Abstract:
In an aspect, tomographically reconstructing a 3D image object corresponding to a data set includes reconstructing a first reconstructed object from the data set, receiving a smoothing map, smoothing the first reconstructed object based on the smoothing map thereby creating a first smoothed object, and outputting the first smoothed object as the 3D image object. 
     In another aspect, smoothing a first object thereby creating a smoothed object having a smoothed value associated to each object point in object space includes receiving the first object, determining, in a series of steps, single-kernel-smoothed objects, wherein each iteration step is associated with a kernel function and includes, determining a start object based on the first object, and smoothing the start object using the kernel function of the iteration step, thereby creating the single-kernel-smoothed object having single-kernel-smoothed values associated to each object point, and constructing the smoothed object from the single-kernel-smoothed values.

Description:
[0001]    An embodiment of the invention relates to image reconstruction, and in particular, to image reconstruction using a pixon method. 
       BACKGROUND 
       [0002]    An overview of different reconstruction methods including a pixon method is given in R. C. Puetter et al., “Digital Image Reconstruction: Deblurring and Denoising,” Annu. Rev. Astro. Astrophys., 2005, 43: 139-194; the pixon method is described in R. C. Puetter et al., “The pixon method of image reconstruction,” Astronomical Data Analysis Software and Systems VIII., edited by D. M. Mehringer, R. L,. Plante D. A. Roberts, Astronomical Society of the Pacific, San Francisco, ASP Conference Series 1999, 172, 307-316, the contents of which are herein incorporated by reference. An application of the pixon method to medical planar imaging is discussed in C. A. Wesolowski et al., “Improved lesion detection from spatially adaptive, minimally complex, pixon® reconstruction of planar scintigraphic images”, Comput. Med. Imaging Graph., 2005, 29, 65-81, the contents of which are herein incorporated by reference. 
       SUMMARY 
       [0003]    An embodiment of the invention is based in part on the recognition that pixon smoothing can be applied externally in tomographic reconstruction. 
         [0004]    In an aspect, tomographically reconstructing a 3D image object corresponding to a data set includes reconstructing a first reconstructed object from the data set and determining a pixon map based on the first reconstructed object and the data set. It further includes pixon smoothing the first reconstructed object based on the pixon map thereby creating a first smoothed object and outputting the first smoothed object as the 3D image object. 
         [0005]    In another aspect, tomographically reconstructing a 3D image object corresponding to a data set includes reconstructing a first reconstructed object from the data set, receiving a smoothing map, smoothing the first reconstructed object based on the smoothing map thereby creating a first smoothed object, and outputting the first smoothed object as the 3D image object. 
         [0006]    In another aspect, a nuclear imaging device for providing a 3D image object includes a detector unit for detecting radiation emitted from within a patient and providing a data set indicative of the detected radiation, a tomographic reconstruction unit configured to reconstruct a first reconstructed object on the basis of the data set and to provide the first reconstructed object as an output object, a pixon smoothing unit configured to receive the first reconstructed object and to smooth the first reconstructed object based on a pixon map that assigns pixon kernel unctions to object points within a 3D object space, thereby creating a first smoothed object, an output port for providing the medical image, and a control unit for controlling which of the output object and the first smoothed object is provided at the output port as the 3D image object. 
         [0007]    In another aspect, smoothing a first object thereby creating a smoothed object having a smoothed value associated to each object point in object space includes receiving the first object, determining, in a series of steps, single-kernel-smoothed objects, wherein each iteration step is associated with a kernel function and includes, determining a start object based on the first object, and smoothing the start object using the kernel function of the iteration step, thereby creating the single-kernel-smoothed object having single-kernel-smoothed values associated to each object point, and constructing the smoothed object from the single-kernel-smoothed values. 
         [0008]    Implementations may include one or more of the following features. Reconstructing a 3D image object may further include determining a quality of the first smoothed object, determining that the quality of the first smoothed object remains outside a limitation of a preset threshold value, updating the pixon map based on the first smoothed object thereby creating an updated pixon map, based on the updated pixon map, pixon smoothing the first smoothed object thereby creating a second smoothed object, and outputting the second smoothed object as the 3D image object. 
         [0009]    Pixon smoothing the first smoothed object may include determining, in a series of iteration steps, intermediate smoothed objects, wherein each iteration step includes receiving an input object, determining a smoothed object based on the input object and the pixon map, and determining a quality of the smoothed object. Pixon smoothing may further include determining that the quality of a first of the intermediate smoothed objects is inside the limitation of a preset threshold value and assigning the first of the intermediate smoothed objects as the first smoothed object. The reconstructed object may be the input object of the first iteration. The intermediate smoothed object determined in an iteration may be the input object for the next iteration. 
         [0010]    Reconstructing a 3D image object may further include reconstructing a second reconstructed object based on the data set and the first smoothed object and outputting the second reconstructed object as the 3D image object. 
         [0011]    Pixon smoothing the first reconstructed object may include a series of iteration steps, wherein each iteration step includes receiving a pixon kernel function, determining a smoothed value of a first object point within object space based on the pixon kernel function, and constructing the first smoothed object by using the smoothed value to determine an entry of the first smoothed object associated to the first object point. 
         [0012]    Determining the smoothed value based on the pixon kernel function may include selecting the first object point to be an object point to which the pixon kernel function is assigned in the pixon map, determining a set of object points associated with the first object point based on the pixon kernel function, and determining the smoothed value based on the values of the object points within the set of object points. 
         [0013]    Determining the smoothed value based on the pixon kernel function may include smoothing the first reconstructed object with the pixon kernel function thereby creating a single-kernel-forward-smoothed object, wherein the single-kernel-forward smoothed object includes as an entry the smoothed value associated to the first object point. 
         [0014]    Smoothing the first reconstructed object may include determining for each of the object points a set of object points based on the pixon kernel function and determining a smoothed value for each object point based on the values of the data points within the set of object data points. 
         [0015]    Smoothing the first reconstructed object may include, based on a value of the pixon map associated to the first object point, determining a smoothing contribution of the pixon kernel function to the pixon smoothed value associated to the first object point, and wherein constructing the first smoothed object may consider the smoothing contribution. 
         [0016]    The pixon map may include a weight for a combination of pixon kernel function and object point. Constructing the first smoothed object may further include weighting the smoothed value with the weight for that combination of pixon kernel function and object point. 
         [0017]    Determining a smoothed value of a first object point may include identifying object points of the first object that are to receive a smoothing contribution from the selected pixon kernel function as indicated in the pixon map, determining contribution factors indicative of an extent to which the selected pixon kernel function contributes to the smoothing of the selected object points, multiplying values of the object points of the first reconstructed object with corresponding contribution factors thereby creating a temporary object, and smoothing the temporary object with the pixon kernel function thereby creating a single-kernel-backward-smoothed object, wherein the single-kernel-backward-smoothed object includes as an entry the smoothed value associated to the first object point. 
         [0018]    Constructing the first smoothed object may include adding the entries of the single-kernel-backward smoothed objects for all pixon functions indicated in the pixon map. 
         [0019]    The first reconstructed object may be reconstructed as a 3D object. Reconstructing the first reconstructed object may include running an algorithm selected from the group consisting of algorithms based on maximum likelihood, algorithms based on an ordered subset expectation maximization, algorithms based on a non-negative least square fit, and algorithms based on an ordered subset non-negative least square fit. 
         [0020]    Reconstructing a 3D image object may further include detecting the data set with a nuclear imaging device. 
         [0021]    The smoothing operation may be based on smoothing selected from the group consisting of smoothing based on pixon smoothing, smoothing based on Fourier filtering, smoothing based on wavelet filtering, smoothing based on filtering with a Wiener filter, and smoothing based on filtering with a fixed filter. 
         [0022]    The reconstruction unit of the nuclear imaging device may be further configured to receive the first smoothed object as the input object for reconstructing a second reconstructed object and to provide the second reconstructed object as the output object. 
         [0023]    The pixon smoothing unit of the nuclear imaging device may be configured to receive the second reconstructed object and to smooth the second reconstructed object thereby creating a second smoothed object. 
         [0024]    The detector unit may include a positron emission tomography detector system and/or a single photon computed tomography detector system and/or a computed tomography detector system. 
         [0025]    Smoothing the first reconstructed object may includes determining, in a series of steps, single-kernel-smoothed objects, wherein each iteration step is associated with a kernel function associated to the smoothing map and includes determining a start object based on the first reconstructed object, smoothing the start object using the kernel function of the iteration step, thereby creating the single-kernel-smoothed object having single-kernel-smoothed values associated to each object point, and constructing the first smoothed object from the single-kernel-smoothed values. 
         [0026]    The first smoothed object may be used as the start object and determining single-kernel-smoothed objects may include receiving contribution factors to the smoothing of the kernel function for each object point and constructing the smoothed object may include weighting the single-kernel-smoothed values with the contribution factors. 
         [0027]    Determining single-kernel-smoothed objects may include, for example, from the smoothing map, receiving contribution factors to the smoothing of the kernel function for each object point and determining the start object may include weighting the values of the first object with the contribution factors. 
         [0028]    Determining single-kernel-smoothed objects may include receiving the kernel function from a set of pixon kernel functions and the contribution factors from a pixon map. 
         [0029]    These general and specific aspects may be implemented using a system, a method, a computer readable medium, or a computer program, or any combination of systems, methods, a computer readable medium, or a computer programs. 
         [0030]    Certain implementations may have one or more of the following advantages. Pixon smoothing can be externally applied to further smooth a reconstructed image based on the statistics of the data set. 
         [0031]    The details of one or more embodiments of the invention are set forth in the accompanying drawings and the description below. Other features, objects, and advantages of the invention will be apparent from the description and drawings, and from the claims. 
     
    
     
       DESCRIPTION OF DRAWINGS 
         [0032]      FIG. 1  is an overview of a nuclear imaging system. 
           [0033]      FIG. 2  is a flowchart illustrating pixon smoothing applied after image reconstruction (post-smoothing). 
           [0034]      FIG. 3  is a flowchart illustrating a first example of a pixon smoothing. 
           [0035]      FIG. 4  is a flowchart illustrating an example of a forward pixon smoothing operation. 
           [0036]      FIG. 5  is a flowchart illustrating a single-kernel function forward update of the forward pixon smoothing operation. 
           [0037]      FIG. 6  is a flowchart illustrating pixon smoothing between two image reconstruction processes (intermediate smoothing). 
           [0038]      FIG. 7  is a flowchart illustrating a combination of post- and intermediate pixon smoothing. 
           [0039]      FIG. 8  is a flowchart illustrating an example of a backward pixon smoothing operation. 
           [0040]      FIG. 9  is a flowchart illustrating a single-kernel function backward update of the backward pixon smoothing operation. 
           [0041]      FIG. 10  is a flowchart illustrating a general concept of a modular reconstruction system. 
       
    
    
       [0042]    Like reference symbols in the various drawings indicate like elements. 
       DETAILED DESCRIPTION 
       [0043]      FIG. 1  shows a nuclear imaging system  100  for tomography with an imaging detector  110 , and a pixon reconstruction unit  120  using a 3D pixon smoothing operation  130  that interacts with a pixon map P. The pixon smoothing operation  130  is based on a pixon method. The pixon method refers to a method that smoothes an object at each point in object space (hereafter an “object point”) by considering an assigned shape or volume for the smoothing. The object space is the space in which the result of the image reconstruction is defined and which corresponds to the 3D volume that was imaged using the nuclear imaging system  100 . A data space is given by the data points measured with the imaging detector  110 . 
         [0044]    Within this application “pixon” is used to indicate that a term, method, object etc. refers to the pixon method, and the use of variably shaped volumes when smoothing an image object. For example, the assigned shapes are defined by pixon kernel functions, and a pixon map P stores the information about which of the pixon kernel functions is assigned to which of the object points. 
         [0045]    The pixon method provides a smoothed image object I in object space that is a reconstruction of a data set D measured in data space and that fulfills statistical conditions of the data set D. The 3D reconstruction in the pixon reconstruction unit  120  includes external pixon smoothing a reconstructed image, by, for example, executing a sequence of reconstructing and smoothing operations. The pixon smoothing operation  130  uses the pixon map P. 
         [0046]    The pixon smoothing operation  130  is spatially adaptive, i.e., the pixon smoothing operation  130  depends on the measured data for every object point. To every object point, one assigns a pixon kernel function, which is the basis for the pixon smoothing operation  130 . Within the pixon reconstruction unit  120 , the pixon map P defines which of the pixon kernel functions is assigned to each of the object points. 
         [0047]    The pixon method is especially suited for reconstructing an object from a measured data set with a low number of counts and an unavoidable noise contribution. Such data sets are produced, for example, with medical imaging techniques in nuclear medicine, which produce 3D images of, for example, a functional process in a patient&#39;s body by using nuclear properties of matter. Examples of such imaging techniques are Positron Emission Tomography (PET) and Single Photon Computed Tomography (SPECT). For these types of nuclear imaging, one administers a radioactive substance to the patient and detects emitted radiation with a detector system, e.g., with a ring detector for PET or with one or several gamma cameras for SPECT. 
         [0048]    Referring to  FIG. 1 , the imaging detector  110  of the nuclear imaging system  100  detects the γ-radiation emitted from the patient. Therefore, it is positioned around or partly around the patient and could be a conventional SPECT or PET detector system. The imaging detector  110  provides the data set D to the pixon reconstruction unit  120 , which uses, for example, a system matrix H to describe the properties of the nuclear imaging system  100 , and an iteratively improved data model to calculate a 3D image object I on the basis of the data set D and the pixon map P. The 3D image object I can then be displayed on a display  140  using well-known volume rendering techniques. 
       Pixon Map Determination 
       [0049]    The pixon method includes a search for the broadest possible pixon kernel functions that consider the largest volume for the smoothing operation at each object point and that together provide an adequate fit of an object, e.g. the 3D image object I, to the data set D. The pixon kernel unctions are determined on the base of a minimum complexity approach and are used within the pixon smoothing operation  130 . An exemplary determination of a pixon map P suited for the case of low count data following a Poisson statistics is described in the co-pending U.S. Patent Application entitled “Determining a pixon map for image reconstruction,” by A. Yahil and H. Vija of even date herewith, the contents of which are herein incorporated by reference. The information about the selected pixon kernel functions is stored in the pixon map P, which assigns to each object point its pixon kernel function. 
       Reconstruction Algorithm 
       [0050]    Iterative image reconstruction methods, such as non-negative least square or Poisson-likelihood algorithms, iteratively fit image models to measured data and thus minimize the effect of noise on the image. The result of a reconstruction algorithm is an approximated image that is fit to the measured data set D according to the rules of the algorithm. Within the pixon reconstruction unit  120 , this approximated image can be used as an input object for the pixon smoothing operation  130 . 
         [0051]    The pixon reconstruction unit  120  represents an image reconstructing approach that uses an image reconstruction algorithm and a pixon smoothing operation together to fit a data model, corresponding to the 3D image object, to the measured data set D. Several examples of applying pixon smoothing operations are described with reference to  FIGS. 2 to 9 . 
       Pixon Smoothing 
       [0052]    Pixon smoothing can be viewed as averaging values of an object over a specific volume defined by the pixon kernel function. The smoothing operation can be written as a matrix operation using a pixon kernel operator K, such that the (smoothed) image object I is given by applying the pixon kernel operator K to a pseudo-image object φ′: 
         [0000]    
       
         
           
             
               I 
               α 
             
             = 
             
               
                 ∑ 
                 β 
               
                
               
                 
                   K 
                   αβ 
                 
                  
                 
                   ψ 
                   β 
                   ′ 
                 
               
             
           
         
       
     
         [0053]    “Pseudo” indicates that the smoothing operation can be understood as a transformation (using the pixon kernel operator K) from a (pseudo-)object space, i.e. the pre-Pixon smoothing space, to the object space of the 3D image object I. Applying the transpose operator of the pixon kernel operator, K T , then projects from the object space back into the pseudo-object space. 
         [0054]    In many cases, the smoothing operation is a convolution operation given by: 
         [0000]    
       
         
           
             
               I 
               α 
             
             = 
             
               
                 ∑ 
                 β 
               
                
               
                 
                   K 
                   
                     α 
                     - 
                     β 
                   
                 
                  
                 
                   ψ 
                   β 
                   ′ 
                 
               
             
           
         
       
     
         [0055]    Convolutions can be calculated, for example, by a direct summation for small pixon kernel functions and by fast Fourier transforms (FFTs) for large kernel functions. If the kernel function can be factorized, a product of operators can be applied to simplify the calculation. 
         [0056]    Kernel functions can be discrete or continuous. They are defined over a volume that surrounds an object point. The volume can be limited (over one or more object points) or extend over the complete object space. Examples for 3D pixon kernel functions include a Gaussian function, an inverted paraboloid, or a function ∫(x;β)=(1+β) −1/β     2   , which approximates the Gaussian and parabolic functions for β-values of zero or infinity, wherein the parameter x can represent the radius or depend on the direction. 
         [0057]    The shapes of the kernel functions can be symmetric, or they can be adjusted in response to a form prevailing in the image object I. Within the shape of the pixon kernel functions, one can weigh the contribution of an object point. A limiting case of a pixon kernel function is the delta-function, in which the pixon smoothed object and the unsmoothed object are identical. 
       External Pixon Smoothing 
       [0058]      FIG. 2  illustrates external pixon smoothing ( 200 ). Using a standard reconstruction algorithm, an input object φ is fitted to the imaging data D (step  210 ). Examples of a reconstruction algorithm include algorithms based on maximum likelihood, based on a penalty function, algorithms based on an ordered subset expectation maximization, algorithms based on a non-negative least square fit, and algorithms based on an ordered subset non-negative least square fit. Details of an algorithm based on a non-negative least square fit are disclosed in the co-pending U.S. Patent Application entitled “NNLS image reconstruction,” by A. Yahil and H. Vija, filed on even date herewith, the contents of which are herein incorporated by reference. In addition the merit function of a reconstruction algorithm can include a penalty function. Examples of a penalty function include a linear (Tikhonov) penalty function, a total variation penalty function, or a penalty function based on maximum entropy or Markov random fields (Gibbs priors). 
         [0059]    In accordance with the above discussed use of the pixon kernel operator K, the resulting estimate of the 3D object is called a pseudo-object φ′. One then determines a pixon map P using the pseudo-object φ′ and the data set D (step  220 ). The pseudo-object φ′ is also the initial object for the pixon smoothing operation (step  230 ), which will be described in more detail in connection with  FIGS. 3 and 4 . During the pixon smoothing operation (step  230 ), one repetitively smoothes each object point of the pseudo-object φ′ over a pixon kernel function assigned by the pixon map P to reach a required quality of the 3D image object I. 
         [0060]      FIG. 3  shows the details associated with the pixon smoothing operation (step  230 ). The pseudo-object φ′ is smoothed using a pixon forward smoothing operation (step  300 ), which results in an updated object I update . An iterative cycle is indicated by increasing an increment, iteration (step  310 ). The number of iterations can be preset or manually assigned. Alternatively, as shown in  FIG. 3 , the number of iterations can be adaptively chosen by calculating a stop-criterion, Q(χ γ   2 ), after completing an iteration step, and determining whether a data model corresponding to the updated object I update  fulfills a preset condition (step  330 ). 
         [0061]    One such condition is a comparison of the stop-criterion, Q(χ γ   2 ), with a threshold, τ, which is stored a the tolerance table  340 . Thus, in such a goodness-of-fit evaluation of the updated object I update , the quality of the pixon smoothed image can be used to end the iteration. Examples for a quality-controlled iterative reconstruction are given in co-pending U.S. Patent Application entitled “Controlling the number of iterations in image reconstruction,” by A. Yahil and H. Vija of even date herewith, the contents of which are herein incorporated by reference. 
       Forward Pixon Smoothing Operation 
       [0062]      FIG. 4  shows the details associated with the pixon forward smoothing operation (step  300 ) of the pseudo-object φ′. Using the pixon map P, one builds a smoothed image by smoothing each object point with the pixon kernel function that is assigned to the object point in the pixon map P. For composing the smoothed image, one smoothes the pseudo-object φ′ by iteratively considering each of the provided pixon kernel functions. Thus, each object point is smoothed with its corresponding pixon kernel function as indicated in the pixon map P. For the first step, an initial image object  10 , with the dimension of the image object and only data points with value zero, and a kernel pointer kdx (identifying the kernel function) are prepared (step  410 ) and provided to a single-kernel function forward update (step  420 ). 
         [0063]    The pseudo-object φ′ and the pixon map P are also input parameters to the single-kernel forward update (step  420 ). Output parameters of the single-kernel forward update (step  420 ) are the unchanged kernel pointer kdx and an updated image object I kdx . At the end of each iteration, one determines whether another pixon kernel function update is necessary (step  430 ), in which case the kernel pointer kdx needs to be increased (step  440 ), or whether all kernel functions have been considered, in which case one assigns the updated image object I kdx  to be the final 3D image object  1 . 
         [0064]      FIG. 5  shows, in detail, the steps in the single-kernel function update (step  420 ) of an image object I kdx-1  as discussed in connection with  FIG. 4 . The image object I kdx-1  comprises smoothed values for all object points, for which the pixon map P indicated smoothing with pixon kernel functions identified by kernel pointers smaller than the current kernel pointer kdx. The pseudo-object φ′ is smoothed with the kernel function indicated by the current kernel pointer kdx (step  500 ). The result is a smoothed pseudo-object φ′ kdx . 
         [0065]    Then, one determines how much a data point is affected by the current kernel function (step  510 ). The corresponding calculation uses the pixon map P and the current value of the kernel pointer kdx to determine a temporary field, temp, which is zero if the object point is not affected. The temporary field, temp, has values between 0 and 1 when two kernel functions are used for smoothing of the object point, and a value of 1 when only the current pixon kernel function is used for smoothing of the object point. For updating each affected object point of the image object I kdx-1 , one adds, to the current value of the image object I kdx-1 , the product of the values of the temporary field, temp, and of the smoothed pseudo-object φ′ kdx  of that object point (step  520 ). The result is the updated image object I kdx . 
         [0066]    There exist a variety of ways to apply pixon smoothing externally to 3D reconstruction algorithms. Single or multiple pixon smoothing can be followed by standard reconstruction using the pixon smoothed object as an initial object for the reconstruction (see  FIG. 6 ). As shown in  FIG. 3 , the pixon smoothing can be applied multiple times until the quality of a corresponding data model fulfills a stop-criterion characterizing the goodness-of-fit of a current data model. Additionally, or as an alternative to the pixon forward smoothing with the operator K, a backward pixon smoothing can be used to generate the object with a transposed pixon operator K T  (see  FIGS. 8 and 9 ). 
         [0067]    For many pixon smoothing operations, the pixon map P defines which of the pixon kernel functions are applied to an object point. The result of applying external pixon smoothing within tomographic reconstruction is an output object I, which is a reconstructed object that fulfills the additional constraints imposed by the pixon method. 
       Intermediate Pixon Smoothing 
       [0068]      FIG. 6  shows a schematic flow chart of intermediate pixon smoothing (step  600 ). An input object φ is smoothed using external pixon smoothing (step  200 ) to create an intermediate pixon smoothed object I i . The intermediate object I i  is an input object for a second standard reconstruction process (step  610 ), which produces the final 3D image object  1 . The reconstruction process (step  610 ) can be of the same or of different type than the standard reconstruction process (step  210 ). If the first reconstruction process is handling the larger part of the reconstruction, the second reconstruction process resembles a final adjustment of the pixon smoothed intermediate object I i . Conversely, if the second reconstruction process handles the larger part of the reconstruction, the external pixon smoothing (step  230 ) can be considered as a preparation of a well-suited input object. 
         [0069]    When pixon smoothing and reconstruction alternate, each standard reconstruction can be accompanied by a determination of an updated pixon map. Thus, the pixon smoothing is always based on the most recent estimate of the object. 
         [0070]    Various combinations of the pixon methods described in  FIGS. 2 to 6  can be employed. An example of a combination of post-smoothing and intermediate smoothing is given in  FIG. 7 , in which an input object φ is first smoothed using (multiple) post-smoothing (step  200 ). This results in an intermediate object I i , which becomes an input object for the pixon intermediate smoothing (step  600 ) that creates the final 3D image object I. 
       Backward Pixon Smoothing Operation 
       [0071]      FIG. 8  shows a backward pixon smoothing operation, which is similar to the forward pixon smoothing of  FIG. 3 . However, unlike the forward pixon smoothing operation, the backward pixon smoothing operation applies the transpose pixon Kernel K T  within the single kernel backward update (step  800 ) instead of the pixon Kernel K within the single kernel forward update (step  320  of  FIG. 3 ). For the smoothing operation, one prepares the initial pseudo-object φ o  and the kernel pointer kdx, which indicates which one of the pixon kernel functions is applied during the single-kernel backward update. An input object I′ and the pixon map P are also used within the update to determine an updated pseudo-object φ kdx . One then evaluates whether to include further kernel function in the smoothing or whether all kernel functions have been considered, in which case the pseudo-object φ kdx  becomes the final pseudo-object φ. 
         [0072]    In  FIG. 9 , a single-kernel backward update implements an application of a transposed pixon Kernel K T . In this case, one begins the procedure by selecting those data points of an input object I′, the smoothed values of which have contributions of the smoothing with a specific pixon kernel function. Specifically, one calculates the entries of a temporary field, temp, by taking the maximum of zero or the difference between one and the modulus of the difference between the corresponding entry of the pixon map P and the value of the kernel pointer kdx (step  900 ). Then, one updates the temporary field, temp, by multiplying the temporary field, temp, with the input image I′ (step  910 ). The updated temporary field, temp, contains only non-zero entries for those data points that are to be updated with the specific kernel function. The updated temporary field, temp, is then smoothed over a pixon kernel function (step  920 ), which is read from a field F of pixon kernel functions using the kernel counter kdx. The result is a smoothed object I′ kdx . As every data point of the updated temporary field, temp, is smoothed over the pixon kernel function, the number of data points of the smoothed object I′ kdx  with non-zero values is in most cases larger than the number of non-zero data points of the updated temporary field, temp. Finally, the updated pseudo-object φ kdx  is created by adding the smoothed object I′ kdx  to the preceding pseudo-object φ kdx-1  (step  930 ). 
       Modular Reconstruction System 
       [0073]      FIG. 10  shows the details of a general concept of a modular reconstruction system. Specific examples were described with respect to  FIGS. 2 to 9 . In general, the pseudo-object φ′ is updated using an iterative algorithm ( 1000 ), which results in an output object I. An iterative cycle is indicated by increasing an increment, iteration (step  1010 ). The number of iterations can be preset or manually assigned. Alternatively, as in  FIG. 3 , a stop-criterion, Q(χ γ   2 ), can be calculated (step  1020 ) after completing an iteration step. In that case, a quality evaluation (step  1030 ) determines whether a data model corresponding to an updated object I update  of each iteration fulfills a preset condition. For example, one can compare whether a stop-criterion that characterizes the quality with a threshold, τ, which is stored in the tolerance table  340 . 
         [0074]    In the iterative algorithm ( 1000 ), one determines the updated object I update  for an input object using an update operation (step  1050 ). The update operation is based on a set of modular operations that include conventional image reconstruction and pixon smoothing. The Pixon smoothing can include its own construction of a pixon map or it can use a previously constructed pixon map. A control mechanism defines the order in which the iterative algorithm applies the modular update operations. The control mechanism can be controlled manually by a user. In addition, or alternatively an automated system can be used that is based, for example, on the quality of the reconstructed image, or the quality of a constructed pixon map. This is, for example, indicated in  FIG. 10  by the arrow pointing from the determination of the stop-criterion, Q(χ γ   2 ) (step  1020 ) to the update operation (step  1050 ). The update operation interacts with evaluation operations like the one based on a stop-criterion as described in connection with  FIG. 3 . 
         [0075]    An examplary series of steps includes a first pixon map construction, followed by a first set of pixon smoothing operations, an iterative image reconstruction with a predefined number of iterations, a second pixon map construction (or an update of the first pixon map), followed by a second set of external pixon smoothing operations. An initial reconstruction can be used to provide an initial pseudo-object to the first pixon smoothing operation. 
         [0076]    A number of embodiments have been described. Nevertheless, it will be understood that various modifications may be made without departing from the spirit of the invention. For example, Pixon smoothing can supplement tomographic reconstruction in different technological fields, such as astronomy, communication technology, materials science and medical imaging for 3D (image) reconstruction. Thus, a pixon map construction and the smoothing operation can be based on data sets measured in these technology fields. 
         [0077]    Examples of reconstruction algorithms include iterative image reconstruction methods, such as non-negative least square or Poisson likelihood algorithms, which iteratively fit image models to the data. An overview of different reconstruction methods is given in R. C. Puetter et al., “Digital Image Reconstruction: Deblurring and Denoising,” Annu. Rev. Astro. Astrophys., 2005, 43: 139-194, the contents of which are herein incorporated by reference. 
         [0078]    The order in which the different pixon kernel functions are used during the smoothing operation can be varied, the step size can be varied, or some pixon kernel functions may be considered only in defined areas of the image. 
         [0079]    The table F of pixon kernel functions may comprise, for example, ten spherical kernel functions. If one does not want to impose symmetry, one may use additionally or alternatively elliptical pixon kernel functions. However, asymmetric kernel functions may increase the computational effort, which one can handle, for example, by using specifically designed hardware. 
         [0080]    The pixon map P can be provided, for example, as a field of variables defining the pixon kernel functions or as a field of indices, which indicate kernel functions within the table F of the pixon kernel functions. 
         [0081]    Various combinations of external pixon smoothing described referring to  FIGS. 2 to 10  can be employed. The pixon smoothing operation may be the calculation of an average of the values of the object points within the volume defined by the corresponding pixon kernel function. The number of iterations can be controlled based on a quality analysis of current objects as shown in  FIG. 3 . Alternatively, a predefined number of iterations assigned by a user or the system configurations can be done. 
         [0082]    Moreover, the smoothing is not restricted to the specific use of a pixon map based on pixon kernel functions to constrain the reconstruction. Instead of a pixon smoothing operation, one could externally apply constraining operations that are based on Fourier filtering, application of a Wiener filter, wavelet filtering and/or application of a fixed filter. For such a constraining operation, the associated filter functions can be stored in a constraining map corresponding to the pixon map. An overview of different smoothing methods is given in R. C. Puetter et al., “Digital Image Reconstruction: Deblurring and Denoising,” Annu. Rev. Astro. Astrophys., 2005, 43: 139-194. Furthermore, the forward and backward smoothing described in connection with  FIGS. 4 ,  5 ,  8 , and  9  represents an independent concept, which one may use for implementing constraining operations based on such a constraining map. 
         [0083]    Instead of being supplied to a renderer for visualization, the output object can be supplied to a record keeping system (e.g., PACS system) or a system for automatic quantitative diagnosing. 
         [0084]    It is to be further understood that, because some of the constituent system components and method steps depicted in the accompanying figures can be implemented in software, the actual connections between the systems components (or the process steps) may differ depending upon the manner in which the disclosed method is programmed. Given the teachings provided herein, one of ordinary skill in the related art will be able to contemplate these and similar implementations or configurations of the disclosed system and method. 
         [0085]    For example, the numerical and symbolic steps described herein can be converted into a digital program executed, e.g., on a digital signal processor according to methods well known in the art. The digital program can be stored on a computer readable medium such as a hard disk and can be executable by a computer processor. Alternatively, the appropriate steps can be converted into a digital program that is hardwired into dedicated electronic circuits within the compressor that executes the steps. Methods for generating such dedicated electronic circuits based on a given numerical or symbolic analysis procedure are also well known in the art. 
         [0086]    Accordingly, other embodiments are within the scope of the following claims.