Patent Publication Number: US-2017372496-A1

Title: Anti-correlated noise filter

Description:
FIELD OF THE INVENTION 
     The following generally relates to filtering image data and finds particular application to spectral computed tomography (CT), but is also amenable to other imaging processing systems. 
     BACKGROUND OF THE INVENTION 
     A computed tomography (CT) scanner includes an x-ray tube that emits radiation. The emitted radiation traverses an examination region with a subject or object located within and is detected by a detector array opposite from the x-ray tube. The detector array detects the radiation traversing the examination region and the subject located therein and generates projection data, e.g., raw detector data or projection images. A reconstructor processes the projection data and reconstructs a volumetric image of the subject or object. 
     A spectral CT scanner includes at least one x-ray tube that emits a high energy spectrum and a low energy spectrum of x-rays, which elicit different structural and spectral characteristics of the subject or object in the generated projection data. That is, the materials of the subject or object are attenuated depending on the energy used. 
     Filters can be used to reduce or suppress noise in either the projection data and/or in the reconstructed image. By reducing the noise in the projection data, e.g., projection space, or in the reconstructed image or image data, e.g., image space, the quality of the image can be improved, e.g., the structural and/or spectral characteristics of the subject or object are more visible. Filtering can also change the characteristics in the image while filtering noise and typically includes parameters or constraints to preserve the characteristics. The filters operate on each image or data independently. Particularly difficult is filtering low-frequency noise while preserving characteristics of the object or subject present at low- frequencies, such as real structures. 
     Noise is introduced by devices used to obtain the data. For example, the imaging device or scanner used to detect the x-ray radiation and generate the projection data introduces noise. Noise can also be introduced in the processing of the data. Basis decomposition processing splits or divides projection data and/or image data into spectral or energy dependent components or basis materials. Examples of decomposed spectral CT projection or image data include photoelectric absorption and Compton-scatter components, water and Iodine components, water and Calcium components, and acetal homopolymer resin, e.g., Delrin® and tin components, and/or other components including three or more basis materials rather than just pairs. The decomposition process introduces noise in pairs known as anti-correlated noise. Anti-correlated noise is introduced between two images or two projection data sets derived from a common image or common projection data set. The anti-correlated noise is negatively correlated between the image pairs or data set pairs, such that when the pairs are combined the noise is cancelled or absent. In other words, the noise introduced in each image or data set is of the same magnitude between pairs with different signs. 
     SUMMARY OF THE INVENTION 
     Aspects described herein address the above-referenced problems and others. The following describes an approach to filtering anti-correlated noise from spectral projection data sets or image data sets. The approach uses an iterative algorithm to jointly process the data sets to reduce or suppress the anti-correlated noise in each of the projection data sets or image data sets while preserving structural and/or spectral aspects. The approach can include other filtering or image processing. In one instance, the approach filters the projection data sets. In another instance, the approach filters the image data sets. 
     In one aspect, an imaging system includes an anti-correlated noise filter, which jointly filters anti-correlated noise from a first portion and a second portion, and the first portion and the second portion include anti-correlated noise. 
     In another aspect, a method of filtering image data includes jointly filtering noise from a first portion and a second portion, and the first portion and the second portion include anti-correlated noise. 
     In another aspect, a non-transitory computer readable storage medium encoded with computer readable instructions, which, when executed by a processor, causes the processor to jointly filter noise from a first portion and a second portion. The first portion and the second portion include anti-correlated noise. The filter operates iteratively according to one of the following functions: 
     
       
         
           
             
               
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     s d   0 =ScaleDown (s 0 , d), p d   0 =ScaleDown (p 0 , d), d is a scale parameter, R(·) is a roughness penalty or regularization term, λ 1  and λ 2  and are weights, Â is the estimated anti-correlated noise image, A is a prior estimate of the anti-correlated noise image, h δ (A)=δ 2 (√{square root over (1+(A/δ) 2 )}−1) is the pseudo-Huber penalty function and δ is the pseudo-Huber parameter; or 
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       BRIEF DESCRIPTION OF THE DRAWINGS 
       The invention may take form in various components and arrangements of components, and in various steps and arrangements of steps. The drawings are only for purposes of illustrating the preferred embodiments and are not to be construed as limiting the invention. 
         FIG. 1  schematically illustrates an example computing system with an anti-correlated noise filter. 
         FIG. 2  schematically illustrates an example of the spectral data manipulation and filtering with the anti-correlated noise filter and a first algorithm. 
         FIG. 3  schematically illustrates an example of the spectral data manipulation and filtering with the anti-correlated noise filter and a second algorithm. 
         FIG. 4  flowcharts a method of filtering anti-correlated noise in image pairs. 
         FIG. 5  flowcharts a method of filtering anti-correlated noise in projection data set pairs. 
     
    
    
     DETAILED DESCRIPTION OF EMBODIMENTS 
     Initially referring to  FIG. 1 , an imaging system  100  includes an imaging scanner, such as a computed tomography (CT) scanner  102 . The scanner is configured to generate projection data and/or images which decompose into anti-correlated portions. The scanner  102  includes one or more radiation sources  104 , such as an x-ray tube, which emits radiation that traverses an examination region  106 . In one instance, a mean or peak emission voltage of the radiation source  104  is switched between an emission voltage of two or more emission voltages (e.g., 80 and 140 kVp, 100 and 120 kVp, etc.). In another variation, the radiation source  104  includes a single broad spectrum x-ray tube. 
     A detector array  108  opposite the radiation source  104  detects the emitted radiation that traverses the examination region  106  and generates projection data  110  indicative of the object or subject in the examination region  106 . Where the radiation source voltage is switched between at least two emission voltages and/or includes two or more x-ray tubes emit radiation at two different emission voltages, the detector array  108  generates projection data  110  for each of the radiation source voltages. For a single broad-spectrum x-ray tube, the detector array  108  includes an energy-resolving detector (e.g., multi-layered, photon counting, etc.) that produces the spectral projection data  110 . 
     The projection data  110  can be represented as projection images, sinograms, and the like. The projection data  110  can include data organization, such as file and/or data set organization, database organizations, object and/or element definition, and the like. The projection data  110  can be stored in random access memory, such as computer memory, local memory, server memory, cloud storage, server storage, local storage, solid state storage, flash storage and the like. In one instance, the projection data  110  is stored in a Picture Archiving and Communication System (PACS)  112 , Radiology Information System (RIS), Hospital Information System (HIS), Electronic Medical Record (EMR) or other system or device communicatively connected to the system  100 . 
     A decomposition unit  114  decomposes the projection data into at least a first portion  116  and a second portion  118 , such as a pair of data sets or data structures. Each portion includes noise, which is anti-correlated with noise in the other portion. For example, the decomposition unit  114  decomposes spectral projection data into photoelectric and Compton-scatter components, water and Iodine components, water and Calcium components, or acetal homopolymer resin, e.g., Delrin® and tin components, and/or other basis material sets. 
     An anti-correlated noise filter  120  jointly processes the first portion  116  and second portion  118 . The anti-correlated noise filter  120  iteratively filters the first portion  116  and second portion  118 . The anti-correlated noise filter  120  receives two projection data sets or image data sets as input and outputs two anti-correlated noise filtered projection data sets or image data sets. The iteration can be performed until a stopping criteria is reached, such as a predetermined number of iterations, a predetermined elapsed time, a difference between the input pairs and the output pairs satisfying a predetermined difference, combinations thereof, and the like. As described in greater detail below, the anti-correlated noise filter  120  jointly filters according to a function of a first algorithm or a second algorithm, which minimizes the anti-correlated noise. The functions are implemented by one or more methods, which filter out the anti-correlated noise in the output projection data sets or image data sets. 
     A reconstructor  122  reconstructs the filtered first portion  116  and second portion  118  into each into an image and/or a combined image. In one instance, the reconstructor  122  reconstructs each of the filtered first portion  116  and the filtered second portion  118 , each into separate images. The separate images can be combined to form a combined image which is displayed on a display device  124 . In another instance, the filtered first portion  116  and filtered second portion  118  are combined in projection space and then reconstructed as one image, which is displayed on the display device  124  and/or stored, such as in the PACS  112 . 
     In one instance, the reconstructor  122  reconstructs the first portion  116  and second portion  118 , e.g., from projection space, into images, e.g., into image space, and the anti-correlated noise filter  120  jointly filters the reconstructed images of the reconstructed first portion  116  and the reconstructed second portion  118 . The filtered reconstructed first portion  116  and the filtered reconstructed second portion  118  can be combined into an image or otherwise manipulated. The combined image is displayed on the display device  124  and/or stored in the PACS  112 . 
     The decomposition unit  114 , the anti-correlated noise filter  120  and the reconstructor  122  are suitably embodied by a data processor  126 , such as a electronic data processor, microprocessor, digital processor, optical processor, and the like, configured to execute computer readable instructions stored in a non-transitory computer readable storage medium or computer readable memory, e.g., software. 
     The processor  126  can also execute computer readable instructions carried by a carrier wave, a signal or other transitory medium to perform the disclosed techniques. The processor  126  can receive parameters through one or more input devices  128 , such as a keyboard, mouse, touch screen, microphone, and the like. The processor  126 , display device  124  and input device  128  can comprise a computing device  130 , such as a desktop computer, laptop, smartphone, body worn device, distributedly connected computing devices, such as a server and a communicatively connected peer or client computer, and the like. 
     With reference to  FIG. 2 , an example of spectral data  200  manipulation and filtering with the anti-correlated noise filter and a first algorithm is schematically illustrated. The spectral data  200  can include either the projection data  110  or reconstructed image data. The decomposition unit  114  decomposes the spectral data  200  into the first portion  116 , such as Compton scatter or scatter, and into the second portion  118 , such as photoelectric or photo. 
     A Structure Propagation (SP) filter  202  can be used to initially filter the decomposed portions individually. The SP filter  202  can be a bi-lateral filter that uses information on edges of the combined image  200  or a weighted combination. The output is an initially filtered first portion  204  and an initially filtered second portion  206 . In a variation, the SP filter  202  is omitted. 
     The anti-correlated noise filter  120  iteratively filters the initially filtered first portion  204  and an initially filtered second portion  206 . In one instance, the initially filtered first portion  204  and an initially filtered second portion  206  are the first portion  116  and the second portion  118 , e.g., without the SP filter  202 . The anti-correlated noise filter  120  uses a minimization function in a first algorithm to iteratively filter the decomposed image volume pair s 0  204 and p 0  206 from the combined projection data or image volume u 0 . An example minimization function is shown in EQUATION 1: 
     
       
         
           
             
               
                 
                   
                     
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                   EQUATION 
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     where R(p) and R(s) are roughness penalties or regularization terms for p and s, respectively, u 0  is an image volume where the correlated noise maximally cancels out with the initially decomposed portions, p 0  and s 0 , e.g., u 0 =p 0 +s 0 , p and s are the filtered image volumes, and λ u , λ p  and λ s  are weights. In one instance, the roughness penalty includes a Huber penalty, such as 
     
       
         
           
             
               
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     and δ is a Huber parameter. R(s) is similarly constructed. In another instance, R(p)is a total variation penalty, such as R(p)=∫| V p|. The constraint λ u  (p+s−u 0 ) ensures that the edges are maintained in the original minimal noise (sum) image volume u 0 , while the constraints λ p  (p−p 0 ) and λ s  (s−s 0 ) reduce the amount of crosstalk between the image volumes p and s. The weights provided by the lambda&#39;s can be adjusted to provide different trade-offs between smoothing and edge preservation and can vary across each image volume. If noise in the initial image volumes p 0  and s 0  is not perfectly anti-correlated,the image volumes can be balanced by scaling both portions, such as b 1 p 0  and b 2 s 0 , where b 1  and b 2  are scaling factors are picked so that the anti-correlated noise completely cancels out in the sum u 0 =b 1 p 0 +b 2 s 0 . 
     The minimization function of EQUATION 1 can be implemented by iterative update functions shown in EQUATIONS 2 and 3: 
     
       
         
           
             
               
                 
                   
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                   EQUATION 
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                       p 
                       
                         i 
                         , 
                         j 
                         , 
                         k 
                       
                       
                         n 
                         + 
                         1 
                       
                     
                     = 
                     
                       
                         
                           
                             λ 
                             
                               i 
                               , 
                               j 
                               , 
                               k 
                             
                             u 
                           
                            
                           
                             ( 
                             
                               
                                 u 
                                 
                                   i 
                                   , 
                                   j 
                                   , 
                                   k 
                                 
                                 0 
                               
                               - 
                               
                                 s 
                                 
                                   i 
                                   , 
                                   j 
                                   , 
                                   k 
                                 
                                 n 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             λ 
                             
                               i 
                               , 
                               j 
                               , 
                               k 
                             
                             p 
                           
                            
                           
                             s 
                             
                               i 
                               , 
                               j 
                               , 
                               k 
                             
                             0 
                           
                         
                         + 
                         
                           
                             δΣ 
                             D 
                           
                            
                           
                             ϕ 
                             D 
                             n 
                           
                            
                           
                             s 
                             D 
                             n 
                           
                         
                       
                       
                         
                           λ 
                           
                             i 
                             , 
                             j 
                             , 
                             k 
                           
                           u 
                         
                         + 
                         
                           λ 
                           
                             i 
                             , 
                             j 
                             , 
                             k 
                           
                           p 
                         
                         + 
                         
                           
                             δΣ 
                             D 
                           
                            
                           
                             ϕ 
                             D 
                             n 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   EQUATION 
                    
                   
                       
                   
                    
                   3 
                 
               
             
           
         
       
     
     where λ u , λ p  and λ 2  are weights, is the initial combined image or projection data, p 0  and s° are the initial input images or projection data, p n  and s n  are current values of the n th  iteration, p n+1  and s n+1  are the next iteration filtered projection data or images, D includes a set of each orthogonal direction {E(ast), W(est), S(outh), N(orth), U(p), and (d)O(wn))}, and i,j,k represent a current voxel in the image or position in projection space volume, and δ is a Huber parameter. The weights, σ of s and φ of p, for each direction, such as E(east), are given by EQUATION 4 and EQUATION 5: 
     
       
         
           
             
               
                 
                   
                     
                       σ 
                       
                         E 
                         , 
                         1 
                       
                       n 
                     
                     = 
                     
                       
                         
                           
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         s 
                                         E 
                                       
                                       - 
                                       s 
                                     
                                     ) 
                                   
                                   2 
                                 
                                 
                                   d 
                                   y 
                                   2 
                                 
                               
                               + 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         s 
                                         N 
                                       
                                       + 
                                       
                                         s 
                                         NE 
                                       
                                       - 
                                       
                                         s 
                                         S 
                                       
                                       - 
                                       
                                         s 
                                         SE 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 
                                   4 
                                    
                                   
                                     d 
                                     x 
                                     2 
                                   
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       s 
                                       U 
                                     
                                     + 
                                     
                                       s 
                                       EU 
                                     
                                     - 
                                     
                                       s 
                                       O 
                                     
                                     - 
                                     
                                       s 
                                       EO 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               
                                 4 
                                  
                                 
                                   d 
                                   z 
                                   2 
                                 
                               
                             
                           
                         
                       
                     
                   
                   ; 
                 
               
               
                 
                   EQUATION 
                    
                   
                       
                   
                    
                   4 
                 
               
             
             
               
                 
                   
                     
                       ϕ 
                       
                         E 
                         , 
                         1 
                       
                       n 
                     
                     = 
                     
                       
                         
                           
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         p 
                                         E 
                                       
                                       - 
                                       p 
                                     
                                     ) 
                                   
                                   2 
                                 
                                 
                                   d 
                                   y 
                                   2 
                                 
                               
                               + 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         p 
                                         N 
                                       
                                       + 
                                       
                                         p 
                                         NE 
                                       
                                       - 
                                       
                                         p 
                                         S 
                                       
                                       - 
                                       
                                         p 
                                         SE 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 
                                   4 
                                    
                                   
                                     d 
                                     x 
                                     2 
                                   
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       p 
                                       U 
                                     
                                     + 
                                     
                                       p 
                                       EU 
                                     
                                     - 
                                     
                                       p 
                                       O 
                                     
                                     - 
                                     
                                       p 
                                       EO 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               
                                 4 
                                  
                                 
                                   d 
                                   z 
                                   2 
                                 
                               
                             
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   EQUATION 
                    
                   
                       
                   
                    
                   5 
                 
               
             
           
         
       
     
     where s and p are the corresponding current voxels, e.g., s i,j,k   n  and p i,j,k   n , and subscripts E, N, NE, S, SE, U, EU, O, EO refer to the voxels or pixels east, north, northeast, south, southeast, up, east up, down, and east down, respectively of the current voxel or pixel, e.g., the voxels in orthogonally adjacent voxels, e.g., E, N, S, U, and O, which are not opposite or west, and two additional voxels east of the current voxel, which are not orthogonal and in the same direction, such as NE and SE, d x is the voxel distance in the North/South direction in the image volume, d y  is the voxel distance in the East/West direction in the image volume, d z  is the voxel distance in the Up/Down direction in the image volume. The weighing is derived analogously for each remaining direction. In each direction D ∈ (E, W, N, S, U, O), the weights σ D,1  and φ D,1  are adjusted to 
     
       
         
           
             
               σ 
               D 
             
             = 
             
               
                 
                   1 
                   
                     
                       d 
                       D 
                     
                      
                     
                       max 
                        
                       
                         ( 
                         
                           
                             σ 
                             
                               D 
                               , 
                               1 
                             
                           
                           , 
                           δ 
                         
                         ) 
                       
                     
                   
                 
                  
                 
                     
                 
                  
                 and 
                  
                 
                     
                 
                  
                 
                   ϕ 
                   D 
                 
               
               = 
               
                 1 
                 
                   
                     d 
                     D 
                   
                    
                   
                     max 
                      
                     
                       ( 
                       
                         
                           ϕ 
                           
                             D 
                             , 
                             1 
                           
                         
                         , 
                         δ 
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
     and used in update equations 2 and 3, where d s , d N :=d x ; d E , d W =d y ; and d U , d O =d z . 
     The output of each iteration is an anti-correlated noise filtered first portion  208  and an anti-correlated second portion  210 , such as the s n+1  and the p n+1  projection data or image volume. The output is used as the input in a next iteration. The anti-correlated noise filtered first portion  208  and the anti-correlated second portion  210  as projection data can be reconstructed into separate images and then combined as an anti-correlated filtered image  212 . The anti-correlated noise filtered first portion  208  and the anti-correlated second portion  210  as reconstructed images can be combined as the anti-correlated filtered image  212 . 
     With reference to  FIG. 3 , an example of the spectral data manipulation and filtering with the anti-correlated noise filter  120  and a second algorithm is schematically illustrated. The spectral data  200  can include either the projection data  110  or image data. The decomposition unit  114  decomposes the spectral data  200  into the first portion  116 , such as the Compton scatter or scatter, and into the second portion  118 , such as the photoelectric or photo. 
     The denoising Structure Propagation (SP) filter  202  can be used to initially filter the decomposed portions individually. In one instance, the SP filter is omitted. The output is an initially filtered first portion  204  and an initially filtered second portion  206 . The anti-correlated noise filter  120  iteratively filters the initially filtered first portion  204  and an initially filtered second portion  206 . In one instance, the initially filtered first portion  204  and an initially filtered second portion  206  are the first portion  116  and the second portion  118 , e.g., without the SP filter  202 . The anti-correlated noise filter  120  uses a minimization function of a second algorithm to iteratively filter the decomposed image pair s 0  204 and p 0  206 from the combined projection data or image u 0 . An example of the constrained minimization function of the second algorithm is shown in EQUATION 6: 
     
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         s 
                         ^ 
                       
                       , 
                       
                         p 
                         ^ 
                       
                     
                     ) 
                   
                   = 
                   
                     
                       arg 
                        
                       
                           
                       
                        
                       
                         
                           min 
                           
                             ( 
                             
                               s 
                               , 
                               p 
                             
                             ) 
                           
                         
                          
                         
                           α 
                            
                           
                               
                           
                            
                           
                             R 
                              
                             
                               ( 
                               s 
                               ) 
                             
                           
                         
                       
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           α 
                         
                         ) 
                       
                        
                       
                         R 
                          
                         
                           ( 
                           p 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   EQUATION 
                    
                   
                       
                   
                    
                   6 
                 
               
             
           
         
       
         
         
           
             subject to the constraints that (s.t.) 
             1. s and p are obtained by removing negatively correlated estimated noise from s 0  and p 0 , respectively; 
             2. {circumflex over (m)} monochromatic image is unchanged; and 
             3. image frequencies outside band frequencies are unchanged,
 
where R(p) and R(s) are roughness penalties or regularization terms for p and s, respectively, such as R(x)=∫| V x|, {circumflex over (m)} is an energy level parameter in keV unit, and α is an algorithm control parameter with a default value, such as equal to 0.5.
 
           
         
       
    
     The minimization constrained by the condition that the virtual monochromatic image or projection data with energy {circumflex over (m)} does not change. The energy {circumflex over (m)} 300 is selected such that its anti-correlated noise is minimized. In one instance, {circumflex over (m)} is defined by: 
     
       
         
           
             
               
                 
                   
                     
                       m 
                       ^ 
                     
                     = 
                     
                       arg 
                        
                       
                           
                       
                        
                       
                         
                           min 
                           m 
                         
                          
                         
                           
                             R 
                              
                             
                               ( 
                               
                                 
                                   
                                     
                                       c 
                                       s 
                                     
                                      
                                     
                                       ( 
                                       m 
                                       ) 
                                     
                                   
                                    
                                   s 
                                 
                                 + 
                                 
                                   
                                     
                                       c 
                                       P 
                                     
                                      
                                     
                                       ( 
                                       m 
                                       ) 
                                     
                                   
                                    
                                   p 
                                 
                               
                               ) 
                             
                           
                           
                             
                               
                                 
                                   c 
                                   s 
                                 
                                  
                                 
                                   ( 
                                   m 
                                   ) 
                                 
                               
                                
                               
                                 R 
                                  
                                 
                                   ( 
                                   s 
                                   ) 
                                 
                               
                             
                             + 
                             
                               
                                 
                                   c 
                                   p 
                                 
                                  
                                 
                                   ( 
                                   m 
                                   ) 
                                 
                               
                                
                               
                                 R 
                                  
                                 
                                   ( 
                                   p 
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   EQUATION 
                    
                   
                       
                   
                    
                   7 
                 
               
             
           
         
       
     
     where R is a regularization function, c s (m) and c p (m) are the coefficients of the s and p, respectively, to obtain the monochromatic image for energy m in keV. 
     In another instance, the spectral virtual monochromatic image, {circumflex over (m)}, is detected by defining a selection region of the combined spectral data  200  using a predetermined threshold value, such as −200 HU. A local standard deviation is calculated for a neighborhood of size ne for the combined spectral data  200 . A set, q of locations is created of the r smallest local standard deviations located in the selection region and an example of {circumflex over (m)} is defined in EQUATION 8: 
     
       
         
           
             
               
                 
                   
                     
                       m 
                       ^ 
                     
                     = 
                     
                       arg 
                        
                       
                           
                       
                        
                       
                         
                           min 
                           m 
                         
                          
                         
                           Σ 
                            
                           
                               
                           
                            
                           
                             localstddev 
                              
                             
                               ( 
                               
                                 
                                   
                                     
                                       
                                         c 
                                         s 
                                       
                                        
                                       
                                         ( 
                                         m 
                                         ) 
                                       
                                     
                                      
                                     s 
                                   
                                   + 
                                   
                                     
                                       
                                         c 
                                         p 
                                       
                                        
                                       
                                         ( 
                                         m 
                                         ) 
                                       
                                     
                                      
                                     p 
                                   
                                 
                                 , 
                                 q 
                                 , 
                                 ne 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   EQUATION 
                    
                   
                       
                   
                    
                   8 
                 
               
             
           
         
       
     
     where the local standard deviation is calculated only over the set q and ne specifies the neighbourhood of the local standard deviation. 
     The minimization function of EQUATION 6 is constrained to process image frequencies only within a band of frequencies, this is obtained as follows: the input p and s images are down scaled by a factor of d, which includes the range (0,1] and 1 is no scaling, 0.5 is scaling by a factor of 2, 0.25 is scaling down by a factor of 4, etc. Scaling down the images combines pixels through averaging and/or interpolation. The following optimization, EQUATION 9, is performed, using for example the lagged diffusivity fixed point iteration algorithm: 
     
       
         
           
             
               
                 
                   
                     A 
                     ^ 
                   
                   = 
                   
                     
                       arg 
                        
                       
                           
                       
                        
                       
                         
                           min 
                           A 
                         
                          
                         
                           α 
                            
                           
                               
                           
                            
                           
                             R 
                              
                             
                               ( 
                               
                                 
                                   s 
                                   d 
                                 
                                 + 
                                 
                                   
                                     
                                       c 
                                       p 
                                     
                                      
                                     
                                       ( 
                                       
                                         m 
                                         ^ 
                                       
                                       ) 
                                     
                                   
                                    
                                   A 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           α 
                         
                         ) 
                       
                        
                       
                         R 
                          
                         
                           ( 
                           
                             
                               p 
                               d 
                             
                             - 
                             
                               
                                 
                                   c 
                                   s 
                                 
                                  
                                 
                                   ( 
                                   
                                     m 
                                     ^ 
                                   
                                   ) 
                                 
                               
                                
                               A 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   EQUATION 
                    
                   
                       
                   
                    
                   9 
                 
               
             
           
         
       
     
     where α is an algorithm control parameter, R is the roughness penalties or regularization terms, c s ({circumflex over (m)}) and c p ({circumflex over (m)}) are the coefficients of the downscaled images s d  and p d , respectively, that enable the algorithm to obtain the monochromic image for energy {circumflex over (m)} keV, d is the downscaling factor, and Â is the estimate of the anti-correlated noise image. The denoised down scaled s and p image are defined in EQUATION 10 and 11: 
       s d   denoised =s d +c p ({circumflex over (m)})Â and p d   denoised =p d −c s ({circumflex over (m)})Â.   EQUATION 10 and 11
 
     The denoised down scaled monochromatic image at energy {circumflex over (m)} in keV doesn&#39;t change, as shown in an example EQUATION 12: 
     
       
         
           
             
               
                 
                   
                     I 
                     
                       d 
                       , 
                       
                         
                           m 
                           ^ 
                         
                          
                         ke 
                          
                         
                             
                         
                          
                         V 
                       
                     
                     denoised 
                   
                   = 
                   
                     
                       
                         
                           
                             c 
                             p 
                           
                            
                           
                             ( 
                             
                               m 
                               ^ 
                             
                             ) 
                           
                         
                          
                         
                           p 
                           d 
                           denoised 
                         
                       
                       + 
                       
                         
                           
                             c 
                             p 
                           
                            
                           
                             ( 
                             
                               m 
                               ^ 
                             
                             ) 
                           
                         
                          
                         
                           s 
                           d 
                           denoised 
                         
                       
                     
                     = 
                     
                       
                         
                           
                             
                               c 
                               p 
                             
                              
                             
                               ( 
                               
                                 m 
                                 ^ 
                               
                               ) 
                             
                           
                            
                           
                             ( 
                             
                               
                                 p 
                                 d 
                               
                               - 
                               
                                 
                                   
                                     c 
                                     s 
                                   
                                    
                                   
                                     ( 
                                     
                                       m 
                                       ^ 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   A 
                                   ^ 
                                 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             
                               c 
                               s 
                             
                              
                             
                               ( 
                               
                                 m 
                                 ^ 
                               
                               ) 
                             
                           
                            
                           
                             ( 
                             
                               
                                 s 
                                 d 
                               
                               + 
                               
                                 
                                   
                                     c 
                                     p 
                                   
                                    
                                   
                                     ( 
                                     
                                       m 
                                       ^ 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   A 
                                   ^ 
                                 
                               
                             
                             ) 
                           
                         
                       
                       = 
                       
                         
                           
                             
                               
                                 c 
                                 p 
                               
                                
                               
                                 ( 
                                 
                                   m 
                                   ^ 
                                 
                                 ) 
                               
                             
                              
                             
                               p 
                               d 
                             
                           
                           + 
                           
                             
                               
                                 c 
                                 s 
                               
                                
                               
                                 ( 
                                 
                                   m 
                                   ^ 
                                 
                                 ) 
                               
                             
                              
                             
                               s 
                               d 
                             
                           
                         
                         = 
                         
                           I 
                           
                             d 
                             , 
                             
                               
                                 m 
                                 ^ 
                               
                                
                               ke 
                                
                               
                                   
                               
                                
                               V 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   EQUATION 
                    
                   
                       
                   
                    
                   12 
                 
               
             
           
         
       
     
     The denoised down scaled monochromatic images are scaled up to generate the {circumflex over (p)} and ŝ output images, in which only image frequencies within a band of frequencies were processed. Examples of scaling up the denoised down scaled monochromatic images are shown in EQUATIONS 13-16: 
       s t =s+ScaleUp(s d +c p  ({circumflex over (m)})Â−ScaleDown (s, d), d)=s+ScaleUp(c p  ({circumflex over (m)})Â, d) and ŝ=s t +ScaleUp (ScaleDown(s−s t , u), u)   EQUATION 13 and 14
 
       p t =p+ScaleUp(p d −c s ({circumflex over (m)})Â−ScaleDown (p,d),d)=p−ScaleUp(c s ({circumflex over (m)})Â, d) and {circumflex over (p)}=p t +ScaleUp (ScaleDown(p−p t , u), u),   EQUATION 15 and 16
 
     where u is a scale factor parameter, such as u=1/16. 
     The operator s u =ScaleUp(s,u) returns the image s u  that is 
     
       
         
           
             1 
             u 
           
         
       
     
     times the size of the image s, and the operator s d  =ScaleDown(s,d) returns the image s d  that is d times the size of the image s d =ScaleDown(s,d) returns the image s d  that is d times the size of the image s. Note that s t  and p t  represent intermediate results in which the image high frequency are preserved, i.e., the high frequencies of the images s t  and p t  are very similar to the high frequencies of the input images s and p, respectively. 
     The output of each iteration is an anti-correlated noise filtered first portion  208  and an anti-correlated second portion  210 , such as the Ŝ and the {circumflex over (P)} projection data or image. The output is used as the input in a next iteration. The anti-correlated noise filtered first portion  208  and the anti-correlated second portion  210  as projection data can be reconstructed into separate images and then combined as an anti-correlated filtered image  212 . The anti-correlated noise filtered first portion  208  and the anti-correlated second portion  210  as reconstructed images can be combined as the anti-correlated filtered image  212 . In another instance, the filter includes minimization of the following two functionals: 
     
       
         
           
             
               
                 
                   
                     
                       A 
                       ^ 
                     
                     = 
                     
                       
                         
                           
                             arg 
                              
                             
                                 
                             
                              
                             min 
                           
                           A 
                         
                          
                         
                             
                         
                          
                         
                           R 
                            
                           
                             ( 
                             
                               
                                 s 
                                 0 
                               
                               + 
                               A 
                             
                             ) 
                           
                         
                       
                       + 
                       
                         R 
                          
                         
                           ( 
                           
                             
                               p 
                               0 
                             
                             - 
                             A 
                           
                           ) 
                         
                       
                       + 
                       
                         ∫ 
                         
                           
                             λ 
                             1 
                           
                            
                           
                             
                               h 
                               δ 
                             
                              
                             
                               ( 
                               A 
                               ) 
                             
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         A 
                         ^ 
                       
                       = 
                       
                         
                           
                             
                               
                                 arg 
                                  
                                 
                                     
                                 
                                  
                                 min 
                               
                                
                               
                                   
                               
                             
                             A 
                           
                            
                           
                             R 
                              
                             
                               ( 
                               
                                 
                                   s 
                                   0 
                                 
                                 + 
                                 A 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           R 
                            
                           
                             ( 
                             
                               
                                 p 
                                 0 
                               
                               - 
                               A 
                             
                             ) 
                           
                         
                         + 
                         
                           ∫ 
                           
                             
                               λ 
                               2 
                             
                              
                             
                               A 
                               2 
                             
                           
                         
                       
                     
                     , 
                   
                 
               
               
                 
                   EQUATION 
                    
                   
                       
                   
                    
                   17 
                    
                   
                       
                   
                    
                   and 
                    
                   
                       
                   
                    
                   18 
                 
               
             
           
         
       
     
     where R(·) is a roughness penalty or regularization term, λ 1  and λ 2  are weights, Â is the estimated anti-correlated noise image, h 67  (A)=δ 2 (√{square root over (1+(A/δ) 2 )}−1) is the pseudo-Huber penalty function and δ is the pseudo-Huber parameter. EQUATION 17 is applied to the p 0  and s 0  images using, for example, a lagged diffusivity fixed point iteration algorithm according to EQUATION 19. 
     
       
         
           
             
               
                 
                   
                     
                       A 
                       ^ 
                     
                     
                       L 
                       1 
                     
                   
                   = 
                   
                     
                       
                         
                           arg 
                            
                           
                               
                           
                            
                           min 
                         
                         A 
                       
                        
                       
                           
                       
                        
                       
                         R 
                          
                         
                           ( 
                           
                             
                               s 
                               0 
                             
                             + 
                             A 
                           
                           ) 
                         
                       
                     
                     + 
                     
                       R 
                        
                       
                         ( 
                         
                           
                             p 
                             0 
                           
                           - 
                           A 
                         
                         ) 
                       
                     
                     + 
                     
                       ∫ 
                       
                         
                           λ 
                           1 
                         
                          
                         
                           
                             
                               h 
                               δ 
                             
                              
                             
                               ( 
                               A 
                               ) 
                             
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   EQUATION 
                    
                   
                       
                   
                    
                   19 
                 
               
             
           
         
       
     
     EQUATION 18 is applied to on the scaled down p 0  and s 0  images, with an optimization performed using, for example, the lagged diffusivity fixed point iteration algorithm according to EQUATION 20. 
     
       
         
           
             
               
                 
                   
                     
                       
                         A 
                         ^ 
                       
                       
                         L 
                         2 
                       
                       d 
                     
                     = 
                     
                       
                         
                           
                             arg 
                              
                             
                                 
                             
                              
                             min 
                           
                           A 
                         
                          
                         
                             
                         
                          
                         
                           R 
                            
                           
                             ( 
                             
                               
                                 s 
                                 d 
                                 0 
                               
                               + 
                               A 
                             
                             ) 
                           
                         
                       
                       + 
                       
                         R 
                          
                         
                           ( 
                           
                             
                               p 
                               d 
                               0 
                             
                             - 
                             A 
                           
                           ) 
                         
                       
                       + 
                       
                         ∫ 
                         
                           
                             λ 
                             2 
                           
                            
                           
                             A 
                             2 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   EQUATION 
                    
                   
                       
                   
                    
                   20 
                 
               
             
           
         
       
     
     where s d   0 =ScaleDown (s 0 , d) and p d   0 =ScaleDown (p 0 , d). The filtered images are obtained by combining the estimates from EQUATIONS 19 and 20 according to EQUATION 21. 
         Â=Â   L     1   +ScaleUp ( Â   L     2     d −ScaleDown ( Â   L     1     , d ), d ),  {circumflex over (p)}=p   0   −Â  and  ŝ=s   0   +Â,    EQUATION 21
 
     where Â is the final estimate for the anti-correlated noise image, {circumflex over (p)} and ŝ are the denoised photo and scatter images, respectively. 
     A third algorithm uses an estimated noise map and filters the anti-correlated noise with an minimization function as defined in EQUATION 22. The estimate noice map, n, is estimated using noise modeling techniques, such as with a Monte Carlo estimate, by analytical methods such as by Wunderlich and Noo in “Image Covariance and Lesion Detectability in Direct Fan-Beam X-Ray Computed Tomorgraph”, or direct extraction techniques, such as described in U.S. Pat. No. 8,938,110. 
         Â=Â   L     1   +ScaleUp( Â   L     1     d   , d )  {circumflex over (p)}=p   0   −Â  and  ŝ=s   0   +Â,    EQUATION 22
 
     where Â is the estimated anti-correlated noise image, {circumflex over (p)} and ŝ are the denoised photo and scatter images, respectively, and p 0  and s 0  are the initial or input photo and scatter images, respectively, d is a scale parameter, and Â L     1    and Â L     2     d  are defined by EQUATION 23 and 24 respectively. 
     
       
         
           
             
               
                 
                   
                     
                       A 
                       ^ 
                     
                     
                       L 
                       1 
                     
                   
                   = 
                   
                     
                       
                         
                           arg 
                            
                           
                               
                           
                            
                           min 
                         
                         A 
                       
                        
                       
                           
                       
                        
                       
                         R 
                          
                         
                           ( 
                           
                             
                               s 
                               0 
                             
                             + 
                             A 
                           
                           ) 
                         
                       
                     
                     + 
                     
                       R 
                        
                       
                         ( 
                         
                           
                             p 
                             0 
                           
                           - 
                           A 
                         
                         ) 
                       
                     
                     + 
                     
                       ∫ 
                       
                         
                           
                             λ 
                             1 
                           
                           n 
                         
                          
                         
                           
                             
                               h 
                               δ 
                             
                              
                             
                               ( 
                               A 
                               ) 
                             
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   EQUATION 
                    
                   
                       
                   
                    
                   23 
                 
               
             
           
         
       
     
     where R is a roughness penalty or regularization term, such as ∫ h δ (| V p|), A is the image, n is the estimated noise map, λ 1  is a weight, and h 67  (A)=δ 2 (√{square root over (1+(A/δ) 2 )}−1) is the pseudo-Huber penalty function and δ is the pseudo-Huber parameter. EQUATION 23 is applied to on the scaled down p 0  and s 0  images, with an optimization performed using, for example, the lagged diffusivity fixed point iteration algorithm according to EQUATION 24. 
     
       
         
           
             
               
                 
                   
                     
                       
                         A 
                         ^ 
                       
                       
                         L 
                         2 
                       
                       d 
                     
                     = 
                     
                       
                         
                           
                             arg 
                              
                             
                                 
                             
                              
                             min 
                           
                           A 
                         
                          
                         
                             
                         
                          
                         
                           R 
                            
                           
                             ( 
                             
                               
                                 s 
                                 d 
                                 0 
                               
                               + 
                               A 
                             
                             ) 
                           
                         
                       
                       + 
                       
                         R 
                          
                         
                           ( 
                           
                             
                               p 
                               d 
                               0 
                             
                             - 
                             A 
                           
                           ) 
                         
                       
                       + 
                       
                         ∫ 
                         
                           
                             ( 
                             
                               
                                 
                                   λ 
                                   2 
                                 
                                 
                                   n 
                                   2 
                                 
                               
                               + 
                               
                                 
                                   λ 
                                   3 
                                 
                                  
                                 F 
                               
                             
                             ) 
                           
                            
                           
                             A 
                             2 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   EQUATION 
                    
                   
                       
                   
                    
                   24 
                 
               
             
           
         
       
     
     where s d   0 =ScaleDown (s 0 +Â L     1   , d) and p d   0 =ScaleDown (p 0 −Â L     1   , d), λ 2  and λ 3  are weights and F is defined by EQUATION 25. 
     
       
         
           
             
               
                 
                   
                     F 
                     = 
                     
                       ScaleDown 
                        
                       
                         ( 
                         
                           
                             
                               σ 
                                
                               
                                 ( 
                                 
                                   
                                     s 
                                     d 
                                     0 
                                   
                                   + 
                                   
                                     
                                       A 
                                       ^ 
                                     
                                     
                                       L 
                                       1 
                                     
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               σ 
                                
                               
                                 ( 
                                 
                                   
                                     p 
                                     d 
                                     0 
                                   
                                   - 
                                   
                                     
                                       A 
                                       ^ 
                                     
                                     
                                       L 
                                       1 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             1 
                             + 
                             
                               
                                 σ 
                                  
                                 
                                   ( 
                                   
                                     
                                       A 
                                       ^ 
                                     
                                     
                                       L 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                               2 
                             
                           
                         
                         ) 
                       
                     
                   
                   , 
                 
               
               
                 
                   EQUATION 
                    
                   
                       
                   
                    
                   25 
                 
               
             
           
         
       
     
     where σ(x) is the local standard deviation of the image  x . 
     With reference to  FIG. 4 , a method of filtering anti-correlated noise in image pairs is flowcharted. It is to be appreciated that the ordering of the acts is not limiting. As such, other orderings are contemplated herein. In addition, one or more acts may be omitted and/or one or more additional acts may be included. 
     At  400 , spectral image data is received, such as the spectral image data  200  described in reference to  FIGS. 2 and 3 . The spectral image data can be received from memory, such as the PACS  112 , or from the reconstructor  122 . 
     The spectral image data is decomposed at  402  into the first decomposed portion, such as Compton scatter, and the second decomposed portion, such as the photoelectric absorption. The decomposed first and second portions include anti-correlated noise. 
     At  404 , each of the decomposed first and second portions can be individually filtered. For example, the first and second portions are each filtered separately with the SP filter  202  using information from the combined image  200  or spectral image data. 
     At  406 , the decomposed pair of images from  402  or the individual filtered pair from  404  are iteratively filtered as a pair using the anti-correlated noise filter  120  as described in reference to  FIG. 1  and as examples in reference to  FIGS. 2 and 3 . The iterative algorithms can be repeated until a stopping criteria is reached, such as a number of iterations, an elapse time, or a threshold change between characteristics of the input images and the output images, a combination, and the like. 
     At  408 , the output images are combined in memory. The output images can be further manipulated, such as other filtering, computations, segmentation, and the like before combining. 
     The combined image is displayed on a display device  124  at  410 . The display can include displaying the output images, i.e., anti-correlated noise filtered images. The combined image can also be output on film by a filmer or the like. 
     The above may be implemented by way of computer readable instructions, encoded or embedded on non-transitory computer readable storage medium, which, when executed by a computer processor(s), cause the processor(s) to carry out the described acts. Additionally or alternatively, at least one of the computer readable instructions is carried by a signal, carrier wave or other transitory medium. 
     With reference to  FIG. 5 , a method of filtering anti-correlated noise in projection data set pairs is flowcharted. At  500 , spectral image data is received, such as the spectral image data  200  described in reference to  FIGS. 2 and 3 . The spectral image data can be received from memory, such as the PACS  112 , or from the imaging scanner, such as the CT scanner  102 . 
     The spectral image data is decomposed at  502  into the first decomposed portion, such as Compton scatter, and the second decomposed portion, such as the photoelectric absorption. The decomposed first and second portions include anti-correlated noise. 
     At  504 , each of the decomposed first and second portions can be individually filtered. For example, the first and second portions are each filtered separately with the SP filter  202  using information from the combined image  200  or spectral image data. 
     At  506 , the decomposed pair of projection data sets from  502  or the individual filtered pair from  504  are iteratively filtered as a pair using the anti-correlated noise filter  120  as described in reference to  FIG. 1  and as examples in reference to  FIGS. 2 and 3 . The iterative algorithms can be repeated until a stopping criteria is reached, such as a number of iterations, an elapse time, or a threshold change between characteristics of the input images and the output images, a combination, and the like. 
     At  508 , the output data sets are combined in memory. The output data sets can be further manipulated, such as other filtering, computations, segmentation, and the like before combining. At  510 , the combined output data set is reconstructed by the reconstructor  122  into an image. Alternatively,  510  and  508  can be reversed, and the output data sets are reconstructed each into an image, and the two reconstructed images are combined. 
     The combined image is displayed on a display device  124  at  512 . The display can include displaying the output images, i.e., anti-correlated noise filtered images. The combined image can also be output on film by a filmer or the like. 
     The above may be implemented by way of computer readable instructions, encoded or embedded on non-transitory computer readable storage medium, which, when executed by a computer processor(s), cause the processor(s) to carry out the described acts. 
     Additionally or alternatively, at least one of the computer readable instructions is carried by a signal, carrier wave or other transitory medium. 
     The invention has been described with reference to the preferred embodiments. Modifications and alterations may occur to others upon reading and understanding the preceding detailed description. It is intended that the invention be constructed as including all such modifications and alterations insofar as they come within the scope of the appended claims or the equivalents thereof.