Patent Publication Number: US-11644587-B2

Title: Pixel summing scheme and methods for material decomposition calibration in a full size photon counting computed tomography system

Description:
BACKGROUND 
     Technical Field 
     The disclosure relates to material decomposition in a full size photon counting computed tomography system. 
     DESCRIPTION OF THE RELATED ART 
     Computed tomography (CT) systems and methods are typically used for medical imaging and diagnosis. CT systems generally create projection images through a subject&#39;s body at a series of projection angles. A radiation source, such as an X-ray tube, irradiates the body of a subject and projection images are generated at different angles. Images of the subject&#39;s body can be reconstructed from the projection images. 
     Conventionally, energy-integrating detectors (EIDs) and/or photon-counting detectors (PCDs) have been used to measure CT projection data. PCDs offer many advantages including their capacity for performing spectral CT, wherein the PCDs resolve the counts of incident X-rays into spectral components referred to as energy bins, such that collectively the energy bins span the energy spectrum of the X-ray beam. Unlike non-spectral CT, spectral CT generates information due to different materials exhibiting different X-ray attenuation as a function of the X-ray energy. These differences enable a decomposition of the spectrally resolved projection data into different material components, for example, the two material components of the material decomposition can be bone and water. 
     Even though PCDs have fast response times, at high X-ray flux rates indicative of clinical X-ray imaging, multiple X-ray detection events on a single detector may occur within the detector&#39;s time response, a phenomenon called pileup. Left uncorrected, pileup effect distorts the PCD energy response and can degrade reconstructed images from PCDs. When these effects are corrected, spectral CT has many advantages over conventional CT. Many clinical applications can benefit from spectral CT technology, including improved material differentiation since spectral CT extracts complete tissue characterization information from an imaged object. 
     One challenge for more effectively using semiconductor-based PCDs for spectral CT is performing the material decomposition of the projection data in a robust and efficient manner. For example, correction of pileup in the detection process can be imperfect, and these imperfections degrade the material components resulting from the material decomposition. 
     In a photon counting CT system, the semiconductor-based detector using direct conversion is designed to resolve the energy of the individual incoming photons and generate measurement of multiple energy bin counts for each integration period. However, due to the detection physics in such semiconductor materials (e.g., CdTe/CZT), the detector energy response is largely degraded/distorted by charge sharing, k-escape, and scattering effects in the energy deposition and charge induction process, as well as electronic noise in the associated front-end electronics. Due to finite signal induction time, at high count-rate conditions, pulse pile-up also distorts the energy response. 
     Due to sensor material non-uniformity and complexity of the integrated detection system, it is very difficult to do accurate modeling of such detector response for a photon-counting detector just based on physics theories or Monte Carlo simulations that are based on certain modeling of the signal induction process, which determines the accuracy of the forward model of each measurement. Also, due to uncertainties in the incident X-ray tube spectrum modeling, additional errors in the forward model are introduced, and all these factors eventually degrade the material decomposition accuracy from the PCD measurements, therefore the generated spectral image quality. 
     Calibration methods have been proposed to solve such problems in the literature. The general idea is to use multiple transmission measurements of various known path lengths to calibrate the forward model such that it agrees with the calibration measurements. Some ideas are applied on estimation of the X-ray spectrum in conventional CT, see Sidky et al., Journal of Applied Physics 97(12), 124701 (2005) and Duan et al., Medical Physics 38(2), February, 2011, and later adopted on PCD measurements to estimate the combined system spectral response, see Dickmann et al., Proc. SPIE 10573, Medical Imaging 2018: Physics of Medical Imaging, 1057311 (Mar. 9, 2018). However, there can be many variations in the detail design and implementation of the calibration method, especially considering the application feasibility in a full 3rd generation CT geometry, which has not been demonstrated or documented in the literature so far. 
     For a typical semiconductor based PCD, the detector usually features a pixelated design, which uses smaller sub-pixels in group as equivalent to a conventional pixel size. This enables high-resolution imaging with smaller pixel sizes, but also requires different calibration schemes with various pixel patterns. Here, the disclosure is focused on the pixelated pattern design of the detector and weighting methods for material decomposition calibration. 
     SUMMARY 
     The embodiments presented herein relate to a two-step calibration method for the polychromatic semiconductor based PCD forward counting model, to account for various pixel summing readout modes for imaging at different resolutions. The method consists of two parts: 1) estimation of the flux independent weighted bin response function S wb (E) using the expectation maximization (EM) method, and 2) estimation of the pileup correction term P b (E, N b , N tot ). Once S wb (E) is estimated from the calibration at plural tube voltage (kVp) settings for each detector pixel, it is saved as a software calibration table in the system. It is then used as an input to estimate the pileup correction terms P b (E, N b , N tot ) at higher flux scans. Both tables are then used for the material decomposition in operational scans to estimate the basis material path lengths. 
     To correct the variation of the detector response due to different PCD sub-pixel summing schemes, the embodiments calibrate forward model parameters based on the various pixel readout modes. 
     Each sub-pixel (or a combined pixel composed of a plurality of sub-pixels) is calibrated separately following the same calibration and processing workflow. This applies to all the air or object scans used to calibrate the forward model and the patient/object scans that use the different pixel readout schemes for imaging with the calibrated tables. 
     To correct anti-scatter grid (ASG) shadow differences on different sub-pixels or combined pixels at different rotation speeds, the calibration is performed at each supported rotation speed as well. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The application will be better understood in light of the description which is given in a non-limiting manner, accompanied by the attached drawings in which: 
         FIG.  1    shows an example of the configuration of a photon-counting type X-ray CT apparatus. 
         FIG.  2    shows an example of a PCD bin response function S b (E) for a photon counting detector. Each curve stands for an example function for each energy bin. 
         FIG.  3    shows an example of a 3×3 sub-pixel pattern with a one dimensional anti-scatter grid. 
         FIG.  4    shows a material decomposition calibration and processing workflow. 
         FIG.  5    shows normalized linear attenuation coefficients for different materials. 
         FIG.  6    shows a schematic of a calibration structure design, where the pileup correction tables P b  are generated and used for each mA individually. 
         FIG.  7    shows a schematic of another calibration structure design, where a universal pileup correction table P b  is generated for the entire mA range. 
         FIGS.  8 A- 8 D  show various summing schemes for decomposition calibration and processing. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     Reference throughout this specification to “one embodiment” or “an embodiment” means that a particular feature, structure, material, or characteristic described in connection with the embodiment is included in at least one embodiment of the application, but do not denote that they are present in every embodiment. 
     Thus, the appearances of the phrases “in one embodiment” or “in an embodiment” in various places throughout this specification are not necessarily referring to the same embodiment of the application. Furthermore, the particular features, structures, materials, or characteristics may be combined in any suitable manner in one or more embodiments. 
       FIG.  1    is a diagram that illustrates an example of the configuration of a photon-counting type X-ray CT apparatus  1 . As illustrated in  FIG.  1   , the photon-counting type X-ray CT apparatus  1  includes a gantry  10 , a bed device  20 , and a console  30 . 
     The gantry  10  is a device that emits X-rays to a subject P (patient), detects the X-rays that are transmitted through the subject P, and outputs them to the console  30 , and it includes X-ray radiation control circuitry  11 , an X-ray generation device  12 , a detector  13 , data acquisition circuitry (DAS: Data Acquisition System)  14 , a rotary frame  15 , and gantry drive circuitry  16 . 
     The rotary frame  15  is an annular frame that supports the X-ray generation device  12  and the detector  13  such that they are opposed to each other with the subject P interposed therebetween and that is rotated at high speed in a circular orbit around the subject P by the gantry drive circuitry  16 . 
     The X-ray radiation control circuitry  11  is a device that serves as a high-voltage generation unit and supplies a high voltage to an X-ray tube  12   a , and the X-ray tube  12   a  generates X-rays by using the high voltage that is supplied from the X-ray radiation control circuitry  11 . Under the control of scan control circuitry  33 , the X-ray radiation control circuitry  11  adjusts the tube voltage or the tube current that is supplied to the X-ray tube  12   a , thereby adjusting the amount of X-rays that are emitted to the subject P. 
     Furthermore, the X-ray radiation control circuitry  11  switches a wedge  12   b . Furthermore, the X-ray radiation control circuitry  11  adjusts the numerical aperture of a collimator  12   c , thereby adjusting the radiation range (the fan angle or the cone angle) of X-rays. Moreover, there may be a case where multiple types of wedges are manually switched by an operator. 
     The X-ray generation device  12  is a device that generates X-rays and emits the generated X-rays to the subject P, and it includes the X-ray tube  12   a , the wedge  12   b , and the collimator  12   c.    
     The X-ray tube  12   a  is a vacuum tube that emits X-ray beams to the subject P by using the high voltage that is supplied by the X-ray radiation control circuitry  11 , and it emits X-ray beams to the subject P in accordance with the rotation of the rotary frame  15 . The X-ray tube  12   a  generates X-ray beams that spread with the fan angle and the cone angle. For example, under the control of the X-ray radiation control circuitry  11 , the X-ray tube  12   a  is capable of continuously emitting X-rays all around the subject P for a full reconstruction or continuously emitting X-rays for a half reconstruction within an emission range (180°+the fan angle) that enables a half reconstruction. Furthermore, under the control of the X-ray radiation control circuitry  11 , the X-ray tube  12   a  is capable of intermittently emitting X-rays (pulse X-rays) at a previously set position (tube position). Furthermore, the X-ray radiation control circuitry  11  is capable of changing the intensity of X-rays, emitted from the X-ray tube  12   a . For example, the X-ray radiation control circuitry  11  increases the intensity of X-rays, emitted from the X-ray tube  12   a , at a specific tube position, and it decreases the intensity of X-rays, emitted from the X-ray tube  12   a , in the area other than the specific tube position. 
     The wedge  12   b  is an X-ray filter that adjusts the amount of X-rays with regard to the X-rays that are emitted from the X-ray tube  12   a . Specifically, the wedge  12   b  is a filter that transmits and attenuates X-rays, emitted from the X-ray tube  12   a , such that X-rays, emitted from the X-ray tube  12   a  to the subject P, has a predetermined distribution. For example, the wedge  12   b  is a filter that is obtained by processing aluminum so as to have a predetermined target angle or a predetermined thickness. Furthermore, the wedge is also called a wedge filter or a bow-tie filter. 
     The collimator  12   c  is a slit that narrows the irradiation range of X-rays, of which the amount of X-rays has been adjusted by the wedge  12   b , under the control of the X-ray radiation control circuitry  11 . 
     The gantry drive circuitry  16  drives and rotates the rotary frame  15  so that the X-ray generation device  12  and the detector  13  are rotated in a circular orbit around the subject P. 
     Each time an X-ray photon enters, the detector  13  outputs the signal with which the energy value of the X-ray photon may be measured. The X-ray photon is, for example, an X-ray photon that is emitted from the X-ray tube  12   a  and is transmitted through the subject P. The detector  13  includes multiple detection elements that output an electric signal (analog signal) of 1 pulse each time an X-ray photon enters. The photon-counting type X-ray CT apparatus  1  counts the number of electric signals (pulses) so as to count the number of X-ray photons that enter each of the detection elements. Furthermore, the photon-counting type X-ray CT apparatus  1  performs arithmetic processing on the signal so as to measure the energy value of the X-ray photon that causes output of the signal. 
     The above-described detection element includes, for example, a scintillator and an optical sensor, such as a photomultiplier tube. In such a case, the detector  13 , illustrated in  FIG.  1   , is an indirect-conversion type detector that converts the incident X-ray photon into scintillator light by using the scintillator and converts the scintillator light into an electric signal by using the optical sensor, such as a photomultiplier tube. Furthermore, there may be a case where the above-described detection element is a semiconductor device of, for example, cadmium telluride (CdTe), cadmium zinc telluride (CdZnTe), or the like. In such a case, the detector  13 , illustrated in  FIG.  1   , is a direct-conversion type detector that directly converts the incident X-ray photon into an electric signal. 
     For example, the detector  13 , illustrated in  FIG.  1   , is a plane detector in which detection elements are arranged in N columns in the channel direction (the direction of the X axis in  FIG.  1   ) and in M columns in the direction of the rotational center axis of the rotary frame  15  (the direction of the Z axis in  FIG.  1   ) where the gantry  10  is not tilted. When a photon enters, the detection element outputs an electric signal of one pulse. The photon-counting type X-ray CT apparatus  1  discriminates among individual pulses that are output from a detection element  131 , thereby counting the number of X-ray photons that enter the detection element  131 . Furthermore, the photon-counting type X-ray CT apparatus  1  performs arithmetic processing based on the intensity of a pulse, thereby measuring the energy value of the counted X-ray photon. 
     The data acquisition circuitry  14  is a data acquisition system (DAS), and it acquires the detection data on X-rays that are detected by the detector  13 . For example, the data acquisition circuitry  14  generates the count data that is obtained by counting the photons (X-ray photons), which come from the X-ray that is transmitted through the subject, for each energy band, and it transmits the generated count data to the console  30  that is described later. For example, if X-rays are continuously emitted from the X-ray tube  12   a  while the rotary frame  15  is rotated, the data acquisition circuitry  14  acquires the group of count data for the entire periphery (360 degrees). The data acquisition circuitry  14  also can acquire data for each view. Furthermore, the data acquisition circuitry  14  transmits each acquired count data in relation to the tube position to the console  30  that is described later. The tube position is the information that indicates the projection direction of the count data. 
     The bed device  20  is a device on which the subject P is placed and, as illustrated in  FIG.  1   , it includes a bed drive device  21  and a top board  22 . The bed drive device  21  moves the top board  22  in the direction of the Z axis to move the subject P into the rotary frame  15 . The top board  22  is a board on which the subject P is placed. Furthermore, in the present embodiment, an explanation is given of a case where the relative position between the gantry  10  and the top board  22  is changed by controlling the top board  22 ; however, this is not a limitation on the embodiment. For example, if the gantry  10  is self-propelling, the relative position between the gantry  10  and the top board  22  may be changed by controlling driving of the gantry  10 . 
     Furthermore, for example, the gantry  10  conducts helical scan to scan the subject P in a helical fashion by rotating the rotary frame  15  while the top board  22  is moved. Alternatively, the gantry  10  conducts conventional scan to scan the subject P in a circular orbit by rotating the rotary frame  15  with the position of the subject P fixed after the top board  22  is moved. Alternatively, the gantry  10  implements a step-and-shoot method to conduct conventional scan at multiple scan areas by moving the position of the top board  22  at a constant interval. 
     The console  30  is a device that receives an operation of the photon-counting type X-ray CT apparatus  1  from an operator and that reconstructs X-ray CT image data by using the projection data that is acquired by the gantry  10 . As illustrated in  FIG.  1   , the console  30  includes input circuitry  31 , a display  32 , the scan control circuitry  33 , preprocessing circuitry  34 , memory circuitry  35 , image reconstruction circuitry  36 , and processing circuitry  37 . 
     The input circuitry  31  includes a mouse, keyboard, trackball, switch, button, joystick, or the like, which is used by an operator of the photon-counting type X-ray CT apparatus  1  to input various commands or various settings, and it transfers the information on the command or setting, received from the operator, to the processing circuitry  37 . For example, the input circuitry  31  receives, from an operator, a capturing condition for X-ray CT image data, a reconstruction condition for reconstructing X-ray CT image data, an image processing condition for X-ray CT image data, or the like. 
     The display  32  is a monitor that is viewed by an operator and, under the control of the processing circuitry  37 , it displays the image data, generated from X-ray CT image data, to the operator or displays a graphical user interface (GUI) for receiving various commands, various settings, or the like, from the operator via the input circuitry  31 . 
     The scan control circuitry  33  controls operations of the X-ray radiation control circuitry  11 , the gantry drive circuitry  16 , the data acquisition circuitry  14 , and the bed drive device  21  under the control of the processing circuitry  37 , thereby controlling data acquisition processing by the gantry  10 . For example, scan control circuitry  33  sends sequence control commands to data acquisition circuitry  14  to control exposure operations, as discussed in more detail below. 
     The preprocessing circuitry  34  performs correction processing, such as logarithmic conversion processing, offset correction, sensitivity correction, or beam hardening correction, on the count data that is generated by the data acquisition circuitry  14 , thereby generating corrected projection data. 
     The memory circuitry  35  stores the projection data that is generated by the preprocessing circuitry  34 . Furthermore, the memory circuitry  35  stores the image data, or the like, which is generated by the image reconstruction circuitry  36  that is described later. Moreover, the memory circuitry  35  appropriately stores processing results of the processing circuitry  37  that is described later. 
     The image reconstruction circuitry  36  reconstructs X-ray CT image data by using the projection data that is stored in the memory circuitry  35 . Here, the reconstruction method includes various methods, and it may be, for example, back projection processing. Furthermore, the back projection processing may include, for example, back projection processing by using a filtered back projection (FBP) method. Alternatively, the image reconstruction circuitry  36  may also use a successive approximation technique to reconstruct X-ray CT image data. Furthermore, the image reconstruction circuitry  36  conducts various types of image processing on X-ray CT image data, thereby generating image data. Then, the image reconstruction circuitry  36  stores, in the memory circuitry  35 , the reconstructed X-ray CT image data or the image data that is generated during various types of image processing. 
     The processing circuitry  37  controls operations of the gantry  10 , the bed device  20 , and the console  30  so as to perform the overall control on the photon-counting type X-ray CT apparatus  1 . Specifically, the processing circuitry  37  controls the scan control circuitry  33  so as to control CT scan that is conducted by the gantry  10 . Furthermore, the processing circuitry  37  controls the image reconstruction circuitry  36  so as to control image reconstruction processing or image generation processing by the console  30 . Furthermore, the processing circuitry  37  performs control such that various types of image data, stored in the memory circuitry  35 , are displayed on the display  32 . 
     Heretofore, the overall configuration of the photon-counting type X-ray CT apparatus  1  according to the first embodiment is explained. Here, each processing function, performed by each of the above-described circuitry, is stored in the memory circuitry  35  in the form of the program that is executable by the computer. Furthermore, each circuitry reads and executes each program from the memory circuitry  35 , thereby performing the above-described various functions. 
     In one example, programs corresponding to the operations of the data acquisition circuitry  14  are stored in the memory circuitry  35  in the form of a program that is executable by a computer. Processor  37  executes the programs for data acquisition circuitry  14  and sends instructions to and controls data acquisition circuitry  14  to acquire data as well as controls the transfer data from data acquisition circuitry  14 . In a second example, data acquisition circuitry  14  includes a processor that reads and executes each program from the memory circuitry  35  to implement the function that corresponds to each program. 
     Furthermore, the word “processor”, used in the above explanations, means for example a central processing unit (CPU), a graphics processing unit (GPU), or a circuit, such as an application specific integrated circuit (ASIC), or a programmable logic device (e.g., a simple programmable logic device: SPLD, a complex programmable logic device: CPLD, or a field programmable gate array: FPGA). The processor reads and executes the program, stored in the memory circuitry, to perform the function. Furthermore, a configuration may be such that, instead of storing a program in the memory circuitry, a program is directly installed in a circuit of the processor. In this case, the processor reads and executes the program, installed in the circuit, to perform the function. Furthermore, with regard to the processors according to the present embodiment, instead of the case where each processor is configured as a single circuit, multiple independent circuits may be combined to be configured as a single processor to implement the function. 
     In a transmission measurement using a photon counting energy-resolving detector (PCD), the forward model can be formulated as below:
 
 N   b ( l   1, . . . ,M )= N   0   ×∫dEw ( E ) S   b ( E )exp(−Σμ m   l   m )  (1)
 
     where, S b (E) stands for the bin response function defined as S b (E)=∫ E     bL     E     bH   dE′R(E, E′), and R(E, E′) is the detector response function, and E bL  and E bH  are the low and high energy thresholds of each counting bin.  FIG.  2    shows an example model of a typical S b (E) for a photon counting detector, where a long tail above the energy window is induced by charge sharing, k-escape and scattering effect. The low energy tail is mostly due to the finite energy resolution from the associated electronic noise. N 0  is the total flux from an air scan, μ m  and l m  are the m th  basis material linear attenuation coefficient and path length, respectively. w(E) is the normalized incident X-ray spectrum. In practise, both w(E) and S b (E) are not exactly known, and they can be combined as one term S wb (E)=w(E)S b (E), called thereafter the weighted bin response function. If S wb (E) can be calibrated through measurements, the decomposition problem at low flux conditions can be well solved. 
     For a high flux scan condition (e.g. a few percent of pulse pileup), pulse pileup introduces additional spectral distortion in the measurement. One way to correct for the pileup effect is to introduce additional correction terms (e.g. Dickmann uses the measured count rate(s) as input). And this type of additional calibration is based on an accurate estimation of the flux independent weighted bin response S wb (E). 
     In typical CT clinical scan conditions, it is common to encounter a few percent or higher pulse pileup for some measurements. The resulting effect is that material decomposition depends on the measured spectrum as well as the flux. 
     For a semiconductor-based PCD, the small pixel design is usually used, in which a smallest readout unit can be grouped into different patterns for the summed readout, and used for imaging at different resolutions. For different pixel summing schemes, the detector response is varied as the charge sharing and cross-talk effect are slightly different due to the different combined-pixel size and shape resulting from the sub-pixel size and shape. Therefore, the calibration and data processing need to account for this variation for more accurate material decomposition results. 
     For a 3 rd  generation CT system, an ASG is often installed to reject scattered photons for cleaner measurements and better image quality. For this small detector pixel design in Photon-counting computed tomography (PCCT), in theory there is no dead detection area like in the conventional scintillator-based detectors, and one can employ the same ASG design or a different one to achieve optimal performance. If the same ASG design is used, then the ASG would produce different shadows across the sub-pixels, introducing even larger difference in the charge sharing and cross-talk effect (see  FIG.  3   ). Hence, it is necessary to consider this variation factor in the forward model calibration as well. 
     In particular,  FIG.  3    shows an example of a 3×3 sub-pixel pattern with 1D ASGs that form a combined pixel. However, different sized sub-pixel patterns can be used, such as n×m sub-pixel patterns or n×n sub-pixels patterns generally where n&gt;=2. For columns c 1  and c 3 , due to the blocked area under ASG plates, the charge sharing and cross-talk effect is greatly reduced from the left/right side, making the effective detector response different from that at the center column c 2 . 
     The disclosure presented herein comprises a two-step calibration method for the PCD forward model for material decomposition. It consists of two parts: 1) estimation of the flux independent weighted bin response function S wb (E) using expectation maximization (EM) method and 2) estimation of the pileup correction term P b  (E, N b , N tot ) which is a function of energy (E) and the measured bin counts (N b , N tot ), where N b  is the individual bin count and N tot  is the total count of all the energy bins. The calibrated forward model can be expressed as:
 
 N   b ( l   1 , . . . ,)= N   0   ∫dES   wb ( E )* P   b ( E,N   b   ,N   tot )exp(−Σμ m   l   m )  (2)
 
     A series of slab measurements with known materials and path lengths are used to calibrate the above forward model. 
     Here, instead of using only two materials, as in prior arts (e.g., see Dickmann), the method uses 2-5 different materials such as polypropylene, water, aluminium, titanium/copper, and k-edge materials to calibrate the weighted bin response function S wb (E) at low flux. With more selective materials used in the calibration, the number of total path lengths is reduced to achieve equivalent or better results. 
     Step 1: With an appropriate tube spectrum modelling to capture the characteristic peaks in the incident spectrum, and a physical model to simulate the photon-counting detector spectral response, an initial guess of S wb (E) can be produced. By using the EM method (e.g., see Sidky), S wb (E) can be reliably estimated for this very ill-conditioned problem based on a few transmission measurements. 
     Here, P b (E, N b , N tot ) is assumed to be constant in Step 1. The calibrated forward model can be simplified to a system of linear equations
 
 N   b ( l   1, . . . ,M )= N   0 ∫ Emin   Emax   dES   wb ( E )exp(−Σμ ml   m )  (3)
 
     Usually, the number of data measurements (M) is much smaller than the number of unknowns (E max ). With the assumption of Poisson distribution of the data acquisition, an iterative EM algorithm can be derived to find the optimal estimation of the unknown energy bin response function S wb (E), as described below. 
     When estimating the bin response function using low flux data acquisition, the pileup effect correction P b  is assumed to be a known term (e.g. constant). So, the model is simplified to
 
 N   b   =N   0   ∫dES   wb ( E )[exp[−Σμ m ( E ) l   m ]]  (4)
 
     Let A j (E)=exp[−Σμ m (E)l m   j ] represent the attenuated path length for j-th measurement. Thus, for each measurement j, we have
 
 N   b   j   =N   0   ∫dES   wb ( E ) A   j ( E )= N   0 Σ E   S   wb ( E ) A   j ( E )  (5)
 
     With M measurements, the data acquisition can be written in the matrix form below 
     
       
         
           
             
               
                 
                   
                     
                       
                         
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     or A·S wb =N b    
     By applying the EM iterative algorithm, the S wb  can be estimated by
 
 S   wb   (k+1)   =S   wb   (k) ⊙(( A   T ·( N   b {circle around (/)}( A·S   wb   (k) ))){circle around (/)}( A   T ·1))  (6)
 
     where
         k: iteration number   ·: matrix multiplication   ⊙: element-wise multiplication   {circle around (/)}: element-wise division   1: vector of ones with size of M×1       

     the updating formula for S wb (E) is given by 
     
       
         
           
             
               
                 
                   
                     
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     Step 2: Once S wb (E) is estimated from the calibration at each tube voltage (kVp) setting for each detector pixel, it is saved as a software calibration table on the system. It will be used as an input to further estimate the pileup correction terms P b (E, N b , N tot ) at higher flux scans. Both tables are then used for the material decomposition in object/patient scans to estimate the basis material path lengths. 
     The calibration tables are updated from time to time based on the system/detector performance variations. This can also be designed as an iterative procedure. If the image quality is not good enough on a quality check phantom, this calibration process is repeated with the updated calibration tables from the last iteration as the initial guess. 
     The high level workflow of the above process is demonstrated in  FIG.  4   . Steps 1) to 4) represent the calibration workflow, and steps 5) to 8) demonstrate how the calibration tables are used in the operational scans of patients/objects to produce spectral images. 
     First, a series of low flux scans on various material slabs are collected at each tube kVp setting, which is the peak potential applied on the X-ray tube. Typical CT systems support a few kVp settings from 70 to 140 kVp which generate different energy spectrums from the X-ray tube for different scan protocol. For a CT scan, both mA and kVp need to be pre-selected before the tube is turned on. Then, the low flux weighted bin response function S wb  is estimated and with the estimated S wb , the high flux slab scans are used to estimate the additional parameters in the pileup correction term P b . With the estimation calibration tables of S wb  and P b  for each detector pixel, the quality of the calibration is checked on a quality phantom, e.g. a uniform water phantom, or phantom with multiple inserts with uniform known materials. The image quality is assessed with predefined standards, and if it is passed, the current calibration tables are saved and then used for the following patient/object scans data processing. Otherwise, the procedure goes through the first three steps again using the last iteration of S wb  and P b  as the initial guess. Here, commonly examined standards are: image CT number accuracy, uniformity, spatial resolution, noise and artifacts. To check the quality of this calibration, these metrics should all be checked, especially the accuracy and artifacts like ring or bands in the image, which indicate the calibration is not good enough. 
     To choose the optimal materials and path lengths for this calibration, one can use the normalized linear attenuation coefficient vs. energy curves, see  FIG.  5    to choose the ones that are different from each other, e.g. polypropylene, water, aluminum, titanium can be a good group of combinations for such calibrations which covers a large range of common materials present in human body. 
     In order to satisfy the low flux condition through the calibration measurement to minimize the pileup effect in the flow diagram, step 1, one can select to use nτ&lt;x, where x˜0.005-0.01 and n is the pixel count rate with the lowest tube flux setting, and τ is the effective dead time of the PCD Application Specific Integrated Circuit (ASIC). By satisfying this condition, one can calculate the shortest path length of each selected calibration material, and the rest of path lengths can either be selected by equal spacing in path length or in resulting measurement count rate. 
     For calibration of the pileup correction term P b  in step 3, the same slab material and path lengths are used for scans at high mA settings. The calibration data can be grouped for each mA and generate different correction tables for each mA setting (see  FIG.  6   ), or include measurements at all flux ranges (e.g., from low to high mA, from high to low mA, or with most frequently used values first) to generate a universal correction table for a continuous mA setting (see  FIG.  7   ). 
     The calibration measurements should be taken with sufficient statistics to minimize the influence of the statistical fluctuation. One non-limiting example is to use &gt;1000 times more statistics as the typical integration period for the calibration data sets to minimize the transferred statistical error in the calibration. Each energy bin b of the calibration measurements will be used to update the corresponding S wb (E) and P b (E, N b , N tot ). 
     Since one can only do limited number of measurements with a few energy bins, the estimation is very ill-conditioned. In this case, a good initial guess is crucial for an accurate estimation as it provides additional constraints for the EM method. One of the design variations to accommodate non-ideal detectors is to allow a more flexible energy window for each bin in the initial guess of S b , especially with small variations in the actual energy threshold setting of the ASIC. By setting the low threshold x keV lower, and high threshold y keV higher, the initial S b  becomes:
 
 S   b ( E )=∫ EbL−x   EbH+y   R ( E,E ′) dE′   (8)
 
where x, y can be chosen between 5 to 10 keV to allow certain variations in the ASIC performance, while still providing additional constraints for the EM problem.
 
     To capture the spectrum variation across the fan beam after bowtie filter and detector response variation across different detector pixels, this calibration process is done pixel by pixel with each bowtie/filter configuration. 
     The design described in the present application employs more than two materials in the calibration, which provides more sensitivity to constraint the weighted bin response function estimation problem of the photon counting detectors. 
     In addition, the method utilizes a different parameterization for the high flux pileup correction terms P b  which is now a function of E, N b  and N tot . The total count term N tot  is introduced for a better approximation of the true pileup phenomena, and can significantly improve the model capability at higher flux condition with fewer parameters. 
     Furthermore, various calibration path length ranges are used at different fan angles to improve the calibration accuracy and efficiency. The slab scans used for the forward model calibration can be selected based on the imaging task to generate the best image quality. 
     Finally, a different scheme is presented to calculate an initial guess of the weighted bin response function by enlarging the energy threshold window, to accommodate non-ideal detector/ASIC performance. 
     To accommodate various detector responses of different summing schemes with ASG, this disclosure presents calibration of the forward model with different sub-pixel summing schemes separately. The calibration tables for each pixel configuration are saved in digital data form and used accordingly in the data processing to reconstruct the images with different resolutions. 
     Steps 1) to 4) in  FIG.  4    are repeated for different pixel readout summing modes. The calibration tables are saved for each mode and then applied for the patient/object scans with the same readout configuration. 
     In a semiconductor-based photon counting detector for CT, a pixel size between 200-500 μm is usually selected for optimal performance. It is a good balance between the charge sharing effect and pulse pileup under CT scan flux conditions. It is smaller than the typical conventional scintillator-based pixel which is usually ˜1×1 mm. Therefore, the readout for image processing can have more than one modes, combining different number of sub-pixels into one or more combined pixels. 
     For the combined-pixel readout mode (N r ×N c ) (i.e., the readout mode for a plurality of sub-pixels that have been combined), the forward model calibration can be done based on the measurement of the sum (or average) of the combined pixel: 
     
       
         
           
             
               
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       FIGS.  8 A- 8 D  show examples of a 3∴3 sub-pixel scheme which forms a combined pixel equivalent to a conventional EID pixel. Four different summing schemes can be used for data readout: 3×3, 1×3, 3×1, 1×1, standing for four possible resolutions of the resulted image. 
     The disclosed calibration scheme for different pixel summing patterns for combined pixels are not limited to a specific sub-pixel pattern or a specific forward model, but can apply to any forward model calibration that is based on measurements of the known materials at different path length samples. Furthermore, the obtained counts for a sub-pixel pattern can be obtained such that an average over an entire sub-pixel pattern is obtained and a normalization factor, based on an estimate of anti-scatter grid (ASG) shadow on the sub-pixel pattern, is applied on the averaged counts. 
       FIGS.  8 A- 8 D  show various summing schemes for decomposition calibration and processing. In particular, in A), summing is performed over the combined pixel, e.g. a 3×3 summing mode. In B), summing is performed over the row direction, e.g. a 1×3 summing mode. In C), summing is performed over the channel direction, e.g. a 3×1 summing mode. Finally, in D), calibration is based on individual sub-pixel. The object scan material decomposition can choose to use one of the summing patterns with the corresponding calibrated tables. 
     By calibrating each sub-pixel and averaging the material decomposition results in the estimated path length sinogram for different combined readout modes, the disclosed method handles the averaging effect in the calibration process directly, avoiding complications in the data weighting after the decomposition step. 
     In order to reduce the calibration time, all the slab calibration data can be acquired using the smallest detector pixel unit (micro-pixel), then combined digitally into different summing patterns in the later on data processing, which can be either carried out in the front-end electronics, or detector/system firmware (FPGA). 
     Numerous modifications and variations of the embodiments presented herein are possible in light of the above teachings. It is therefore to be understood that within the scope of the claims, the disclosure may be practiced otherwise than as specifically described herein.