system and the aperture moves within this plane, perpendicular to the optical axis. This kind of scanning aperture may be achieved using a high NA lens with an additional small scanning aperture that restricts the light passing through the optical system. The aperture in such a scanning aperture system may be considered as selecting a region in Fourier space represented by the dashed circle in  FIG. 10B  outside which the spectral content is blocked. The size of the dashed circle in  FIG. 10B  corresponds to the small aperture of a low NA lens. The transverse wavevector (k x   i , k y   i ) may be considered as representing the shifted position of the aperture rather than the transverse wavevector of angled illumination. It is noted that a spatial light modulator in the Fourier plane may be used rather than a scanning aperture to achieve the same effect. 
     A general overview of a process  500  that can be used to generate a higher resolution image of a specimen by Fourier Ptychographic imaging is shown in  FIG. 5 . The process  500  includes various steps some of which may be manually performed, or automated, and certain processing steps, that may be performed using the computer system  1800 . Such processing is typically controlled via a software applications executable by the processor upon the computer  1801  to perform the Ptychographic imaging. 
     In the process  500 , at step  510 , a specimen may optionally be loaded onto the microscope stage  114 . Such loading may be automated. In any event, a specimen  102  is required to be positioned for imaging. Next, at step  520 , the specimen may be moved to be positioned such that it is within the field of view of the microscope  101  around its focal plane. Such movement is optional and where implemented may be manual, or automated with the stage under control of the computer  1801 . Next, with a specimen appropriately positioned, steps  540  to  560  define a loop structure for capturing and storing a set of images of the specimen for a predefined set of illumination configurations. In general this will be achieved by illuminating the specimen from a specific position or at a specific angle. In the case that the variable illuminator  108  is formed of a set of LEDs such as an LED matrix, this may be achieved by switching on each individual LED in turn. The order of illumination may be arbitrary, although it is preferable to capture images in the order in which they will be processed (which may be in order of increasing angle of illumination). This minimises the delay before processing of the captured images can begin if the processing is to be started prior to the completion of the image capture. The predetermined set of illumination configurations that may be used will be discussed further with reference to  FIGS. 11 to 16 . 
     Step  550  sets the next appropriate illumination configuration, then at step  560  a lower resolution image  104  is captured on the camera  103  and stored on data storage  106  ( 1810 ). The image  104  may be a high dynamic range image, for example a high dynamic range image formed from one or more images captured over different exposures times. Appropriate exposure times can be selected based on the properties of the illumination configuration. For example, if the variable illuminator is an LED matrix, these properties may include the illumination strength of the LED switched on in the current configuration. 
     Step  570  checks if all the illumination configurations have been selected, and if not processing returns to step  540  for capture at the next configuration. Otherwise when all desired configurations have been captures, the method  500  continues to step  580 . At step  580  the processor  1805  operates to generate a higher resolution image from the set of lower resolution captured images  104 . This step will be described in further detail with respect to  FIG. 6  below. The higher resolution image is then optionally output at step  590 , completing process  500 . Output of the higher resolution image may include storage of the image on a non-transitory computer readable medium, display of the image on the display device  1814 , printing the image on the printer  1815 , or communication of the image for remote storage, display or printing. 
     A method  600 , used at step  580  to generate a higher resolution image  110  from the set of lower resolution captured images  104  will now be described in further detail below with reference to  FIG. 6 . The method  600  is preferably performed by execution of a software application by the processor  1805  operating upon images stored in the HDD  1810 , whilst using the memory  1806  for intermediate temporary storage. 
     Method  600  starts at step  610  where the processor  1805  retrieves a set of captured images  104  of the specimen  102  and partitions each of the captured images  104 .  FIGS. 7A and 7B  illustrate a suitable partitioning of the images. The rectangle  710  in  FIG. 7A  represents a single lower resolution capture image  104  of size formed by a width  720  and a height  730 . The sizes would typically correspond to the resolution (e.g. 5616 by 3744 pixels) of the sensor in the camera  103 . In step  610 , the rectangle  710  may be partitioned into equal sized square regions  740  on a regular grid with an overlap between each pair of adjacent partitions  745 . The cross hashed partition  750  is adjacent to partition  755  on the right and  760  below, and an expanded view of these three partitions is shown in  FIG. 7B . Each partition has size  765  by  775  where suitable sizes may both be 150×150 pixels. The overlapping regions in the x- and y-dimensions are illustrated by  770  and  780  for which a suitable size may be 10 pixels. 
     The overlapping regions may take different sizes over the capture images  104  in order for the partitioning to cover the field of view exactly. Alternatively, the overlapping regions may be fixed in which case the partitioning may omit a small region around the boundary of the capture images  710 . The size of each partition and the total number of partitions may be varied to optimise the overall performance of the system in terms of memory use and processing time. A set of partition images is formed corresponding to the geometry of a partition region applied to each of the set of lower resolution capture images. For example, the partition  750  may be selected from each capture image to form one such set of partitions. 
     Steps  620  to  640  define a loop structure that processes the sets of partitions of the lower resolution images in turn. The sets of partitions may be processed in parallel for faster throughput. Step  620  select the next set of lower resolution partitions of the capture images. Step  630  then generates a higher resolution partition image from the set of partition images. Each higher resolution partition image may be temporarily stored in memory  1806  or  1810 . This step will be described in further detail with respect to  FIG. 8  below. Each higher resolution partition image is essentially a partition corresponding to a corresponding region  740  of each of the lower resolution capture images, but at a higher resolution. Step  640  checks if all sets of partition images of the lower resolution capture images have been processed, and if so processing continues to step  650 , otherwise processing returns to step  620 . 
     At step  650 , the set of higher resolution partition images are combined to form a single higher resolution image  110 . A suitable method of combining the images may be understood with reference to  FIG. 7A . A higher resolution image corresponding to the capture image field of view covered by the partition sets is defined, where the higher resolution image is upscaled relative to the capture image by the same factor as the upscaling of the higher resolution partition images relative to the lower resolution capture partition images. Each higher resolution partition image is then composited by the processor  1805  onto the higher resolution image at a location corresponding to the lower resolution partition location upscaled in the same ratio. Efficient compositing methods exist that may be used for this purpose. Ideally, the compositing should blend the content of the adjacent high resolution partition images in the overlapping regions given by the upscaled equivalent of regions  745 . This completes the processing of method  600 . 
     Method  800 , used at step  630  to generate a higher resolution partition image from set of lower resolution partition images, will now be described in further detail below with reference to  FIG. 8 . The method  800  is preferably implemented using software executable by the processor  1805 . 
     First at step  810 , a higher resolution partition image is initialised by the processor  1805 . The image is defined in Fourier space, with a pixel size that is preferably the same as that of the lower resolution capture images transformed to Fourier space by a 2D Fourier transform. It is noted that each pixel of the image stores a complex value with a real and imaginary component. The initialised image should be large enough to contain all of the Fourier space regions corresponding to the variably illuminated lower resolution capture images, such as the region illustrated by the dashed circle in  FIG. 10B . The transverse wavevectors (k x   i , k y   i ) corresponding to an LED matrix with 169 LEDs are illustrated in  FIG. 11B . In this case the higher resolution partition image needs to large enough to contain an appropriate Fourier space region around each of the transverse wavevectors. For the case of an objective lens, with circular Fourier space regions of radius r k , the higher resolution partition image should cover the convex hull of the set of transverse wavevectors in  FIG. 11B  dilated by the radius of the regions r k . 
     It is noted that in alternative implementations, the higher resolution partition image may be generated with a size that can dynamically grow to include each successive Fourier space region as the corresponding lower resolution capture image is processed. 
     Once the higher resolution partition image has been initialised in step  810 , steps  820  to  870  loop over a number of iterations. The iterative updating is used to resolve the underlying phase of the image data to reduce errors in the reconstructed high-resolution images. The number of iterations may be fixed, preferably somewhere between 4 and 15, or it may be set dynamically by checking a convergence criteria for the reconstruction algorithm. 
     Each iteration starts at step  820 , then step  830  determines an appropriate order for processing the set of partition images of the lower resolution capture images for the current iteration. The order may be defined by indexing each lower resolution capture image according to the order of capture. For a total of N capture images, the indices take the range i=1, . . . N. 
     A number of suitable orderings may be defined based on the set of transverse wavevectors (k x   i , k y   i ) corresponding to the image captures. The transverse wavevectors may correspond to the angle of illumination, or the position of a scanning, or otherwise modifiable aperture, such as spatial light modulator (LCD mask). Transverse wavevectors corresponding to a number of different configurations are illustrated in  FIGS. 11A to 16F  and are discussed below. The choice of processing order may depend on the configuration of the system, such as the selection of a particular arrangement of the light sources in the illuminator  108 , and the iteration number. 
     A square-ascending order, as known and used, is defined based on concentric squares around the DC point (k x =k y =0). Capture images corresponding to transverse wavevectors on smaller squares are processed prior to those on larger squares. In terms of the transverse wavevectors this corresponds to processing images in order of increasing value of the maximum of the modulus of the transverse wavevectors, which may be expressed as k sq =max(|k x |, |k y |). If more than one wavevector is on the same square (i.e. has the same value of k sq ) then those wavevectors are ordered according to the angle of the transverse wavevector relative to a line from the origin such as the x- or y-axis. For example, capture images on the same concentric square may be ordered according to increasing or decreasing angle around the z-axis relative to the x-axis, as seen in  FIG. 4 , being in the plane  420 . 
     A preferred implementation makes use of processing in both ascending and descending directions. 
     For a square lattice arrangement of transverse wavevectors, the ascending-square sort order is illustrated in  FIG. 19A . The dots represent the set of transverse wavevectors, with the central dot  1910  corresponding to a transverse wavevector that is near to zero (which may be referred to as the DC image). The central dot  1910  corresponds to the transverse wavevector of the first selected capture image, after which the order of selection of the transverse wavevectors follows the line path  1915  around concentric squares of transverse wavevectors in an anti-clockwise fashion to an outer transverse wavevector  1920 . The descending-square processing order follows the same path  1915  but in reverse, starting at an outer wavevector  1920  and working in to the centre  1910 . 
     An ascending-radial processing order may be defined in a similar fashion to the ascending-square processing order but based on concentric circles around the DC point rather than concentric squares. In terms of the transverse wavevectors this corresponds to processing images in order of increasing transverse radial wavevector, which may be expressed as k rad =√{square root over (k x   2 +k y   2 )}. As for the ascending-square order, if more than one wavevector is on the same circle (i.e. has the same value of k rad ) then those wavevectors may be ordered according to the angle of the transverse wavevector around the z-axis relative to a line from the origin such as the x-axis. 
     For a concentric radial lattice arrangement of transverse wavevectors, the ascending-radial processing order is illustrated in  FIG. 19B . The first selected wavevector  1930  is at the centre of the grid with a transverse wavevector near to zero, after which the order of selection of the transverse wavevectors follows a line path  1935  around concentric circles of transverse wavevectors in an anti-clockwise fashion to an outer transverse wavevector  1940 . The descending-radial processing order follows the same path  1935  but in reverse, starting at an outer wavevector  1940  and working in to the centre  1930 . 
     For a spiral lattice arrangement of transverse wavevectors, the ascending-radial processing order is illustrated in  FIG. 19C . The first selected wavevector  1950  is at the centre of the grid, after which the order of selection of the transverse wavevectors follows a spiral path  1955  outwards in an anti-clockwise fashion to an outer transverse wavevector  1960 . The descending-radial processing order follows the same path  1955  but in reverse, starting at the outer wavevector  1960  and working in to the centre  1950 . 
     It is noted that in the illustrations, the ascending-square and descending-square order is shown for a square lattice of transverse wavevectors, and the ascending-radial and descending-radial orders are shown for a concentric lattice and spiral arrangement. The square and radial orders are easier to visualise when the underlying lattice and processing order selection are based on similar geometry. However either processing order may be used for any lattice. 
     The above described two types of processing order: ascending and descending. An ascending processing order is typically started near the centre of the lattice, or equivalently a small transverse wavevector, and proceeds outwards, while a descending processing order is typically starting near the outside of the lattice, or equivalently an large transverse wavevector, and proceeds inwards. Variants of the ascending-square and ascending-radial processing may be defined that follow the basic pattern of an ascending order through most of the sequence. Similarly, variants of the descending-square and descending-radial ordering may be defined that follow the basic pattern of a descending processing order through most of the sequence. These variants may be defined based on a rule defined in terms of the positions of LEDs rather than transverse wavevectors. The selected processing order may be defined differently for different partitions of the reconstruction image. 
     As described above, the processing order may be selected based on the iteration. For example, the first iteration might use an ascending processing order, and the final iteration might use a descending processing order. In between the first and last order it may be advantageous to use ascending then descending on subsequent iterations. For example, an even number of iterations may be used, with the first and subsequent odd iterations using an ascending processing order, and the second and all other even iterations using a descending processing order. 
     A typical sequence based on the ascending-square and descending-square processing order might be a total of 10 iterations for which the 1 st , 3 rd , 5 th , 7 th  and 9 th  iterations use an ascending-square order and the 2 nd , 4 th , 6 th , 8 th , and 10 th  iterations use a descending-square order. A typical sequence based on the ascending-radial and descending-radial processing order might be a total of 10 iterations for which the 1 st , 3 rd , 5 th , 7 th  and 9 th  iterations use an ascending-radial order and the 2 nd , 4 th , 6 th , 8 th , and 10 th  iterations use a descending-radial processing order. Alternative sequences may combine different processing orders for different iterations and/or different partitions. 
     The order for the first iteration may match the illumination configuration order selected at step  540  so that the reconstruction algorithm performed at step  580  may start as soon as the first image is captured, and before all of the lower resolution images are captured at step  560 . 
     Next, steps  840  to  860  step through the images of the ordered set of partition images of the lower resolution capture images from step  830 . Step  840  selects the next image from the set, then step  850  updates the higher resolution partition image based on the currently selected lower resolution partition image of the set. This step will be described in further detail with respect to  FIG. 9  below. Processing then continues to step  860  which checks if all images in the set have been processed, then returns to step  840  if they have not or continues to step  870  if they have. From step  870 , processing returns to step  820  if there are more iterations to perform, or continues to step  880  if the iterations are complete. 
     The final step  880  of method  800  is to perform an inverse 2D Fourier transform on the higher resolution partition image to transform it back to real space. 
     Method  900 , used at step  850  to update the higher resolution partition image based on a single lower resolution partition image will now be described in further detail below with reference to  FIG. 9 . 
     In step  910 , the processor  1805  selects a spectral region in the higher resolution partition image corresponding to the currently selected partition image from a lower resolution capture. This is achieved as illustrated in  FIG. 10B  which shows the Fourier space representations of a specimen, a dashed circle representing the spectral region  1005  associated with a single capture image, and a transverse wavevector shown by the solid arrow that corresponds to the configuration of the illumination. The spectral region  1005  may be selected by allocating each pixel in the higher resolution partition image as inside or outside the circular region, and multiplying all pixels outside the region by zero and those inside by 1. Alternatively, interpolation can be used for pixels near the boundary to avoid artefacts associated with approximating the spectral region geometry on the pixel geometry. In this case, pixels around the boundary may be multiplied by a value in the range 0 to 1. 
     It is noted that if the variable illuminator  108  does not illuminate with plane waves at the specimen  102 , then the angle of incidence for a given illumination configuration may vary across the specimen, and therefore between different partitions. This means that the set of spectral regions corresponding to a single illumination configuration may be different for different partitions. 
     Optionally, the signal in the spectral region may be modified in order to handle aberrations in the optics. For example, the spectral signal may be multiplied by a phase function to handle certain pupil aberrations. The phase function may be determined through a calibration method, for example by optimising a convergence metric (formed when performing the generation of a higher resolution image for a test specimen) with respect to some parameters of the pupil aberration function. The pupil function may vary over different partitions as a result due to slight differences in the local angle of incident illumination over the field of view. 
     Next, at step  920 , the image data from the spectral region is transformed by the processor  1805  to a real space image at equivalent resolution to the lower resolution capture image partition. The spectral region may be zero-padded prior to transforming with the inverse 2D Fourier transform. The amplitude of the real space image is then set to match the amplitude of the equivalent (current) lower resolution partition image at step  930 . The complex phase of the real space image is not altered at this step. The real space image is then Fourier transformed at step  940  to give a spectral image. Finally, at step  950 , the signal in the spectral region of the higher resolution partition image selected at step  910  is replaced with the corresponding signal from the spectral region in the spectral image formed at step  940 . It is noted that in order to handle boundary related artefacts, it may be preferable to replace a subset of the spectral region that does not include any boundary pixels. If the signal in the spectral region was modified to handle aberrations at step  910 , then a reverse modification should be performed as part of step  950  prior to replacing the region of the higher resolution partition image at this stage. 
     First Exemplary Implementation 
       FIGS. 11A, 11C and 11E  illustrate spatial arrangements of light sources as projected on a plane perpendicular to the optical axis. The corresponding transverse wavevectors are shown in  FIGS. 11B, 11D, and 11F  respectively.  FIG. 11A  shows the prior art arrangement of light sources as a regular square lattice on an LED matrix, with a LED spacing corresponding to a fraction of 0.40 of the acceptance angle θ F  at the centre of the arrangement. The corresponding set of transverse wavevectors shown in  FIG. 11B  are not evenly spaced, having an increased spacing in the centre compared to the outside of the arrangement. 
       FIG. 11D  shows an alternative set of transverse wavevectors which are regularly or evenly spaced with a light source spacing corresponding to a fraction of 0.5 of the acceptance angle θ F . In order to achieve this arrangement, the light sources are positioned so that they form the arrangement shown in  FIG. 11C  on a projected plane perpendicular to the optical axis. The density of light sources is larger in the centre compared to the outside of the arrangement. By corollary, the density of positions of illumination drops substantially to zero outside the circular region established by illumination afforded within the optical system. 
     A further modification may be made by applying a transform to the desired set of transverse wavevectors.  FIG. 11F  shows a set of transverse wavevectors that have been modified in this way, and  FIG. 11E  shows the corresponding arrangement on a projected plane perpendicular to the optical axis. 
     A variety of suitable transforms exist, some examples being defined in terms of the radial coordinates, (k r , k θ ), of the transverse wavevector which are defined such that k x +jk y =k r e jk     θ    and may be calculated as follows: 
         k   r =√{square root over (( k   x ) 2 +( k   y ) 2 )},
 
         k   θ =arctan 2( k   x   ,k   y ),   (6)
 
     A suitable transform is to scale the radial component of the transverse wavevector according to a power law, for example: 
     
       
         
           
             
               
                 
                   
                     
                       k 
                       r 
                     
                     → 
                     
                       
                         
                           k 
                           0 
                         
                         4 
                       
                        
                       
                         
                           ( 
                           
                             
                               4 
                                
                               
                                 k 
                                 r 
                               
                             
                             
                               k 
                               0 
                             
                           
                           ) 
                         
                         γ 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     where a suitable value for the parameter γ is 1.15 if the spacing of the light sources corresponds to a fraction of 0.55 of the acceptance angle θ F . The Cartesian transverse wavevectors are then simply given by k x =k r  cos θ and k y =k r  sin θ. Other suitable transforms may be defined in terms of simple nonlinear functional forms such as polynomial, rational, trigonometric, logarithmic, or combinations of these. According to Equations (6) and (7), positions of illumination on the plane (e.g.  11 E,  12 E,  14 E,  15 E,  16 E) map to 2D wavevectors in a Fourier reconstruction space such that the density is greater towards the wavevector corresponding to the DC term of the Fourier reconstruction (e.g. respectively  11 F,  12 F,  14 F,  15 F,  16 F). The density of light sources increases in lower radial wavevectors in the central region of Fourier space. This is seen for example in  FIGS. 11F, 12F, 14F, 15F , and 
     In general, a set of illumination configurations corresponding to  FIGS. 11A and 11B  will be referred to as (prior art) arrangement (P), however the number of light sources and parameters of the arrangement may differ from the illustrations. Similarly, an arrangement corresponding to  FIGS. 11E and 11F  will be referred to as (A1). The arrangements illustrated in  FIGS. 11A to 11F  may be used in an FPM system such as that illustrated in  FIG. 1 . The arrangements in  FIGS. 11C to 11F  can be advantageous for improved accuracy of reconstruction in terms of the performance over the arrangement in  FIGS. 11A and 11B . 
     Second Exemplary Implementation 
       FIGS. 12A, 12C and 12E  illustrate spatial arrangements of light sources as projected on a plane perpendicular to the optical axis. The corresponding transverse wavevectors are shown in  FIGS. 12B, 12D, and 12F  respectively. The positions corresponding to most of the light sources, and therefore also the transverse wavevectors, are the same as those in the corresponding images in  FIG. 11A to 11F . Note with respect to  FIG. 12D  that the transverse wavevectors are substantially evenly-spaced. In the arrangements shown in  FIGS. 12A to 12F , however, the set of light sources is selected based on a cutoff at a specific radial wavevector. This arrangement may be referred to as a circular support. 
     The configuration illustrated in  FIGS. 12A and 12B  will be referred to as (A2), however the number of light sources and parameters of the arrangement may differ from the illustrations. The arrangements illustrated in  FIG. 12  may be used in an FPM system such as that illustrated in  FIG. 1 , and may be advantageous in terms of the system performance when compared with the equivalent arrangements in  FIG. 11 . 
     Third Exemplary Implementation 
       FIGS. 13A and 13B  illustrate two alternative spatial arrangements of light sources for a variable illuminator  108  that can be advantageous in terms of the system performance compared to some of the arrangements shown in  FIGS. 11 and 12 . The illumination angles formed by the arrangements of  FIGS. 13A and 13B  form substantially regular patterns when defined in terms of polar coordinates, rather than the Cartesian coordinates that form the natural basis for defining the square lattice structure shown in  FIG. 2A . The polar coordinate system is defined in the spatial domain by a radial coordinate that depends on the magnitude of the distance of the light source from the optical axis as projected on a plane perpendicular to the optical axis and an angular coordinate that corresponds to the angle of the light source around the optical axis in the projected plane. In the Fourier domain the polar coordinates are the radial coordinates of the transverse wavevector, (k r , k θ ), defined in equation 6. 
       FIG. 13A  shows a concentric arrangement  1310  for a variable illuminator  108  including light sources  1320  ( 220 ) positioned in a number of concentric rings or circles, where the rings are equally spaced in the radial coordinate. The number of light sources on each ring is proportional to the index of the concentric ring, with an additional light source at the centre  1315 , being a position of illumination or circle with a radial distance of zero (0). In the example shown, the spacing of the concentric rings is marked  1325 . The number of light sources in a first innermost ring  1330  is 4, then 8 in the second ring  1335 , and 4i in the i th  concentric ring. The light sources are equally spaced in angle on each ring. As such, the positions of illumination are spaced evenly on concentric circles such that the number of angular locations selected around each circle increases monotonically with its radius. This configuration can be expressed as the set of light source positions given by x i,j =r i  cos θ i,j  and y i,j =r i  sin θ i,j  with: 
     
       
         
           
             
               
                 
                   
                     r 
                     i 
                   
                   = 
                   
                     i 
                      
                     
                         
                     
                      
                     Δ 
                      
                     
                         
                     
                      
                     r 
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       θ 
                       
                         i 
                         , 
                         j 
                       
                     
                     = 
                     
                       
                         2 
                          
                         
                             
                         
                          
                         π 
                          
                         
                             
                         
                          
                         j 
                       
                       
                         i 
                          
                         
                             
                         
                          
                         
                           N 
                           θ 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                     
                 
               
             
           
         
       
     
     where the indices take the ranges i=0, . . . , N r  and j=0, . . . , max(0, iN θ −1), and θ 0,0  takes the value zero. The number of rings is defined by N r  and the number of additional light sources per concentric ring is given by N θ . For the example in  FIG. 13A , the parameters are N r =8 and N θ =4. A suitable spacing for the concentric rings  1325  corresponds to a fraction of between 0.3 and 0.45 of the acceptance angle θ F . 
       FIG. 13B  shows a spiral arrangement  1340  for a variable illuminator  108  incorporating light sources  1350  ( 220 ). The positions are selected at a set of indices such that the radius and angle are proportional to the square root of the index. This configuration can be expressed as the set of light source positions given by x i =r i  cos θ i  and y i =r i  sin θ i  with: 
     
       
      
       r 
       i 
       =S 
       r 
       √{square root over (i)},  
      
     
       θ i   =s   θ   √{square root over (i)},   (9)
 
     for i=0, . . . , (N−1), where N is the total number of light sources. Suitable parameters for the design are given by S r  corresponding to a fraction of 0.325 of the acceptance angle θ F  and S θ =0.3. 
     As mentioned above, the concentric and spiral arrangements form substantially regular patterns, when defined in polar coordinates. In the concentric arrangement, the light sources are equally spaced in angle on each concentric ring. In the spiral arrangement, the angle is proportional to square root of the index of the light source. 
     Other arrangements are possible based on these models. For example, the concentric arrangement may be modified such that the number of light sources on each concentric ring in the concentric arrangement varies in a nonlinear manner, or in irregular steps, while maintaining the equal angular spacing on each ring. Alternatively, a pattern may be formed by combining a number of discrete polar arrangements together with different parameter values (preferably without including multiple light sources at the centre). Interesting arrangements useful for Fourier ptychography may be formed from a set of spirals placed at different angles to each other to achieve improved accuracy or efficiency. 
       FIGS. 14A, 14C and 14E  illustrate spatial arrangements of light sources as projected on a plane perpendicular to the optical axis based on a concentric arrangement (e.g.  FIG. 13A ). The corresponding transverse wavevectors are shown in  FIGS. 14B, 14D, and 14F  respectively. These arrangements may be used in an FPM system such as that illustrated in  FIG. 1  and offer improvements in performance over the arrangement in  FIGS. 11A and 11B  with respect to accuracy and/or efficiency. 
       FIG. 14A  shows a concentric arrangement of light sources projected on a plane perpendicular to the optical axis based on a concentric arrangement. The corresponding set of transverse wavevectors shown in  FIG. 14B  are not evenly spaced, having an increased spacing in the centre compared to the outside of the arrangement. The spacing  1325  of concentric rings corresponds to a fraction of 0.35 of the acceptance angle θ F  at the centre of the arrangement. 
       FIG. 14D  shows an alternative set of transverse wavevectors which form a regular concentric arrangement defined in the transverse wavevector space. In order to achieve this arrangement, the light sources are positioned so that they form the arrangement shown in  FIG. 14C  on a projected plane perpendicular to the optical axis. The density of light sources is larger in the centre compared to the outside of the arrangement. The spacing  1325  of concentric rings corresponds to a fraction of 0.45 of the acceptance angle θ F . 
     A further modification may be made by applying a transform to the desired set of transverse wavevectors.  FIG. 14F  shows a set of transverse wavevectors that have been modified in this way, and  FIG. 14E  shows the corresponding arrangement on a projected plane perpendicular to the optical axis. A variety of suitable transforms exist, as discussed above with reference to  FIG. 11F . The spacing  1325  of concentric rings corresponds to a fraction of 0.45 of the acceptance angle θ F  and the parameter γ is 1.05 for a nonlinear (power law) transform defined by equation (7). For the arrangements illustrated in  FIGS. 14E and 14F , the number of light sources and the precise parameterisation of the arrangement may differ from the illustrations. Use of the power law provides for positions of illumination on the plane map to 2D wavevectors in a Fourier reconstruction space such that the density is greater towards the wavevector corresponding to the DC term of the Fourier reconstruction. 
     It is noted that a subset of the concentric or spiral arrangements may be selected that are non-circular in its extent. For example, the set of light sources falling within a square geometry may be selected.  FIGS. 15A to 15F  illustrate three such arrangements that are based on the arrangements in  FIGS. 14A to 14F  but with selection based on a square geometry. For the arrangements illustrated in  FIGS. 15A and 15B , the number of light sources and the precise parameterisation of the arrangement may differ from the illustrations. 
       FIGS. 16A, 16C and 16E  illustrate spatial arrangements of light sources as projected on a plane perpendicular to the optical axis based on a spiral arrangement ( FIG. 13B ). The corresponding transverse wavevectors are shown in  FIGS. 16B, 16D, and 16F  respectively. These arrangements may be used in an FPM system such as that illustrated in  FIG. 1  and offer improvements in performance over the arrangement in  FIGS. 11A and 11B  with respect to accuracy and/or efficiency. 
       FIG. 16A  shows a spiral arrangement of light sources projected on a plane perpendicular to the optical axis based on a spiral arrangement. The corresponding set of transverse wavevectors shown in  FIG. 16B  are not evenly spaced, having an increased spacing in the centre compared to the outside of the arrangement. Suitable parameters for the design are given by S r  corresponding to a fraction of 0.325 of the acceptance angle θ F  and S θ =0.3 at the centre of the arrangement. 
       FIG. 16D  shows an alternative set of transverse wavevectors which form a regular spiral arrangement defined in the transverse wavevector space. In order to achieve this arrangement, the light sources should be positioned so that they form the arrangement shown in  FIG. 16C  on a projected plane perpendicular to the optical axis. The density of light sources becomes larger toward the centre compared to the outside of the arrangement. Suitable parameters for the configuration are given by S r  corresponding to a fraction of 0.325 of the acceptance angle θ F  and S θ =0.3. 
     A further modification may be made by applying a transform to the desired set of transverse wavevectors.  FIG. 16F  shows a set of substantially regularly-spaced transverse wavevectors that have been modified in this way, and  FIG. 16E  shows the corresponding arrangement on a projected plane perpendicular to the optical axis. A variety of suitable transforms exist, as discussed above with reference to  FIG. 11F . Suitable parameters for this configuration are given by k r  corresponding to a fraction of 0.35 of the acceptance angle θ F , k θ =0.3 and the parameter γ is 1.05 for a nonlinear transform defined by equation (7). 
     Fourth Exemplary Implementation 
     In some applications, it may be advantageous to switch on multiple light sources at one time and capture lower resolution images on the camera  103 . The computer processing required to generate the higher resolution image would be different in this case, owing to a need for additional processing from a non-adjacent sources and hence angles, however similar advantages over prior art variable illumination arrangements may be obtained. 
     Advantage 
     Estimates of the comparative performance of the above arrangements may be quantified using simulations of an FPM system with different variable illumination arrangements corresponding to different sets of illumination configurations. A large image of a histopathology slide may be used to simulate an infinitesimally thin specimen, and it is assumed that the specimen is in focus so that the effects of depth are small and may be ignored. Each low resolution capture image may be synthesised by selecting a small aperture in Fourier space corresponding to a low NA lens at a wavevector offset position corresponding to the angle of illumination. The low NA lens acts as a low resolution optical element to filter light in the imaging system. Spatial padding and a suitable windowing function may be used in the synthesis of these images to avoid artefacts at the image boundaries. The Tukey and Planck-taper window functions are suitable window functions for this purpose. The synthesised capture image is selected from the region at the centre of the synthesised image for which the window function is flat and takes the value 1. 
     The capture images are processed according to method  600  ( 580 ) for a fixed number of iterations and the reconstructed image may be compared to the true image. Metrics such as mean square error and structural similarity (SSIM) are suitable for the comparison. 
       FIG. 17  shows plots of the SSIM index against the simulated number of light sources for a number of reconstruction algorithms described herein. Although each plot consists of a number of discrete points, a straight line interpolation is included between the points. The reconstruction algorithms are referred to as AS (ascending-square,  FIG. 19A  from  1910  out), AR (ascending-radial,  FIG. 19B  from  1930  out), ADS (ascending-descending-square,  FIG. 19A  from  1910  out and then back on successive iterations), ADR (ascending-descending-radial,  FIG. 19B  from  1930  out and then back on successive iterations). For the same number of light sources, the ADS and ADR approaches show an improved SSIM compared to AD and AR over a substantial part of the plot range. This means that for a given target reconstruction accuracy (SSIM score), the number of light sources required would be less for arrangements implemented according to ADS and ADR relative to those implemented according to AD and AR. 
     It is possible to estimate the reduction in the number of light sources required to achieve a given score using the interpolation data shown in  FIG. 17 . For example, for 196 light sources, the reconstruction algorithm AS has an SSIM of 0.89. The estimated number of light sources to achieve the same SSIM for the other arrangements are given in Table 1 below. For reconstruction algorithm AR, the number of light sources is reduced to 193, for ADS the number of light sources reduces to 166, and for ADR the number reduces to 164. Based on the shape of the curves in  FIG. 17 , this advantageous reduction in the number of light sources increases further with increasing SSIM. 
     
       
         
           
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                 Estimated required number of light sources and 
               
               
                 % reduction to achieve given SSIM for FPM simulation 
               
               
                 using different reconstruction algorithms. 
               
            
           
           
               
               
               
               
               
               
            
               
                   
                 Configuration 
                 AS 
                 AR 
                 ADS 
                 ADR 
               
               
                   
                   
               
               
                   
                 Number of light sources 
                 196 
                 193 
                 166 
                 164 
               
               
                   
                 to achieve SSIM = 0.892 
               
               
                   
                 % Change relative to 
                 — 
                 −1.5% 
                 −15% 
                 −16% 
               
               
                   
                 arrangement AS 
               
               
                   
                   
               
            
           
         
       
     
     It is noted that the advantage estimates described above with reference to  FIG. 17  correspond to the case of plane wave illumination. If the variable illuminator is an LED matrix positioned relatively close to the specimen then the incident illumination cannot be considered to form a plane wave at the specimen and the mapping from position to wavevector would vary across the transverse dimensions of the specimen. This would alter the arrangement in wavevector space, which would in turn change the performance of the FPM system. 
     Furthermore, it is noted that it the above variable illuminator arrangements may be substantially achieved using an LED matrix with a very dense arrangement of LEDs on a regular grid. For each LED position in the design, an LED from the LED matrix may be selected that is close to the position of the corresponding light source in the variable illuminator arrangement. This essentially uses a subsampling of the LED matrix light sources to illuminate the specimen to thereby use that subset of sources that are close to the desired position in the illuminator arrangement. 
     INDUSTRIAL APPLICABILITY 
     The arrangements described are examples of apparatus for Fourier ptychographic imaging and are applicable to the computer and data processing industries, and particularly for the microscopic inspection of matter, including biological matter. For example, specific arrangements according to the present disclosure provide for reducing the number of light sources to achieve a similar imaging effect as prior arrangements, or to provide improved performance using comparable numbers of light sources. 
     The arrangements disclosed, particularly through the control of the illuminator  108  (via  118 ) and the camera  103  (via  120 ) provide for the computer  105 , when appropriately programmed, to implement the Fourier ptychographic imaging system. More specifically, the application program  1833  can be configured to control the illuminator and camera to cause the capture of the images  104  and then to process the images  104  as described to form a desired (higher resolution) image of the specimen. 
     The foregoing describes only some embodiments of the present invention, and modifications and/or changes can be made thereto without departing from the scope and spirit of the invention, the embodiments being illustrative and not restrictive.