Patent Publication Number: US-11653892-B2

Title: Counting response and beam hardening calibration method for a full size photon-counting CT system

Description:
BACKGROUND 
     Technical Field 
     The disclosure relates to counting response and beam hardening calibration 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 measurements of multiple energy bin counts for each integration period. Compared to the conventional scintillator based EID, the PCD measurement takes equal weight for each registered photon in a few different energy bins. It allows the generation of counting images with better signal-to-noise ratio as well as spectral images with multiple basis materials. 
     However, due to the detection physics in such semiconductor materials (e.g. CdTe/CZT), the detector energy response is largely deviated from ideal response 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. With the 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. 
     A counting image measurement uses all the registered photons in the detector above certain energy threshold. The accurate modeling of the detector response is required to resolve the attenuated line integral. Calibration methods have been proposed to solve similar problems in literature. The general idea is to use multiple transmission measurements of various known path lengths to modify the forward model such that it agrees with the calibration measurements, see Sidky et al., Journal of Applied Physics 97(12), 124701 (2005); Duan et al., Medical Physics 38(2), February, 2011; and Dickmann et al., Proc. SPIE 10573, Medical Imaging 2018: Physics of Medical Imaging, 1057311 (Mar. 9, 2018). A calibration method is developed to generate spectral image using multiple energy bin measurement in material decomposition. For counting image measurement, the model parameterization could be different, and likely be simplified. The beam hardening effect, described below, needs to be calculated and corrected based on the calibrated counting response. 
     Most CT reconstruction algorithms assume that the x-ray source is monochromatic. In reality, the x-ray source is polychromatic and the attenuation of x-rays through tissue is frequency (i.e., energy) dependent. Higher energy photons are attenuated less than lower energy photons, thus the x-rays reaching the detector are “harder” than those that left the source. 
     If not accounted for, artifacts because of the above effect, hereinafter called beam hardening artifacts, will appear in the reconstructed images. The primary contributors to beam hardening are soft tissues with densities close to water, as well as bone or iodine contrast in cardiac scans. 
     Artifacts may include cupping (for example, with soft tissue beam hardening) as well as dark streaks and bands which can affect clinical diagnosis. Bone beam hardening artifacts may include dark streaks or bands between high density bone structures, such as bones in the scull. A dark band present in heart muscle due to beam hardening in a cardiac scan may be interpreted as ischemia, for example. 
     Theoretically, beam hardening correction is not complicated. However, with the advanced CT configurations present in current scanners (for example bowtie filters for dose reduction, wide-angle cone beam geometry for fast volume image acquisition; etc.) beam hardening correction for clinical CT becomes more challenging. 
     A particularly challenging case is cardiac imaging where a high-density CT contrast agent (typically iodinated and contrast agent) is injected into the patient. In this case, there are three primary beam hardening sources: soft tissue, bone, and iodine. Most current beam hardening methods can only correct for two materials, not three. 
     In addition, with the development of novel contrasts targeting specific diseases in molecular CT imaging, it is possible to inject more than one contrast for clinical diagnoses, which requires a multi-material beam hardening correction method. 
     Numerous methods exist to counteract the effects of beam hardening. Hardware filters, such as bowtie filters, along with linearizing correction procedures are commonly used to reduce beam hardening effects. 
     Herein, a PCD detector counting response model and a calibration method, including a workflow for corresponding beam hardening correction, is described. 
     SUMMARY 
     The embodiments presented herein relate to a two-step calibration method for the polychromatic PCD forward counting model. First, estimation of the flux independent weighted counting response function S w (E) using the expectation maximization (EM) method is performed, followed by the estimation of the pileup correction term P(E,N tot ) based on the estimated S w (E). Once S w (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. Then it is used as an input to estimate the pileup correction terms P(E,N tot ) at higher flux scans. Both tables are then used for the calculation of the beam hardening correction to generate the counting image. The beam hardening calibration may use water only correction by estimating the water equivalent path lengths directly from the counting sinogram, or use an iterative method by estimating 2 basis material path lengths from the reconstructed image for a more accurate beam hardening calculation. 
    
    
     
       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 a counting response and beam hardening calibration processing workflow. 
         FIG.  4    shows a schematic of a calibration structure design, where the pileup correction tables P b  are generated and used for each mA individually. 
         FIG.  5    shows a schematic of another calibration structure design, where a universal pileup correction table P b  is generated for the entire mA range. 
     
    
    
     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     ⁡     (     1     1   ,   …   ,   M       )       =       N   0     ×     ∫     dE   ⁢           ⁢     w   ⁡     (   E   )       ⁢       S   b     ⁡     (   E   )       ⁢     exp   ⁡     (       -     Σμ   m       ⁢           ⁢     l   m       )               ,           (   1   )               
where S b (E) denotes the bin response function defined as S b (E)=∫ EbL   EbH R(E,E′)dE′ where R(E,E) is the detector response function, and E bL  and E bH  are the low and high energy threshold of each counting bin.  FIG.  2    shows an example model of a typical (E) function for a PCD, 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 t     h    basis material linear attenuation coefficient and path length, respectively. w(E) is the normalized incident X-ray spectrum. In practice, 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.
 
     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 and this type of additional calibration is based on an accurate estimation of the flux independent weighted bin response S wb  (E). The forward model can be expressed as 
     
       
         
           
             
               
                 
                   
                     
                       N 
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                     ⁡ 
                     
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                         l 
                         
                           1 
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                           M 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
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                     ⁢ 
                     
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                             dES 
                             wb 
                           
                           ⁡ 
                           
                             ( 
                             E 
                             ) 
                           
                         
                         * 
                         
                           
                             P 
                             b 
                           
                           ⁡ 
                           
                             ( 
                             
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                               , 
                               
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                                 tot 
                               
                             
                             ) 
                           
                         
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                             ( 
                             
                               
                                 - 
                                 
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                               ⁢ 
                               
                                   
                               
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     P b  can be parameterized as a function of the energy, bin counts N b  and the total count N tot . 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. 
     For a counting measurement, the total count N tot  can be formulated as: 
     
       
         
           
             
               
                 
                   
                     N 
                     tot 
                   
                   = 
                   
                     
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                             ( 
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                             ( 
                             
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     where S w (E) is the flux independent weighted counting response function 
     If assuming only the total count N tot  is available in the measurement, P is now a function of only E and N tot , to account for both the counting and spectral related correction. For an ideal monochromatic measurement at energy E 0 , the measurement is formulated as: 
     
       
         
           
             
               
                 
                   
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                     tot 
                   
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     The difference between Eqs. (3) and (4) constitutes the beam hardening correction for the polychromatic measurement. How to calibrate the counting detector response and calculate the beam hardening correction is addressed by the present disclosure. 
     Here, a two-step calibration method for the polychromatic PCD forward counting model is presented. It comprises two parts: 
     Step 1: estimation of the flux independent weighted counting response function S w (E) using the state of the art expectation maximization (EM) method; 
     Step 2: estimation of the pileup correction term P(E,N tot ) which can be a function of energy (E) and the measured total count N tot . Once S w (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 estimate the pileup correction terms P(E,N tot ) at higher flux scans. Both tables will be used for the calculation of the beam hardening correction to generate the counting image. 
     For most materials, the attenuation coefficient can be decomposed into two basis materials, such as water and bone, or water and iodine. The forward counting model can be formulated as: 
     
       
         
           
             
               
                 
                   
                     
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                           exp 
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   - 
                                   
                                     μ 
                                     1 
                                   
                                 
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                                   1 
                                 
                               
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                                   2 
                                 
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     where μ 1  and μ 2  are the linear attenuation coefficients for the chosen basis materials, e.g. water and bone, or water and iodine and l 1  and l 2  are the corresponding line integrals of the basis materials. 
     Flat slabs with selected basis materials and thicknesses will be placed in the scan field of view to cover the entire detector. The basic measurements will be done by stationary scans where the X-ray tube is parked at multiple fixed locations. 
     With an appropriate tube spectrum modelling to capture the characteristic peaks in the incident spectrum, and a physics based model to simulate the photon-counting detector spectral response, an initial estimate of S w (E) can be produced. By using EM method, see Sidky, S w (E) can be reliably estimated for this ill-conditioned problem based on a few transmission measurements. 
     A design variation for the pileup correction term P is to also use the measured bin counts N b  in the parameterization, and P(E,N tot ,N b ) is calibrated using the same method as the above. The number of energy bins can be set from 2-6. 
     To calculate the beam hardening correction, with the calibrated detector response S w  and P, one can just estimate the water equivalent path length sinogram based on the projection measurement, as water is a good approximation for most scanning object of human body. 
     For a more precise beam hardening correction, one can estimate the path lengths sinogram of the selected basis materials, e.g. water and bone, or water and iodine, using an iterative beam hardening correction workflow: the initial sinogram from the object scan is first generated without beam hardening correction, and an initial image is generated. The image is then segmented into basis material images. Then, a forward projection is applied to calculate the estimated basis material path lengths for each detector and each view. With the estimated path lengths, the beam hardening correction is calculated and applied on the original sinogram. The corrected sinogram will be reconstructed for the next iteration image, and go through another round of basis material path length estimation and correction if needed. 
     The high level workflow is illustrated in  FIG.  3   . Steps 1) to 3) represent the counting response calibration workflow, and steps 4) to 10) represent the BHC workflow for two different BHC methods (Method 1 uses the water only correction by estimating the water equivalent path lengths directly from the counting sinogram. Method 2 uses an iterative method by estimating 2 basis material path lengths from the reconstructed image for a more accurate beam hardening calculation. This can be used in cases when Method 1 is not sufficient, e.g. head scans). The image quality is assessed with predefined standards, and if it is passed, the current corrected sinogram is saved and then used for the following patient/object scans data processing. Otherwise, the procedure goes through steps 7) to 9) in  FIG.  3   . 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. 
     Detector Response Model 
     For the PCD counting response at low flux, S w  is calibrated with low flux transmission measurements with different materials and pathlengths. This captures the count rate independent response of the counting measurement. Different from the individual bin counts required to perform material decomposition and generate spectral images, the calibration for S w  only requires a total count measurement for which the photon is registered when it passes an energy threshold. Typically, one can set the threshold between 20 to 40 keV. 
     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 w  is estimated and with the estimated S w , the high flux slab scans are used to estimate the additional parameters in the pileup correction term P. Finally, the estimation calibration tables of S w  and P for each detector pixel are saved. 
     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 the PCD counting response at high flux, a parameterization based on the photon energy E and the measured total counts N tot  is used. The implementation of the front-end electronics (ASIC) can be approximated as a nonparalyzable model. Given an effective counting deadtime τ, the measured count can be formulated as 
     
       
         
           
             
               
                 
                   
                     N 
                     tot 
                   
                   = 
                   
                     
                       N 
                       in 
                     
                     
                       1 
                       + 
                       
                         
                           N 
                           in 
                         
                         ⁢ 
                         τ 
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     Where N in  is the incident total count on the detector surface. With an ideal detector counting response, N in  can be expressed as: 
     
       
         
           
             
               
                 
                   
                     N 
                     in 
                   
                   = 
                   
                     
                       N 
                       0 
                     
                     ⁢ 
                     
                       ∫ 
                       
                         dE 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           exp 
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   - 
                                   
                                     
                                       μ 
                                       1 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       E 
                                       ) 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   l 
                                   1 
                                 
                               
                               - 
                               
                                 
                                   
                                     μ 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     E 
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   l 
                                   2 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     To employ this approximation with enough flexibility to account for certain deviation from the ideal nonparalyzable model due to complication in the real detection process and practical ASIC implementation, P(E,N tot ) is modelled as the following form: 
     
       
         
           
             
               
                 
                   
                     P 
                     ⁡ 
                     
                       ( 
                       
                         E 
                         , 
                         
                           N 
                           tot 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       + 
                       
                         Σ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           a 
                           n 
                         
                         ⁢ 
                         
                           E 
                           n 
                         
                       
                       + 
                       
                         Σ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           b 
                           n 
                         
                         ⁢ 
                         
                           N 
                           tot 
                           n 
                         
                       
                       + 
                       
                         Σ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           c 
                           nm 
                         
                         ⁢ 
                         
                           E 
                           n 
                         
                         ⁢ 
                         
                           N 
                           tot 
                           m 
                         
                       
                     
                     
                       1 
                       + 
                       
                         
                           N 
                           in 
                         
                         ⁢ 
                         τ 
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
             
               
                 
                   
                     N 
                     in 
                   
                   = 
                   
                     
                       N 
                       tot 
                     
                     
                       1 
                       - 
                       
                         
                           N 
                           tot 
                         
                         ⁢ 
                         τ 
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     The polynomial terms based on N tot  and E are introduced to account for variation from the nonparalyzable model, and are determined based on the high flux transmission measurements. 
     One parameterization variation is to include N b  in P to better capture the spectral dependent response, so that: 
     
       
         
           
             
               
                 
                   
                     P 
                     ⁡ 
                     
                       ( 
                       
                         E 
                         , 
                         
                           N 
                           tot 
                         
                         , 
                         
                           N 
                           b 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       + 
                       
                         Σ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           a 
                           n 
                         
                         ⁢ 
                         
                           E 
                           n 
                         
                       
                       + 
                       
                         Σ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           b 
                           n 
                         
                         ⁢ 
                         
                           N 
                           tot 
                           n 
                         
                       
                       + 
                       
                         Σ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           c 
                           nm 
                         
                         ⁢ 
                         
                           E 
                           n 
                         
                         ⁢ 
                         
                           N 
                           tot 
                           m 
                         
                       
                       + 
                       
                         Σ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           d 
                           n 
                         
                         ⁢ 
                         
                           N 
                           b 
                         
                       
                     
                     
                       1 
                       + 
                       
                         
                           N 
                           in 
                         
                         ⁢ 
                         τ 
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     Estimation of Dead Time τ 
     For the counting of dead time τ of each detector pixel, one can assume a universal value based on the ASIC design for every pixel. But it is possible that the counting performance is varied from pixel to pixel due to the actual design implementation. One can first estimate the effective dead time τ i  for each pixel i based on the measurements. 
     A set of air scans from low to high flux (mA) can first be collected, with each bowtie configuration. τ i  for each pixel can be estimated based on the counting curve measurement using the ideal nonparalyzable model, then used for the estimation of the pileup response term P(E,N tot ) or P(E,N tot ,N b ). 
     The counting curve can also be measured with a certain material path length, e.g. 5-15 cm of water, and capture certain spectrum related counting response in P. 
     When estimating τ using those measurements, any numerical method can be used to find the solution by minimizing the weighted error square of the slab measurement 1 to j, 
     
       
         
           
             
               
                 
                   Σ 
                   j 
                 
                 ( 
                 
                   
                     
                       ( 
                       
                         
                           N 
                           tot 
                         
                         - 
                         
                           N 
                           mod 
                         
                       
                       ) 
                     
                     2 
                   
                   
                     σ 
                     tot 
                     2 
                   
                 
                 ) 
               
               j 
             
             . 
           
         
       
     
     Two different calibration methods for the P term are illustrated in  FIG.  4    and  FIG.  5   . The method of  FIG.  4    calibrates P at a discrete current value mA and generates different correction tables for each mA setting, and the method of  FIG.  5    calibrates P for the entire mA range (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, which is a more challenging design that may need to use more parameters in P. 
     Beam Hardening Correction 
     The beam hardening correction (BHC) here is defined as the difference between eqs. (3) and eq. (4). The attenuation line integral p mono  is defined as the logged counts for a monochromatic measurement, which is the input sinogram for FBP reconstruction: 
     
       
         
           
             
               
                 
                   
                     p 
                     mono 
                   
                   = 
                   
                     
                       
                         ( 
                         
                           log 
                           ⁢ 
                           
                             
                               N 
                               tot 
                             
                             
                               N 
                               0 
                             
                           
                         
                         ) 
                       
                       mono 
                     
                     = 
                     
                       - 
                       
                         ∫ 
                         
                           
                             μ 
                             ⁡ 
                             
                               ( 
                               
                                 E 
                                 0 
                               
                               ) 
                             
                           
                           ⁢ 
                           dl 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     The logged count for a polychromatic measurement with the PCD response is:
 
 p   poly =(log  N   tot   /N   0 )poly=log(ƒ dES   w ( E )* P ( E,N   tot )*exp(−ƒμ( E ) dl ))  (12)
 
     For BHC method 1, one can estimate the water equivalent path length l w  by: 
     
       
         
           
             
               
                 
                   
                     l 
                     w 
                   
                   = 
                   
                     
                       
                         ( 
                         
                           log 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             N 
                             tot 
                             ′ 
                           
                           ⁢ 
                           
                             / 
                           
                           ⁢ 
                           
                             N 
                             0 
                           
                         
                         ) 
                       
                       ⁢ 
                       poly 
                     
                     
                       
                         μ 
                         w 
                       
                       ⁡ 
                       
                         ( 
                         
                           E 
                           0 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     Where N′ tot_poly  is the detector response corrected total counts: 
     
       
         
           
             
               
                 
                   
                     N 
                     
                       tot 
                       ⁢ 
                       _ 
                       ⁢ 
                       poly 
                     
                     ′ 
                   
                   = 
                   
                     
                       N 
                       tot 
                     
                     
                       ∫ 
                       
                         
                           
                             dES 
                             w 
                           
                           ⁡ 
                           
                             ( 
                             E 
                             ) 
                           
                         
                         * 
                         
                           P 
                           ⁡ 
                           
                             ( 
                             
                               E 
                               , 
                               
                                 N 
                                 tot 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     From eqs. (8) and (9), for detector i and view j, 
     
       
         
           
             
               
                 
                   
                     
                       p 
                       mono 
                     
                     ⁡ 
                     
                       ( 
                       
                         i 
                         , 
                         j 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       - 
                       
                         
                           μ 
                           w 
                         
                         ⁡ 
                         
                           ( 
                           
                             E 
                             0 
                           
                           ) 
                         
                       
                     
                     ⁢ 
                     
                       
                         l 
                         w 
                       
                       ⁡ 
                       
                         ( 
                         
                           i 
                           , 
                           j 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       p 
                       poly 
                     
                     ⁡ 
                     
                       ( 
                       
                         i 
                         , 
                         j 
                       
                       ) 
                     
                   
                   = 
                   
                     log 
                     ⁡ 
                     
                       ( 
                       
                         ∫ 
                         
                           
                             
                               dES 
                               w 
                             
                             ⁡ 
                             
                               ( 
                               E 
                               ) 
                             
                           
                           * 
                           
                             P 
                             ⁡ 
                             
                               ( 
                               
                                 E 
                                 , 
                                 
                                   
                                     N 
                                     
                                       tot 
                                       poly 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       i 
                                       , 
                                       j 
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                           * 
                           
                             exp 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   - 
                                   
                                     
                                       μ 
                                       w 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       E 
                                       ) 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     l 
                                     w 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       i 
                                       , 
                                       j 
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     Then, the beam hardening correction is calculated by: 
     
       
         
           
             
               
                 
                   
                     BHC 
                     ⁡ 
                     
                       ( 
                       
                         i 
                         , 
                         j 
                       
                       ) 
                     
                   
                   = 
                   
                     log 
                     ⁡ 
                     
                       ( 
                       
                         
                           ∫ 
                           
                             
                               
                                 dES 
                                 w 
                               
                               ⁡ 
                               
                                 ( 
                                 E 
                                 ) 
                               
                             
                             * 
                             
                               P 
                               ⁡ 
                               
                                 ( 
                                 
                                   E 
                                   , 
                                   
                                     
                                       N 
                                       
                                         tot 
                                         ⁢ 
                                         _ 
                                         ⁢ 
                                         poly 
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         i 
                                         , 
                                         j 
                                       
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                             
                             * 
                             
                               exp 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     - 
                                     
                                       
                                         μ 
                                         w 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         E 
                                         ) 
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       l 
                                       w 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         i 
                                         , 
                                         j 
                                       
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         - 
                         
                           ( 
                           
                             
                               - 
                               
                                 
                                   μ 
                                   w 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     E 
                                     0 
                                   
                                   ) 
                                 
                               
                             
                             ⁢ 
                             
                               
                                 l 
                                 w 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
     and the corrected polychromatic line integral is obtained: 
     
       
         
           
             
               
                 
                   
                     
                       p 
                       corr 
                     
                     ⁡ 
                     
                       ( 
                       
                         i 
                         , 
                         j 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           log 
                           ⁢ 
                           
                             
                               N 
                               tot 
                             
                             
                               N 
                               0 
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         poly 
                         ⁡ 
                         
                           ( 
                           
                             i 
                             , 
                             j 
                           
                           ) 
                         
                       
                     
                     - 
                     
                       BHC 
                       ⁡ 
                       
                         ( 
                         
                           i 
                           , 
                           j 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     17 
                     ⁢ 
                     a 
                   
                   ) 
                 
               
             
           
         
       
     
     In one embodiment, the water path length l w  can be calculated “on the fly” and the correction applied as in Eq. (17) and (17a). 
     In another embodiment, BHC(i,j) can be precalculated and resorted such that the input is the uncorrected count N tot_poly  and air flux N 0 , and the output is the corrected count: 
     
       
         
           
             
               
                 
                   
                     
                       p 
                       corr 
                     
                     ⁡ 
                     
                       ( 
                       
                         i 
                         , 
                         j 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           log 
                           ⁢ 
                           
                             
                               N 
                               tot 
                             
                             
                               N 
                               0 
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         poly 
                         ⁡ 
                         
                           ( 
                           
                             i 
                             , 
                             j 
                           
                           ) 
                         
                       
                     
                     - 
                     
                       
                         BHC 
                         ′ 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               N 
                               
                                 tot 
                                 ⁢ 
                                 _ 
                                 ⁢ 
                                 poly 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 i 
                                 , 
                                 j 
                               
                               ) 
                             
                           
                           , 
                           
                             N 
                             0 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     17 
                     ⁢ 
                     b 
                   
                   ) 
                 
               
             
           
         
       
     
     For BHC method 2, using a two-basis material based beam hardening correction, the basis material path lengths l 1 , l 2  are estimated from the initial uncorrected image, for view i, channel j. The correction is calculated as below: 
     
       
         
           
             
               
                 
                   
                     BHC 
                     ⁡ 
                     
                       ( 
                       
                         i 
                         , 
                         j 
                       
                       ) 
                     
                   
                   = 
                   
                     log 
                     ( 
                     
                       
                         ∫ 
                         
                           
                             
                               dES 
                               w 
                             
                             ⁡ 
                             
                               ( 
                               E 
                               ) 
                             
                           
                           * 
                           
                             P 
                             ⁡ 
                             
                               ( 
                               
                                 E 
                                 , 
                                 
                                   
                                     N 
                                     
                                       tot 
                                       ⁢ 
                                       _ 
                                       ⁢ 
                                       poly 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       i 
                                       , 
                                       j 
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                           * 
                           
                             exp 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   
                                     - 
                                     
                                       
                                         μ 
                                         1 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         E 
                                         ) 
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       l 
                                       1 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         i 
                                         , 
                                         j 
                                       
                                       ) 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     
                                       μ 
                                       2 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       E 
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       l 
                                       2 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         i 
                                         , 
                                         j 
                                       
                                       ) 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       - 
                       
                         ( 
                         
                           
                             
                               - 
                               
                                 
                                   μ 
                                   1 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     E 
                                     0 
                                   
                                   ) 
                                 
                               
                             
                             ⁢ 
                             
                               
                                 l 
                                 1 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 ) 
                               
                             
                           
                           - 
                           
                             
                               
                                 μ 
                                 2 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   E 
                                   0 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               
                                 l 
                                 2 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 ) 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
     E 0  is usually selected between 70-80 keV. Then the corrected projection sinogram p corr (i,j) is obtained for further reconstruction steps: 
     
       
         
           
             
               
                 
                   
                     
                       p 
                       corr 
                     
                     ⁡ 
                     
                       ( 
                       
                         i 
                         , 
                         j 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           log 
                           ⁢ 
                           
                             
                               N 
                               tot 
                             
                             
                               N 
                               0 
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         poly 
                         ⁡ 
                         
                           ( 
                           
                             i 
                             , 
                             j 
                           
                           ) 
                         
                       
                     
                     - 
                     
                       BHC 
                       ⁡ 
                       
                         ( 
                         
                           i 
                           , 
                           j 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
     Using the described methods, the beam hardening term is mostly based on calibration of a known material with minimum model dependence. 
     In one non-limiting embodiment of method 2, the basis material path lengths l 1  and l 2  can be estimated by segmenting the initial image into separated basis material images IMG 1  and IMG 2 , and calculating l 1  and l 2  by forward projecting through IMG 1  and IMG 2 . 
     In another non-limiting embodiment of method 2, l 1  and l 2  can be estimated from IMG 1  and IMG 2  using two-dimensional Fast Fourier Transform (FFT). 
     The disclosed counting detector response method is a novel calibration method specified for photon counting CT that combines the calibration and correction of X-ray spectrum, the PCD detector response, and the beam hardening effect. Compared to the conventional beam hardening calibration method, the disclosed method does not depend on the modelling of the X-ray tube spectrum and detail geometry of the beam pre-filtration, and the calculations are mostly based on calibration measurements for each individual pixel, and hence in practice can achieve better accuracy and better image quality. 
     In one non-limiting embodiment, measurements with multiple materials and known path lengths are used to calibrate the photon counting detector counting response of the forward model. The beam hardening corrections are then applied to the measured projection data sinogram and the corrected sinograms are reconstructed to the counting image at the selected single energy. 
     The count rate independent weighted total count response estimation employs EM method using low flux calibration data. The pulse pileup correction terms in the model are estimated at each high flux with the estimated weighted total count response function. The calibration tables are generated pixel by pixel, and can be scan protocols specific (kVp, mA, collimation, rotation speed etc.). 
     For the low flux weighted count response function estimation, an initial estimate with lower energy window than the default hardware setting can be used to accommodate any variation of the energy threshold in real detector performance. 
     An energy E and total count N tot  based polynomial parameterization of the pileup model is designed to capture the detail detector counting performance based on the ideal non-paralyzable counting model. 
     A variation of the pileup model design is to include energy E, total count N tot , and the individual energy bin count N b  in the polynomial parameterization. The number of energy bins used can be between 2 to 6. 
     The dead time in the base non-paralyzable model can be set universally according to the default hardware setting (ASIC) or estimated pixel by pixel through a set of measurements from low to high mA. The measurements can be taken without scanning object (air scan) or with certain slab, e.g. 5-15 cm thick water, to cover the relevant flux range during a typical object scan. 
     The calibration scans at each kVp are conducted with individual bowtie/filter configuration and the tables are generated for each configuration used in object scans. 
     The calibration path length range is designed to cover the targeted scan object size/shape. The calibration path length samples or range can be the same for all the detector channels across the fan beam, or a sub-group of the samples can be used to target the path length range of the object scan for different detector channels located at different fan angle. The samples that are used for the calibration and the resulted tables can also be imaging task specific. 
     With the calibrated detector response, the beam hardening correction can be calculated on the fly with the initial estimated path length and the measured air flux N 0 , object scan counts N tot , or N tot  and N b . 
     The beam hardening correction can also be pre-calculated for each detector pixel at each estimated path length l and the count N tot , (or with N b ), or for the measured air flux N 0 , N tot  (or with N b ). 
     In another non-limiting embodiment, an iterative beam hardening correction can also be applied using estimated basis material path lengths l 1 , l 2 . The basis material can be first segmented in the uncorrected images, and then the basis material path lengths for each view and each channel are estimated through a forward projection. This operation can go through multiple iterations until the beam hardening effect is minimized to acceptable level. 
     In addition to the total count image, the disclosed method can be applied to any energy bin measurement within the applicable energy threshold (e.g. 15 keV to max kVp of the tube). Each energy bin image can be reconstructed using the same approach with different beam hardening corrections. 
     Obviously, numerous modifications and variations of the present disclosure are possible in light of the above teachings. It is therefore to be understood that within the scope of the appended claims, the disclosure may be practiced otherwise than as specifically described herein.