Abstract:
A computer-implemented method of assessing the quality of a functional image for an object includes causing a computer to execute instructions for providing a signal distribution of values N generating a transformed distribution by calculating, for each value N, a transformed value X=√{square root over (N+¼)}, reconstructing the functional image from the signal distribution, deriving an expected distribution of expected values λ from the functional image, generating a residual distribution by calculating, for each value N, a residual values ΔX=X−√{square root over (λ)}, and outputting the residual distribution.

Description:
TECHNICAL FIELD 
     This invention relates to medical imaging, and in particular, to image presentation/quality assessment in functional imaging. 
     BACKGROUND 
     Medical imaging of metabolic and biochemical activity within a patient is known as functional imaging. Functional imaging techniques include, for example, nuclear imaging such as planar nuclear imaging, Single Photon Computed Tomography (SPECT), Positron Emission Tomography (PET), and functional computed tomography (fCT). The reconstruction of a functional image from data acquired by functional imaging is often difficult because the data can be characterized by a small signal rate and a low signal-to-noise ratio. For nuclear imaging, for example, the count rate is limited by the amount of a radioactive substance that can be administered without harming the patient. In addition, a functional image does not necessarily provide structural information. Thus, one often evaluates a functional image with the help of a structural image. 
     An overview of SPECT systems and PET systems as well as iterative image reconstruction for emission tomography is given in chapter 7, chapter 11, and chapter 21 of M. Wernick and J. Aarsvold, “Emission tomography: the fundamentals of PET and SPECT,” Elsevier Academic Press, 2004, the contents of which are herein incorporated by reference. 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. 
     Square root transforms and their mathematical effect on variance analysis are disclosed in M. S. Bartlett, “The square root transformation in analysis of variance,” J. Roy Stat. Soc. Suppl., 3, 68-78, 1936, F. J. Anscombe, “The transformation of Poisson, binomial and negative-binomial data,” Biometrika. 35, 236-254, 1948, and L. D. Brown and L. H. Zhao, “A test for the Poisson distribution,” Sankhya: Indian J. Stat., 64, 61 1-625, 2002. An application of a square root transform for presenting scintigraphic images is disclosed in A. H. Vija et al., “Statistically based, spatially adaptive noise reduction of planar nuclear studies.” Proc. SPIE, 5747, 634-645, 2005. 
     SUMMARY 
     The invention is based in part on the recognition that the quality of a reconstructed image can be assessed by calculating residuals that are characterized by a reduced dependence on the signal strength, while maintaining an almost constant noise level. Such residuals can be well suited for image quality assessment because significant deviations of the residual can indicate, for example, problems with the reconstruction. 
     For Poisson-distributed events, residuals with an essentially constant standard deviation can be derived by transforming the detected events N (signal distribution) into a transformed distribution according to X=√{square root over (N+¼)}, deriving the expected events for the reconstructed image, and calculating the residual as the difference between the transformed distribution and the square root of the expected events. 
     In one aspect, the invention features computer-implemented methods of assessing the quality of image reconstruction of a functional image for an object, the methods include causing a computer to execute instructions for providing a signal distribution of values N, generating a transformed distribution by calculating, for each value N, a transformed value X=√{square root over (N+¼)}, reconstructing the functional image from the signal distribution, deriving an expected distribution of expected values λ from the functional image, generating a residual distribution by calculating, for each value N, a residual values ΔX=X−√{square root over (λ)}, and outputting the residual distribution. 
     In another aspect, methods for assessing the quality of a functional image for an object include acquiring image data for reconstructing the functional image, causing a computer to execute instructions for deriving a signal distribution of values N from the image data, generating a transformed distribution by calculating, for each value N, a transformed value X=√{square root over (N+¼)}, reconstructing the functional image from the signal distribution, deriving an expected distribution of expected values λ from the functional image, generating a residual distribution by calculating, for each value N, a residual values ΔX=X−√{square root over (λ)}, and outputting the residual distribution. 
     In another aspect, functional imaging devices include a detector unit for detecting a functional signal from a measured object within a detecting area and providing image data indicative of the detected functional signal, and an image processing unit for deriving an image from the image data, the image processing unit configured to derive a signal distribution of values N from the image data, generate a transformed distribution by calculating, for each value N, a transformed value X=√{square root over (N+¼)}, reconstruct a functional image from the signal distribution, derive an expected distribution of expected values λ from the functional image, generate a residual distribution by calculating, for each value N, a residual values ΔX=X−√{square root over (λ)}, and output the residual distribution. 
     Implementations may include one or more of the following features. 
     In some embodiments, outputting the residual distribution can include displaying the residual distribution. Outputting the transformed distribution can include simultaneously displaying the quality image and the reconstructed functional image. 
     In some embodiments, the methods can further include deriving the signal distribution from measured functional image data. 
     In some embodiments, providing a signal distribution can include providing a signal distribution that is Poisson-distributed. 
     In some embodiments, providing a signal distribution can include providing a signal distribution that corresponds to measured raw counts of a scintigraphic image. 
     Then, the methods can further include acquiring a scintigraphic image. 
     Reconstructing the functional image from the measured raw counts can include executing a smoothing operation. 
     In some embodiments, providing a signal distribution can include providing a signal distribution that corresponds to measured raw counts of at least one of planar nuclear imaging, single photon computed tomography, positron emission tomography, and functional computed tomography. 
     In some embodiments, the methods can further include acquiring the signal distribution by at least one of planar nuclear imaging, single photon computed tomography, positron emission tomography, and functional computed tomography. 
     In some embodiments, reconstructing of the functional image can include performing an iterative update operation. 
     In some embodiments, the signal distribution can be defined in a two-dimensional data space, the functional image can be defined in a three- or higher dimensional object space, and reconstructing the functional image includes performing projections between object space and data space. 
     In some embodiments, calculating the residual values can provide the residual values in object space. 
     In some embodiments, the methods can include assessing a quality of the reconstructed functional image based on the residual distribution and adapting the reconstruction based on the assessed quality. Adapting the reconstruction can include controlling the number of iterations. In addition, or alternatively, adapting the reconstruction can include modifying a control parameter of the reconstruction. 
     Acquiring the image data can include performing at least one of planar nuclear imaging, single photon computed tomography, positron emission tomography, and functional computed tomography. 
     The functional imaging devices can further include a display device, and the image processing unit can be further configured to display the residual distribution as an image on the display device to, e.g., a user. 
     In some embodiments, the image processing unit can further be configured to adapt the reconstruction of the functional image based on an assessment of the quality of the residual distribution. 
     In some embodiments, the image processing unit can further be configured to adapt the reconstruction by controlling the number of iterations. 
     In some embodiments, the image processing unit can further be configured to adapt the reconstruction by modifying a control parameter of the reconstruction. 
     Examples of image data include data from nuclear imaging (e.g., from SPECT or PET) and data from functional CT of the measured object. For nuclear imaging, the detected and expected events correspond to detected and expected counts of photons. 
     The detector unit of the functional imaging device can include a detector system selected from the group consisting of a positron emission tomography detector system and a single photon computed tomography detector system. 
     The functional imaging device can further include an input device. 
     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 
         FIG. 1  is a schematic illustration of a functional imaging device. 
         FIG. 2  is a simplified flowchart of image processing and quality assessment of processed images. 
         FIG. 3  is a side-by-side presentation of a scinitgraphic image and an image based on a first transformed distribution. 
         FIG. 4  is a side by side presentation of a reconstructed image and a quality image based on residuals between a second transformed distribution and expected counts based on the reconstruction. 
         FIG. 5  is a flowchart illustrating quality controlled image reconstruction. 
     
    
    
     Like reference symbols in the various drawings indicate like elements. 
     DETAILED DESCRIPTION 
     Imaging techniques in medicine can generate two-, three- or higher dimensional images of functional processes in a patient&#39;s body by, for example, using nuclear properties of matter. Examples of these functional imaging techniques include PET, SPECT, and fCT, for which one often administers a radioactive substance to a 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 planar nuclear imaging and SPECT. 
     For planar nuclear imaging, a detector remains stationary with respect to the patient to detect emitted radiation in one detector position only, corresponding to one projection and acquires a planar, two-dimensional (2D) nuclear image (herein also referred to as scintigraphic image). For tomographic imaging, one detects emitted radiation with multiple detector positions (projections) and reconstructs the acquired data into a three-dimensional (3D) image using a tomographic reconstruction algorithm. Planar and tomographic imaging can include additional dimensions such as the time and energy of the detected photons. 
     Because too much radiation can be harmful to the patient, the flux of detected nuclear radiation, i.e., the number of counts per unit time, is limited. As a result, functional imaging systems usually have to derive a functional image from a limited number of detected events. 
     An additional difficulty arises in the evaluation of functional images because nuclear counts are Poisson-distributed and the standard deviation (noise level) of counts varies with signal strength. Thus, it can be difficult to judge visually the statistical significance of a feature in an image (e.g., a scintigraphic image) that is based on nuclear activity and accordingly Poisson-distributed detector counts. 
     As an alternative to using Poisson counts N for presenting a scintigraphic image or for evaluating a tomographic image, one can use a transformation of the counts into an X-variable, which is generally defined as
 
 X=√{square root over (N+c)},  
 
where N is a Poisson-distributed variable and c is a transform-specific constant that provides transform-specific features to the distribution of the X-variable and associated statistical parameters such as the standard deviation of the X-variable. As further explained below, those transform-specific features can affect the visual impression of a presented image and/or allow a quality assessment of a functional image derived from Poisson-distributed counts.
 
     Specifically, a nearly constant variance of the X-variable can be achieved by a transformation with the constant c having a value of ⅜, i.e., X=√{square root over (N+⅜)}. The advantage of such a defined X-variable is that the standard deviation of the noise of the X-variable, for counts N≧2, is almost independent of the signal strength (here the number of counts). In contrast, the standard deviation of the noise of the Poisson-distributed counts N, which is √{square root over (N)}, varies with the signal strength. A constant noise level allows easier visual assessment of the statistical significance of signal changes. Consequently, the use of the above X-variable with c=⅜ for presenting functional images can enable one to more accurately interpret the images. 
     Moreover, a nearly constant variance for a residual defined as ΔX=X−√{square root over (λ)} for a detector element at a spatial position can be achieved for a transformation with the constant c having a value of ¼, i.e., X=√{square root over (N+¼)}. In the residual, λ is the expected count at that spatial position derived for a reconstructed functional image. 
     Assessing the quality of image processing of, for example, planar and tomographic nuclear images can be based on evaluating the distribution of the residuals ΔX. Using a value of ¼ for the constant c minimizes the dependence of the residuals on the signal for counts N≧2, while maintaining an almost constant variance of the residuals ΔX. The residuals ΔX are suited for the task of quality assessment. Significant deviations of the residuals ΔX in a residual distribution may indicate problems with the reconstruction. 
     Conventional quality assurance of nuclear images is often done by eye, i.e., by subjectively assessing whether the processed image “makes sense” based on medical or other knowledge. Quality assurance can also include the inspection of difference image of the difference between the counts displayed or predicted by the processed image and the actual counts obtained. 
     In addition to a subjective evaluation of the above introduced residual distribution of the residual ΔX=X−√{square root over (λ)}, those residuals allows an automated evaluation to provide feedback to the image reconstruction. The automated evaluation can support a decision about, for example, the necessity of an additional iteration step. Alternatively or in addition, the automated evaluation can control the image reconstruction by modifying control parameters of, for example, a reconstruction algorithm. Controlling image reconstruction is applicable for image reconstruction in two, three, or more dimensions. 
     As an example from the field of nuclear imaging, the concepts of presenting a functional image and assessing the quality of a reconstructed functional image are described in connection with  FIGS. 1 and 2 . Exemplary images and distributions for planar imaging are shown in  FIGS. 3 and 4 , and the application of quality assessment to control image reconstruction is explained in connection with  FIG. 5 . 
     Referring to  FIG. 1 , a functional imaging system  100  includes an imaging detector  110 , an image processing unit  120 , an evaluation unit  130 , and a display  140 . As shown in  FIG. 2 , the imaging detector acquires a signal distribution related to functional activity within a field-of-view of the imaging detector  110  (step  101 ). The acquired signal distribution can include a number of events N i  at a point i in data space. For example, in the case of nuclear imaging, the imaging detector  110  comprises an array of detector elements that detect the nuclear γ-radiation, which is emitted from a patient after administering the radioactive substance. The signal distribution associates then a count of detected photons (events N i ) with each detector element of the detector  110 . Examples for an imaging detector  110  include a conventional SPECT detector system and a PET detector system, which can be positioned around or partly around the patient. 
     The image processing unit  120  receives from the detector  110  the signal distribution, e.g., in form of a data set D of the detected events N i , and derives from it a planar or tomographic functional image I. This functional image I is an estimate of a functional distribution of the detected (functional) activity associated with the functional processin the examined volume. The functional distribution is defined in an object space (step  102 ). The object space can be 2D in planar imaging and 3D or more dimensional in tomographic imaging. The functional image I can be displayed to a user on the display  140  as a 2D or 3D image (step  103 ). 
     For planar imaging, the image processing unit  120  can perform some image processing steps (e.g., one or more pixon smoothing operations) on the data set D to correct errors or enhance the quality of the functional image I. For tomographic imaging, the reconstruction is in general more complex and can, for example, use a system matrix to describe the nuclear imaging system  100  and an iteratively improved data model to calculate a tomographic image object as a functional image from the data set D. 
     The evaluation unit  130  is configured to calculate a transformed distribution T c  of values X i =√{square root over (N i +c)} from the data set D (step  104 ). For planar imaging, a transformed distribution T c=3/8  can be displayed to the user as a transformed image either alone or in combination with the functional image (step  105 ).  FIG. 3  shows a planar scintigraphic image  201  based on detected nuclear events N i  next to a transformed image  204  of transformed values X i,c=3/8  =√{square root over (N i +⅜)}. The use of a constant noise level in generating transformed image  204  modifies the visual impression and assists a user in diagnosing the medical condition of the patient. 
     Alternatively, or additionally, the evaluation unit  130  can be configured to calculate a transformed distribution T c=1/4  with transformed values X i,c=1/4 =√{square root over (N i +¼)} and to derive (or receive from the image processing unit) expected events λ i  from the reconstructed functional image I. For planar imaging, the smoothed image data D can essentially be treated as if they already represented the expected events λ i , while for 3D and higher dimensional reconstruction the functional image, reconstructed in a 3D or higher dimensional object space, is first projected into data space to derive the expected events λ i . The evaluation unit  130  is then further configured to calculate a residual distribution R from the transformed distribution T c=1/4  and the expected events λ i  (step  106 ). Specifically, a residual of the residual distribution R at a point i in data space is defined by ΔX i =X i,c=1/4 −√{square root over (λ i )}. An image of the residual distribution R can be displayed to a user on the display  140  (step  107 ). 
     For planar imaging, data space and object space coincide so that the residual distribution also contains information in object space. Accordingly, a deviation in the residual distribution R can be directly related to a position in the functional image and therefore the object. A side-by-side presentation of the residual distribution and the reconstructed image can then help to identify image related causes for deviations in the residual distribution.  FIG. 4  shows a residual distribution  205  next to a reconstructed planar nuclear image  202 . The reconstructed planar nuclear image  202  was derived by pixon-smoothing a raw count image that was derived from a planar nuclear imaging measurement. 
     The residual distribution can further be used for automated quality assessment (step  108 ) and controlling of parameters employed in the image reconstruction (step  109 ). The below introduced parameter of a preset cut-off value applied in pixon smoothing is an example of a parameter of a reconstruction algorithm that can be controlled, e.g., increased or decreased, by evaluating the residual distribution. 
     Pixon smoothing is based on a pixon map, which is derived from a search for a broadest possible pixon kernel function at each point in the object space such that the set of kernel functions collectively support an adequate fit of an object to the measured data set D. In particular, the pixon map assigns to each object point a specific pixon kernel function. 
     The construction of the pixon map is one example of using information gained from the residual distribution to control the reconstruction of the functional image. In brief, one constructs the pixon map by iteratively considering each of the pixon kernel functions individually, calculating a goodness-of-fit for every object point of an input object, and comparing the goodness-of-fit with a preset cut-off value. If the calculated goodness-of-fit of an object point fulfills a preset condition, one or more pixon kernel functions are assigned to that object point. The information about the set of assigned kernel functions is stored in the pixon map. 
     For image reconstruction, pixon smoothing and the generation of a pixon map are described in more detail in U.S. patent application Ser. No. 11/931,084, filed Oct. 31, 2007 and entitled “EXTERNAL PIXON FOR TOMOGRAPHIC IMAGE RECONSTRUCTION TECHNICAL FIELD,” U.S. patent application Ser. No. 11/931,195, filed Oct. 31, 2007 and entitled “RECONSTRUCTING A TOMOGRAPHIC IMAGE,” and U.S. patent application Ser. No. 11,931,030, filed Oct. 31, 2007 and entitled “DETERMINING A PIXON MAP FOR IMAGE RECONSTRUCTION,” and U.S. Patent Applications filed on even date herewith and entitled “DETERMINING A MULTIMODAL PIXON MAP FOR TOMOGRAPHIC-IMAGE RECONSTRUCTION” by A. Yahil and H. Vija. The contents of all the preceding patent applications are incorporated herein by reference. 
     In general, an iterative reconstruction algorithm can impose quality control on the resulting image, as shown in  FIG. 5 . The control is based on assessing the quality of the iteratively updated reconstructed image. During the iteration process iterations of which are indicated by an increase of an iteration counter (step  300 ), an updated object ψ update  is used as an input object for the update operation (step  310 ) of the next iteration step (the first iteration step uses an initial input object ψ). Thus, each iteration step begins with an improved estimate of the object. As iteration progresses, the updated object ψ update  converges to a final distribution. In the context of PET or SPECT imaging, this distribution represents the distribution of a radioactive substance administered to the patient. 
     The update operation (step  310 ) can be, for example, an update operation of an ordered-subset-expectation-maximization algorithm, a non-negative least-squares algorithm, or a reconstruction algorithm using pixon smoothing. Moreover, the operation (step  310 ) can be an update operation of a multimodal image reconstruction as described in U.S. Patent Applications filed on even date herewith and entitled “MULTIMODAL IMAGE RECONSTRUCTION” by A. Yahil and H. Vija, the contents of which are incorporated herein by reference. 
     In many practices, the update operation (step  310 ) is subject to various control parameters, such as the cut-off parameter introduced above for pixon smoothing, the type of pixon kernel functions, and the number of iterations. 
     Referring again to  FIG. 5 , the update operation (step  310 ) is followed by a test operation of the quality of the updated object ψ update  (step  320 ). Specifically, the test operation calculates and analyzes the distribution of the residuals R i =X i,c=1/4 −√{square root over (λ i )} based on the estimated events in data space derived for the reconstructed image and the X-variable for c=¼. The test operation identifies, for example, the extent and the number of regions of residuals above a threshold value. 
     The algorithm then compares the test result with a quality-criterion and determines whether another iteration step should be performed (step  330 ). Additionally, or alternatively, the algorithm can adaptively modify the reconstruction (step  340 ). Such modification can include modifying the number of iterations or the number of parameters and their values, and any combinations thereof. The control can be performed, for example, in a similar manner to that described in the U.S. patent application Ser. No. 11/930,985, filed Oct. 31, 2007 and entitled “CONTROLLING THE NUMBER OF ITERATIONS IN IMAGE RECONSTRUCTION” by A. Yahil and H. Vija, the contents of which are incorporated herein by reference. 
     If no further iteration or modification of the update operation is necessary, the iteration is stopped and the currently calculated image object is assigned as the output of the reconstruction, i.e., as image object I. 
     A number of embodiments of the invention have been described. Nevertheless, it will be understood that various modifications may be made without departing from the spirit and scope of the invention. For example, the control of the reconstruction can be applied to a wide variety of reconstruction processes, including but not limited to maximum-likelihood-expectation-maximization algorithms, maximum a posteriori reconstruction algorithms, and Bayesian reconstruction algorithms. 
     The signal distribution can be discrete distributions, e.g., in nuclear imaging. 
     The updated object provided as image object I does not need to be the most recently updated object but could be the updated object with the highest quality. This could, in some cases, be an object that precedes the last object. 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. 
     The source of the functional signal can be an object or patient positioned within the detecting area of the functional imaging system. 
     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 as well as hardware, 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. 
     For example, the numerical and symbolic steps described herein can be converted into a digital program executed 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. 
     Accordingly, other embodiments are within the scope of the following claims.