Abstract:
A method for electronically cleansing a virtual object formed from acquired image data converted to a plurality of volume elements is provided. The present method allows individual volume elements, or voxels, to represent more than one material type. The method includes defining a partial volume image model for volume elements representing a plurality of material types based, at least in part, on the measured intensity value of the volume element. The material mixture for each of the volume elements representing a plurality of material types can be estimated using the observed intensity values and the defined partial volume image model. The volume elements representing a plurality of material types can then be classified in accordance with the estimated material mixture. For electronic colon cleansing, the method includes removing at least one classification of volume elements when displaying the virtual object.

Description:
STATEMENT OF PRIORITY AND RELATED APPLICATIONS 
     This application claims priority to U.S. Provisional Application 60/741,494 filed on Nov. 30, 2005, entitled An Improved Electronic Colon Cleansing Method for Detection of Colonic Polyps by Virtual Colonoscopy, the disclosure of which is hereby incorporated by reference in its entirety. 
    
    
     STATEMENT OF GOVERNMENT RIGHTS 
     This invention was made with government support under grant number CA082402 awarded by the National Cancer Institute. The government has certain rights in the invention. 
    
    
     BACKGROUND OF THE INVENTION 
     Colorectal cancer is a leading cause of cancer-related deaths in the United States. More than 90% of colon cancers develop from adenomatous polyps, removal of which can dramatically reduce the risk of death. Accepted guidelines recommend the screening of adults who are at average risk for colorectal cancer, since the detection and removal of adenomas has been shown to reduce the incidence of cancer and cancer-related mortality. Currently available detection methods include fecal occult blood tests, sigmoidoscopy, barium enemas, and fiber optic colonoscopies (OC). Unfortunately, most people do not follow this advice because of the discomfort and inconvenience of the traditional optical colonoscopy. To encourage people to participate in screening programs, virtual colonoscopy (VC), also known as computed tomographic colonography (CTC), has been proposed and developed to detect colorectal neoplasms by using a computed tomography (CT) or magnetic resonance imaging (MRI) scan. VC is minimally invasive and does not require sedation or the insertion of a colonoscope. Compared to OC, VC has the potential to become a common screening method in terms of safety, cost, and patient compliance. VC exploits computers to reconstruct a 3D model of the CT scans taken of the patient&#39;s abdomen, and creates a virtual fly-through of the colon to help radiologists navigate the model and create an accurate, efficient diagnosis. Previously known systems and methods for performing virtual colonoscopy are described, for example, in U.S. Pat. Nos. 5,971,767, 6,331,116 and 6,514,082, the disclosures of which are incorporated by reference in their entireties. 
     It has been demonstrated that the performance of VC can compare favorably with that of traditional OC. As is required with traditional optical colonoscopies, the colon needs a thorough cleansing before the VC and computer aided detection (CAD) of polyps. However, even with a rigorous cleansing of the colon, remaining stool and fluid residues may mimic polyps, thereby dramatically reducing the efficiency of the VC and CAD. 
     Electronic colon cleansing (ECC) may be used to improve the efficiency of VC and CAD by effectively removing colonic material from the acquired images. Preparing the colon for ECC varies slightly from traditional VC and CAD without ECC. A preliminary step of ECC is tagging the colonic material with a contrast agent. With the addition of a contrast agent, the tagged stool and fluid have an enhanced image density compared to the density of the colon/polyp tissues. By segmenting the colon images and recognizing patterns, ECC methods can be used to identify the enhanced colonic material and produce a “clean” colon model for both VC navigation and CAD analysis. A known ECC approach is to apply simple thresholds to the image data for segmentation and then removing certain tagged material. However, the effectiveness of the threshold approach to ECC may be impaired by a partial volume (PV) effect at various boundary regions, such as the air-colonic material interface and the colon/polyp tissue-colonic material interface. 
     Various approaches to mitigate the PV effect have been explored in the art. For example, a ray-based detection technique exists, which utilizes a predefined profiled pattern to detect interfaces. Other techniques use morphological and linear filters, image gradient information, and a priori models to mitigate the PV effect. Each of the previous methods relies on the assumption that limited information may be derived from an image voxel, which limits the effectiveness of the solution. As a result, improved techniques for addressing the PV effect are desired. An improvement is particularly important in CAD of polyps because the mucosa region of the colon, in which polyps often reside, is a boundary region which is often obscured by the PV effect. Thus, it is an object of the present method to provide an improved electronic cleansing process and, in particular, to provide an electronic cleansing process which results in a clean mucosa layer well-suited for CAD techniques. 
     SUMMARY OF THE INVENTION 
     In accordance with the present invention, a method for electronically cleansing a virtual object formed from acquired image data converted to a plurality of volume elements is provided. The present method allows individual volume elements, or voxels, to represent more than one material type. The method includes defining a partial volume image model for volume elements representing a plurality of material types based, at least in part, on the measured intensity value of the volume element. The material mixture for each of the volume elements representing a plurality of material types can be estimated using the observed intensity values and the defined partial volume image model. The volume elements representing a plurality of material types can then be classified in accordance with the estimated material mixture. For electronic colon cleansing, the method includes removing at least one classification of volume elements when displaying the virtual object. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a simplified flow chart illustrating the steps in one exemplary embodiment of the present method of electronic colon cleansing; 
         FIG. 2  is a simplified flow chart further representing the process of developing a PV or mixture image model, suitable for use with the present method; 
         FIG. 3  is a simplified flow chart further illustrating an exemplary process of PV segmentation of the mixture image in accordance with the present method; 
         FIG. 4A  is an exemplary CT image slice of the colon, illustrating a layer of enhanced colonic materials. The dotted vertical line indicates a sample path for the density profile of  FIG. 4C ; 
         FIG. 4B  is an exemplary CT image slice of the colon that has been partially cleansed, leaving an interface layer intact; 
         FIG. 4C  is an exemplary graph that shows the corresponding density profile of the exemplary CT image slice along the vertical line, illustrated in  FIG. 4A ; 
         FIG. 4D  is an exemplary graph that shows the corresponding density profile of the exemplary CT image slice in  FIG. 4B , along the vertical line illustrated in  FIG. 4A ; 
         FIG. 5A  is an exemplary CT image slice illustrating the mixture-based PV segmentation result from the corresponding image slice of  FIG. 4A ; 
         FIG. 5B  is an exemplary graph that shows the corresponding mixture segmentation profiles for  FIG. 5A  for air, muscle and bone/tagged materials along the sample dotted vertical line which is the same as the line shown in  FIG. 4A ; 
         FIG. 5C  is an exemplary CT image slice that illustrates the border of volume S e  (the entire colon lumen enclosure) by a white line enclosing the entire colon lumen; 
         FIG. 5D  is the exemplary CT image slice of  FIG. 5C  that further shows the regions in the enclosed volume of the colon lumen, which results from segmentation and grouping; 
         FIG. 6A  is an exemplary CT image slice (originally shown in  FIG. 4A ); 
         FIG. 6B  is the CT image slice of  FIG. 6A  further showing the cleansed CT image after altering the density values and in which all the tagged colonic materials are removed; 
         FIG. 6C  is the CT image slice of  FIG. 6A  that further illustrates the final cleansed CT image having a fully restored mucosa layer and a cleansed colon lumen; 
         FIG. 6D  is an exemplary graph that shows the corresponding density profile of the CT image slice along the vertical line in  FIG. 6A ; 
         FIG. 6E  is a graph that shows the corresponding density profile of the CT image slice in  FIG. 6B  along the vertical line shown in  FIG. 6A ; 
         FIG. 6F  is a graph that shows the corresponding density profile of the CT image slice in  FIG. 6C  along the vertical line shown in  FIG. 6A ; 
         FIG. 7  is a table that shows the various material mixtures that might be present in a voxel when four material types are present in a CT image; 
         FIG. 8  is a simplified flow chart illustrating the steps of the exemplary process of further classifying the voxels as parts of the volume of interest after PV segmentation. 
     
    
    
     DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS 
     An overview of the present method is illustrated by reference to the simplified flow chart in  FIG. 1 . An exemplary embodiment of the present invention assumes that, after proper bowel preparation, appropriate 2D image data has been acquired, such as through the use of a spiral CT scan, MRI scan, or other suitable method known in the art of virtual colonoscopy (step  100 ). From the 2D image data, the volume of interest (VOI), such as the colon tissue volume, is extracted in a manner generally known in the art (step  105 ). For example, a sliced spiral CT scanner can be used with clinically available protocols to cover the entire abdominal volume during a single breath hold. Suitable detector collimation can be 5 mm and the images can be reconstructed as 1 mm thick slices of 512×512 array size. An example of a CT image slice is illustrated in  FIG. 4A . The exemplary 2D image in  FIG. 4A  has five distinct areas: air ( 400 ), an interface layer ( 405 ), colonic material ( 410 ), a mucosa layer ( 415 ), and colon tissue/bone ( 420 ). The present methods operate to classify the voxels in a region of interest, such as the colon, using a mixture-based image segmentation algorithm and to remove or reclassify the classification of colonic materials for a clean colon lumen and mucosa layer, resulting in the “cleansed” image in  FIG. 6C . 
     Returning to  FIG. 1 , in step  110 , the density values of each voxel are used to create a mixture image model, also represented herein as a PV image model. Further, in step  115 , the mixture image model is used in a PV segmentation algorithm, described in further detail below, to initially classify the various voxels of the acquired colon image. In step  120 , to address any misclassified voxels, the classified voxels are reclassified through a series of dilation and erosion operations in a manner generally known in the art, which are described in further detail below. Depending on a voxel&#39;s reclassification, in step  125 , the voxel may be grouped in different regions of similar density value. In step  130 , depending on its region, a voxel&#39;s density value can be altered, described in further detail below, to further classify the individual voxels as part of the volume of interest and subsequently restored. Further in step  130 , voxels grouped in some boundary regions are subjected to an iterative segmentation algorithm, described in further detail below, to further classify the voxels as part of the volume of interest. 
     Turning to steps  110  and  115 , prior to the virtual colon cleansing, the image voxels can be initially classified as containing one or more materials by using a mixture image segmentation algorithm, also known as a PV image segmentation algorithm. Mixture image segmentation operates to determine the material mixtures and the material model parameters from the acquired image data, which include observed density or intensity values. To achieve the goals of mixture image segmentation, the observed density or intensity values must relate in some way to the material mixtures and the material model parameters. The mixture image model in step  110  is used to establish the relationship between the measured intensity values and the mixture properties of each voxel so that there may be effective mixture image segmentation in step  115 . 
       FIG. 2  is a flow chart further illustrating the steps of creating a mixture image model. First, in step  200 , a relationship is established between what is observed, the measured intensity values for each voxel, and what is not observed, the density contribution of each material represented by a voxel. In the present exemplary embodiment, the acquired image density distribution Y can be represented by a column vector [y 1 , y 2 , . . . , y N ] T , where the observed density of an individual voxel i is y i  and N is the total number of voxels in the image. If the acquired image density {y i } may possibly contain K material types within each voxel i, then there may be a possibility of K tissue types, where each material type k has a contribution x ik  to the observed density value y i  for an individual voxel i. Therefore, the observed density value y i  can be represented by the relationship 
               y   i     =       ∑     k   =   1     K     ⁢       x   ik     .             
The expression states that the sum of K density contributing elements x ik , for a particular voxel i, constitutes the observed density y i  for a particular voxel i.
 
     If there are multiple material types in at least some of the voxels, then a statistical determination of each material type&#39;s contribution to the individual voxel&#39;s intensity value is performed (step  205 ). Under the present exemplary embodiment, the unobservable density contributing element x ik  is expected to follow a Gaussian statistical distribution with mean μ ik  and variance σ ik   2  Voxel i may be fully filled by material type k; in which case x ik  may become the observable variable y i  in the present exemplary embodiment. If, however, voxel i is partially filled by material type k, then m ik  may be the fraction of material type k inside voxel i. When there is a possibility of multiple material types, the resulting relationship can be expressed by 
                 μ   ik     ⁢     =   def     ⁢         m   ik     ⁢     μ   k     ⁢           ⁢   and   ⁢           ⁢     σ   ik   2       ⁢     =   def     ⁢       m   ik     ⁢     σ   k   2           ,           ⁢     
     ⁢       where   ⁢           ⁢       ∑     k   =   1     K     ⁢           ⁢     m   ik         =   1     ,     0   ≤     m   ik     ≤     1   ⁢           ⁢   and   ⁢           ⁢     (       μ   k     ,     σ   k     ,   2       ,     μ   ik     ,     σ   ik     ,   2         )       ≥   0.           
The image density y i  at voxel i can be expressed as
 
               y   i     =         ρ   i     ⁢       ∑     k   =   1     K     ⁢       m   ik     ⁢     μ   k           +     ɛ   i             
which follows a Gaussian distribution, where ∈ i  is Gaussian noise associated with y i  at voxel i with a mean of zero and a variance expressed as
 
               σ     y   i     2     =         ∑     k   =   1     K     ⁢     σ   ik   2       =       ∑     k   =   1     K     ⁢       m   ik     ⁢       σ   k   2     .                 
Notation ρ i  reflects the bias field or inhomogeneity effect at voxel i which is a result of a non-uniform RF field across the body and tissue response to the local magnetic field. In a non-MRI scan, however, the ρ i  factor may be removed.
 
     The probability distribution of sampling {y i }, given the parameters {m ik , ρ i , μ k σ k   2 }, can be expressed as: 
                           Pr   ⁡     (       Y   |   M     ,   ρ   ,   μ   ,   σ     )       =       ∏     i   =   1     N     ⁢           ⁢     Pr   ⁡     (         y   i     |     m   i       ,   ρ   ,   μ   ,   σ     )                     =       ∏     i   =   1     N     ⁢           ⁢       1       2   ⁢   π   ⁢       ∑     k   =   1     K     ⁢           ⁢       m   ik     ⁢     σ   k   2               ⁢     exp   ⁡     [     -         (       y   i     -       ρ   i     ⁢       ∑     k   =   1     K     ⁢           ⁢       m   ik     ⁢     μ   k             )     2       2   ⁢       ∑     k   =   1     K     ⁢           ⁢       m   ik     ⁢     σ   k   2               ]                         (   1   )               
where M=[m 1 , m 2 , . . . , m N ] T , m i =[m i1 , m i2 , . . . , m iK ] T , μ=[μ 1 , μ 2 , . . . , μ K ] T , and σ 2 =[σ 1   2 , σ 2   2 , . . . , σ K   2 ] T . The probability distribution of sampling {x ik }, given the parameters {m ik , ρ i , μ k , σ k   2 } can be expressed as:
 
                     Pr   ⁡     (       X   |   m     ,   ρ   ,   μ   ,   σ     )       =       ∏     i   ,     k   =   1         N   ,   K       ⁢           ⁢       1       2   ⁢   π   ⁢           ⁢     m   ik     ⁢     σ   k   2           ⁢     exp   ⁡     [     -         (       x   ik     -       ρ   i     ⁢     m   ik     ⁢     μ   k         )     2       2   ⁢     m   ik     ⁢     σ   k   2           ]                   (   2   )               
where X=[x 1 , x 2 , . . . , x N ] T  and x i =[x i1 , x i2 , . . . , x iK ] T . Taken together, equations (1) and (2) represent a PV or mixture image model, which may be used in of mixture image segmentation in step  115  of  FIG. 1 .
 
     The mixture image segmentation determines the composition of material mixtures {m ik }, 
               (         ∑     k   =   1     K     ⁢     m   ik       =       1   ⁢           ⁢   and   ⁢           ⁢   0     ≤     m   ik     ≤   1       )     ,         
and 0≦m ik ≦1), and the material model parameters {ρ i , μ k , σ k   2 } from the acquired image data {y i }. As described below, an expectation-maximization (EM) algorithm, which is generally known in the art, can be used to estimate the mixture parameters. The task of determining the mixture parameters {m ik , ρ i , μ k , σ k   2 }, given the acquired image data {y i }, can be specified by an a posteriori probability, which first requires an a priori distribution of {m ik } and {ρ i }.
 
     In image processing applications, a Markov random field (MRF) a priori regularization can be use to determine a maximum a posteriori probability (MAP) solution. The MRF model for {m ik } can be expressed in the form: 
                     Pr   ⁡     (       m   i     |     N   i       )       =       1   Z     ⁢     exp   (       -   β     ⁢       ∑       k   =   1     ,     j   ∈     N   i         K     ⁢         α   ij     ⁡     (       m   ik     -     m   jk       )       2                     (   3   )               
where N i  denotes the neighborhood of voxel i, β is a parameter controlling the degree of the penalty on the mixture M, α ij  is a scale factor reflecting the difference among various orders of the neighboring voxels, and Z is the normalization factor for the MRF model.
 
     In the present exemplary embodiment, only the first-order neighborhood system is considered and α ij  is the same for the six first-order neighbors when the image has a uniform spatial resolution in three dimensions (e.g., α ij =1). When the axial resolution is two times lower than the transverse resolution, the α ij  for the two neighbors in the axial direction is two times smaller than the α ij  for the four neighbors in the transverse plane (e.g., α ij =1 in transverse plane and α ij =0.5 in axial direction). 
     A similar MRF can be specified for {ρ i } as: 
                     Pr   ⁢           ⁢     (       ρ   i     |     N   i       )       =       1   Z     ⁢     exp   ⁡     [         -     γ   1       ⁢       ∑     j   =   1     R     ⁢           ⁢       (       D   j     *   ρ     )     i   2         -       γ   2     ⁢       ∑     j   ,     l   =   1       R     ⁢           ⁢       (       D   j     *     D   l     *   ρ     )     i   2           ]                 (   4   )               
where R equals 2 for 2D slice images and 3 for 3D volume images. Notation D is the standard forward finite difference operator along the corresponding directions. The symbol * denotes the one dimensional discrete convolution operator. The first-order regularization term (associated with γ 1 ) penalizes a large variation in the bias field and the second-order regularization term (associated with γ 2 ) penalizes the discontinuities in the bias field {ρ i }. Both parameters γ 1  and γ 2  play a similar role as β, in the MRF for {m ik }, by controlling the degree of smoothness of the bias field. The conditional expectation in the EM algorithm, given the observed data {y i } and the estimate {M (n) , μ (n) , σ 2(n) } in the n th  iteration, can be expressed by:
 
                     Q   (     M   ,   ρ   ,   μ   ,     σ   |     M     (   n   )         ,     ρ     (   n   )       ,     μ     (   n   )       ,     σ     (   n   )         )     =       E   [         ln   ⁢           ⁢     Pr   (       X   |   M     ,   ρ   ,   μ   ,   σ     )       |   Y     ,     M     (   n   )       ,     ρ     (   n   )       ,     μ     (   n   )       ,     σ     (   n   )         ]     =         -   1     /   2     ⁢       ∑       i   =   1     ,     k   =   1         N   ,   K       ⁢     {     [           ⁢               ln   ⁡     (     2   ⁢   π   ⁢           ⁢     m   ik     ⁢     σ   k   2       )       +       1       ω   ik     ⁢     σ   k   2         ⁢     (       x   ik     2   ⁢     (   n   )         -     2   ⁢     ρ   i     ⁢     m   ik     ⁢     μ   k     ⁢     x   ik     (   n   )         +       ρ   i   2     ⁢     m   ik   2     ⁢     μ   k   2         )         ]     +     2   ⁢   β   ⁢       ∑     j   ∈     N   i         ⁢           ⁢         α   ij     ⁡     (       m   ik     -     m   jk       )       2         +     2   ⁡     [         γ   1     ⁢       ∑     j   =   1     R     ⁢           ⁢       (       D   j     *   ρ     )     i   2         +     ]         }                     (   5   )               
where the conditional means for x ik  and x ik   2  can be expressed by:
 
     
       
         
           
             
               
                 
                   
                     
                       
                         
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     The maximization in the EM algorithm determines the estimate in the (n+1) th  iteration, which maximizes the conditional expectation of equation (5). For the mixture model parameter {μ k }, the differential is 
                   ∂     Q   ⁡     (   .   )         /     ∂     μ   k            μ   k     =     μ   k     (     n   +   1     )                 =   0     ,         
which leads to:
 
                     μ   k     (     n   +   1     )       =         ∑     i   =   1     N     ⁢           ⁢     x   ik     (   n   )             ∑     i   =   1     N     ⁢           ⁢       ρ   i     (   n   )       ⁢     m   ik     (   n   )                     (   8   )               
For the other mixture model parameter {σ k   2 }, the relationship may be expressed by:
 
                     σ   k     2   ⁢     (     n   +   1     )         =       1   N     ⁢       ∑     i   =   1     N     ⁢           ⁢           x   ik     2   ⁢     (   n   )         -     2   ⁢     ρ   i     (   n   )       ⁢     m   ik     (   n   )       ⁢     μ   k     (   n   )       ⁢     x   ik     (   n   )         +       ρ   i     (   n   )       ⁢     m   ik     2   ⁢     (   n   )         ⁢     μ   k     2   ⁢     (   n   )               m   ik     (   n   )         .                 (   9   )               
Maximizing the conditional expectation function Q(.) with respect to the tissue mixture parameter {m ik }, under the conditions of
 
                 ∑     k   =   1     K     ⁢     m   ik       =   1         
and 0≦m ik ≦1, does not generally have a closed-form solution like equations (6) and (7) have. When only a single acquired image {y i } is available, under the condition of
 
                   ∑     k   =   1     K     ⁢     m   ik       =   1     ,         
the solution for the maximization of Q(.), if it exists, limits voxel i to having a maximum of two tissue types. When noise is present in the image, additional constraints may be needed for a regularization solution. Adding the MRF to equation (5) causes the conditional expectation of the posteriori to have a quadratic form when the distribution is at the n th  iteration and the variance σ ik   2 =m ik σ k   2  is fixed for the (n+1) th  iterated estimate. By maximizing the quadratic form Q(.)+Pr(m i |N i ) with respect to the material mixture parameter {m ik }, under the conditions of
 
                 ∑     k   =   1     K     ⁢     m   ik       =   1         
and 0≦m ik ≦1, a closed-form solution can be derived. In the case where each voxel is limited to a maximum of two tissue types, i.e., m i2 =1−m i1 , the resultant relationship can be expressed as:
 
                     m     i   ⁢           ⁢   1       (     n   +   1     )       =                 x     i   ⁢           ⁢   1       (   n   )       ⁢     σ     i   ⁢           ⁢   2       2   ⁢     (   n   )         ⁢     μ   1     (   n   )         +       σ     i   ⁢           ⁢   1       2   ⁢     (   n   )         ⁢     μ   2     2   ⁢     (   n   )           -                   x     i   ⁢           ⁢   2       (   n   )       ⁢     σ     i   ⁢           ⁢   1       2   ⁢     (   n   )         ⁢     μ   2     (   n   )         +     4   ⁢     βσ     i   ⁢           ⁢   1       2   ⁢     (   n   )         ⁢     σ     i   ⁢           ⁢   2       2   ⁢     (   n   )         ⁢       ∑     j   ∈     N   i         ⁢           ⁢       α   ij     ⁢     m     j   ⁢           ⁢   1       (   n   )                           μ   1     2   ⁢     (   n   )         ⁢     σ     i   ⁢           ⁢   2       2   ⁢     (   n   )           +       μ   2     2   ⁢     (   n   )         ⁢     σ     i   ⁢           ⁢   1       2   ⁢     (   n   )           +     4   ⁢     βσ     i   ⁢           ⁢   1       2   ⁢     (   n   )         ⁢     σ     i   ⁢           ⁢   2       2   ⁢     (   n   )         ⁢       ∑     j   ∈     N   i         ⁢           ⁢     α   ij                     (   10   )               
The bias field parameter {ρ i } can be expressed by:
 
                         ∑     k   =   1     K     ⁢           ⁢         μ   k     (   n   )       ⁢     x   ik     (   n   )           σ   k     2   ⁢     (   n   )             =               ρ   i     ·       ∑     k   =   1     K     ⁢           ⁢           m   ik     (   n   )       ⁢     μ   k     2   ⁢     (   n   )             σ   k     2   ⁢     (   n   )           ⁢     m   ik     (   n   )       ⁢     μ   k     2   ⁢     (   n   )               +                       γ   1     ⁡     (       H   1     *   ρ     )       i     +         γ   2     ⁡     (       H   2     *   ρ     )       i       ⁢                     ⁢     
     ⁢   where   ⁢     
     ⁢               H   1     =     [         0         -   1         0             -   1         4         -   1             0         -   1         0         ]               H   2     =     [         0       0       1       0       0           0       2         -   8         2       0           1         -   8         20         -   8         1           0       2         -   8         2       0           0       0       1       0       0         ]             .             (   11   )               
Solving equation (11) for {ρ i } can be performed using equation (10).
 
     Equations (8), (9), (10), and, if needed, (11) provide a MAP-EM solution for mixture image segmentation when the mixture in each voxel is resolved to a maximum of two material types per voxel. 
       FIG. 3  is a simplified flow chart illustrating a PV segmentation process of step  120  using the MAP-EM solution of the PV image model derived above. Mixture image segmentation begins with an initial segmentation operation in step  305  involving a simple threshold operator, as known in the art, for hard segmentation. From the hard segmentation of step  300 , each voxel is initially classified as one of four initial tissue types (air, soft tissue, muscle, and bone/TM) or model parameter sets {μ k   (0) , σ k   2(0) } using the labeled voxel density values. The initial tissue mixture {m ik   (0) } assumes that each voxel is 100% occupied by a single predominant tissue type. Applying the initial estimates of step  300 , the MAP-EM solution set forth above in Equations 8-11 is then applied in step  305 . The results of consecutive iterations of the MAP-EM solution are tested for convergence to an acceptable value in step  310 . In one embodiment, convergance is assumed when the following termination criterion is satisfied: 
               Max   ⁡     (                μ   k     (     n   +   1     )         μ   k     (   n   )         -   1              k   =   1     ,   2       )       &lt;     γ   .           
When the maximum difference between the means of each tissue class at the n th  and the (n+1) th  iterations is less than the specified threshold γ, the iteration process is terminated. In the present exemplary embodiment, γ is set to be 0.05.
 
     Since the MAP-EM solution of Equations (8) through (11) only provides a closed form solution for identifying two material types per voxel, and since there is no limitation on the number of material types in an entire image, an additional process is used to constrain the analysis to two material types between iterations of the MAP-EM solution of Step  305 . In the present exemplary embodiment where the colon represents the volume of interest, the voxels of an image may be generally classified as containing four material types with different image densities, i.e., K=4: (i) air in the colon lumen and lungs, (ii) fat or soft tissues, (iii) muscle, and (iv) bone or tagged colonic materials. When there are four tissue types (the air, soft tissue, muscle, and bone/TM) expected to be present in an image, there are a total of 15 possible tissue mixtures, as shown in  FIG. 7 . This knowledge of the region of interest can be used to constrain the voxel mixture to a maximum of two components. 
     Steps  315  through  325 , which are described below, can be used to limit each voxel to two material type mixtures which are compatible with the above described MAP-EM solution. In step  315 , the mixture estimate of each voxel is evaluated and the most dominant component is selected as the voxel type. A nearest neighbor analysis, such as evaluating the 6 first-order neighbors, can then be applied to each voxel to determine the tissue types of the voxel neighborhood in step  320 . The mixture for the voxel is then assigned as the two predominant types in the neighborhood in step  325 . For example, if three voxels of the first-order neighborhood where classified as air, two as bone/tagged materials and one was classified as muscle, the mixture for the voxel would be air and bone/TM. In the event that two tissue types are equally represented in the nearest neighbor analysis, the weights of the tissue types in the neighborhood voxels can be added and those components with the highest sum can be considered the predominant tissue types. For example, if two voxels were classified as air, two as muscle, and two voxels in the neighborhood classified as bone/TM, the values of air, muscle and bone/TM components in the mixtures of each voxel derived from the MAP-EM solution can be added to determine which are the two predominant tissue types in step  325 . After the two predominant tissue types are identified in step  325 , the PV image segmentation process returns to step  310  for another iteration of the MAP-EM solution. The process repeats until convergance of the MAP-EM solution is detected in step  310 , as described above. 
     The PV image segmentation process described above is applicable to the case where each voxel can be resolved to a maximum of two material types per voxel (i.e., N=2 in step  325 ). However, the present method can be extended to mixtures of three or more material types per voxel (i.e., N=3 or more in step  325 ). Regardless of the number of material types, equations (8) and (9) of the MAP-EM solution described above are applicable. However, when a voxel is to be resolved to mixtures of three or more material types, equation (10) no longer provides a solution for mixture image segmentation and a new expression for the mixture parameter must be derived. 
     With three material types, for example, the conditional expectation can be expressed as: 
                       Q   i     ≈       1   2     ⁢       ∑   k     ⁢           ⁢     {         1       (       m   ik     ⁢     σ   k   2       )       (   n   )         ⁡     [         -   2     ⁢     ρ   i     (   n   )       ⁢     x   ik     (   n   )       ⁢     m   ik     ⁢     μ   k     (   n   )         +       ρ   i     2   ⁢     (   n   )         ⁢         m   ik   2     ⁡     (     μ   k     (   n   )       )       2         ]       +     2   ⁢   β   ⁢       ∑     r   ∈     N   i         ⁢           ⁢         α   ir     ⁡     (       ω   ik   2     -     2   ⁢     m   ik     ⁢     m   rk     (   n   )           )       2           }           =         1   2     ⁢       ∑   k     ⁢           ⁢     {       m   ik   2     ⁡     (           (       ρ   i     (   n   )       ⁢     μ   k     (   n   )         )     2         (       m   ik     ⁢     σ   k   2       )       (   n   )         +     2   ⁢   β   ⁢       ∑     r   ∈     ɛ   i         ⁢           ⁢     α   ir           )       }         -       ∑   k     ⁢           ⁢     {       m   ik     ⁡     (           ρ   i     (   n   )       ⁢     x   ik     (   n   )       ⁢     μ   k     (   n   )             (       m   ik     ⁢     σ   k   2       )       (   n   )         +     2   ⁢   β   ⁢       ∑     r   ∈     ɛ   i         ⁢           ⁢       α   ir     ⁢     m   rk     (   n   )               )       }                 (   12   )               
In matrix form, this be represented by:
 
               Q   i     ≈         1   2     ⁢       (           m     i   ⁢           ⁢   1             m     i   ⁢           ⁢   2             m     i   ⁢           ⁢   3             )     ⁡     [           T   0           T   1           T   2               T   3           T   4           T   5               T   6           T   7           T   8           ]       ⁢     (           m     i   ⁢           ⁢   1                 m     i   ⁢           ⁢   2                 m     i   ⁢           ⁢   3             )       -       (           b   0           b   1           b   2           )     ⁢     (           m     i   ⁢           ⁢   1                 m     i   ⁢           ⁢   2                 m     i   ⁢           ⁢   3             )               
where T is a diagonal matrix that may be represented by:
 
             T   =       [           T   0         0       0           0         T   4         0           0       0         T   8           ]     ⁢           ⁢   with                           T   0     =           (       ρ   i     (   n   )       ⁢     μ   1     (   n   )         )     2         (       m     i   ⁢           ⁢   1       ⁢     σ   1   2       )       (   n   )         +     2   ⁢   β   ⁢       ∑     r   ∈     ɛ   i         ⁢     α   ir             ,       T   4     =           (       ρ   i     (   n   )       ⁢     μ   2     (   n   )         )     2         (       m     i   ⁢           ⁢   2       ⁢     σ   2   2       )       (   n   )         +     2   ⁢   β   ⁢       ∑     r   ∈     ɛ   i         ⁢     α   ir             ,                   T   8     =           (       ρ   i     (   n   )       ⁢     μ   3     (   n   )         )     2         (       m     i   ⁢           ⁢   3       ⁢     σ   3   2       )       (   n   )         +     2   ⁢   β   ⁢       ∑     r   ∈     ɛ   i         ⁢     α   ir             ,       b   0     =           ρ   i     (   n   )       ⁢     x     i   ⁢           ⁢   1       (   n   )       ⁢     μ   1     (   n   )             (       m     i   ⁢           ⁢   1       ⁢     σ   1   2       )       (   n   )         +     2   ⁢   β   ⁢       ∑     r   ∈     ɛ   i         ⁢       α   ir     ⁢     m     r   ⁢           ⁢   1       (   n   )                 ,                   b   1     =           ρ   i     (   n   )       ⁢     x     i   ⁢           ⁢   2       (   n   )       ⁢     μ   2     (   n   )             (       m     i   ⁢           ⁢   2       ⁢     σ   2   2       )       (   n   )         +     2   ⁢   β   ⁢       ∑     r   ∈     ɛ   i         ⁢       α   ir     ⁢     m     r   ⁢           ⁢   2       (   n   )                 ,       b   2     =           ρ   i     (   n   )       ⁢     x     i   ⁢           ⁢   3       (   n   )       ⁢     μ   3     (   n   )             (       m     i   ⁢           ⁢   3       ⁢     σ   3   2       )       (   n   )         +     2   ⁢   β   ⁢       ∑     r   ∈     ɛ   i         ⁢       α   ir     ⁢       m     r   ⁢           ⁢   3       (   n   )       .                           
With the limitation set at a maximum three material types per voxel, the solution for the mixture parameter {m ik } becomes:
 
                   {               m     i   ⁢           ⁢   1       +     m     i   ⁢           ⁢   2       +     m     i   ⁢           ⁢   3         =   1                     m     i   ⁢           ⁢   1       ⁡     (       T   0     -     T   1       )       +       m     i   ⁢           ⁢   2       ⁡     (       T   3     -     T   4       )       +       m     i   ⁢           ⁢   3       ⁡     (       T   6     -     T   7       )         =       b   0     -     b   1                         m     i   ⁢           ⁢   1       ⁡     (       T   0     -     T   2       )       +       m     i   ⁢           ⁢   2       ⁡     (       T   3     -     T   5       )       +       m     i   ⁢           ⁢   3       ⁡     (       T   6     -     T   8       )         =       b   0     -     b   2                       (   13   )               
The same derivation can be repeated to obtain similar equations for the solution of the mixture parameter {m ik } when the limitation is four or more material types per voxel.
 
     Referring again to  FIG. 1 , after the iterative PV or mixture image segmentation of step  115  is completed, the voxels in the colon lumen may be classified as air, mixture of air with tissue, mixture of air with tagged materials, or mixture of tissue with tagged materials. Once the voxels are initially classified by mixture image segmentation, the present electronic colon cleansing (ECC) method applies this information to remove undesired components and to present a clean colon lumen and mucosa layer. 
     In most cases, PV image segmentation of  FIG. 3  provides correct tissue/material mixtures within the interface and mucosa layers. In some cases, however, PV image segmentation may identify an incorrect mixture component in these layers. In step  120 , voxels in the colon lumen may be reclassified to reduce such misclassification.  FIG. 5B  shows an example of such incorrectly segmented mixtures in voxels around location  30  along the vertical profile  530 . The peak around location  30  along the vertical profile (the interface layer) indicates a misidentification of voxels  530 . Around location  30 , the graph indicates that the voxel contains a mixture of muscle material, colon air, and bone/TM. However, at the air/tagged materials interface layer, there is no possibility of muscle being present, therefore such a classification can be considered a misclassification. 
     In order to avoid such incorrect identifications, the present exemplary embodiment utilizes the well-known dilation-erosion strategy, such as described in “Digital Image Processing,” Gonzalez et al., Addison-Wesley (1992), to reclassify potentially misclassified voxels. In the reclassification strategy of step  120 , as implemented in the present exemplary embodiment, the colon air segmentation volume may be represented by S a , which consists of the air component {m air } in the voxel array. Further, the tagged colonic material segmentation volume may be represented by S t . For each volume, a 3D dilation operation may be applied by a three-cubic strel matrix:
 
 D   a =Dilation( S   a ,strel), D   t =Dilation( S   t ,strel),strel=[1 1 1] T ·[1 1 1].  (14)
 
A new volume S e  may be constructed by applying the erosion operation on the sum of D a  and D t , i.e., S e =Erosion ((D a +D t ), strel). After the reclassification of the voxels, the voxels are classified correctly as air, a mixture of air/TM, or TM.
 
     Step  125  may include grouping of like voxels in regions according to their mixture type. The grouping of voxels facilitates the subsequent steps of ECC by establishing the various regions in the acquired image according to the group. The new erosion volume S e , established in step  120 , covers the entire colonic space including the tagged material region and the interface layer, as shown by the enclosed white cylindrical boarder  535  in  FIG. 5C . Based on the three volumes D a , D t , and S e , the entire colonic space may be divided into three regions according to the following criteria
 
 P   m ={voxel i   |S   i   e &gt;0 ,D   i   a &gt;0 ,D   i   t &gt;0}
 
 P   a ={voxel i |vowel i   ∉P   m   ,S   i   e &gt;0 ,S   i   a &gt;0 ,S   i   t =0}
 
 P   t ={voxel i |vowel i   ∉P   m   ,S   i   e   &gt;S   i   e =0 ,S   i   t &gt;0}  (15)
 
where P a  represents the colonic air space of region  1515  in  FIG. 5D , P t  reflects the tagged material region III  525  in  FIG. 5D , and P m  indicates the interface layer region II  520 .
 
     Step  130  may include further classifications of certain voxels as different parts of the volume of interest (VOI). Referring to  FIG. 8 , in the present exemplary embodiment, any voxels comprising the air space may be classified as part of the lumen in the VOI  800 . In the present exemplary embodiment where the VOI is a colon, the open space would be the colon lumen. Following the air space classification, various interface layers can be identified  805 . Interface layers may include zones at the border of different regions, which include voxels having a mixture of material types representative of the neighboring materials. In the present exemplary embodiment, where the VOI is a colon, there may be two interface layers. Referring to  FIG. 4A , the first interface layer is an enhanced mucosa layer (“the mucosa layer”)  415  consisting of a mixture of colon tissues and tagged materials, represented in  FIG. 4C  and  FIG. 4D . The present ECC method identifies the enhanced mucosa layer and removes the portion with tagged colonic materials. 
     The second interface layer  405  in the present exemplary embodiment is the layer that can be presented between the air layer and the tagged materials layer as seen in  FIG. 4A  and  FIG. 4C . Due to the PV effect and other potential errors such as from the scanner, patient motion, and image reconstruction procedure, the image density at the interface layer  405  may vary from the low end of about −900 HU, for air, to the high end of about 400 HU, for enhanced residues, as depicted in  FIG. 4C . A simple threshold approach, as known in the art, can be applied to remove a substantial portion of tagged colonic materials. However, since the second interface layer  405  exhibits density values which overlap the range of colon tissues, it is difficult to distinguish the voxels of colonic materials in this layer from that of the colon tissues based on density value alone. As illustrated in  FIG. 4B , applying a simple threshold, such as 200 HU, removes the majority of tagged colonic materials, but an interface layer  425  is retained. 
     Referring to  FIG. 5D , three regions may be identified as parts of the colon or colon lumen. Region I consists entirely of the colonic air space and, as stated above, may be classified as the colon lumen in step  800 . However, before voxels in regions II and III may be classified as part of the colon, their image density values I i   new  may be altered by a density altering function in step  810 : 
                     I   i   new     =     {                   I   i   original     +       (       μ   1     -     μ   4       )     ·     m     i   ⁢           ⁢   4         +                 (         I   _     i     -     I   i   original       )     ·     m     i   ⁢           ⁢   4                   if             I   _     i     ≥     I   i   original                         I   i   original     +       (       μ   1     -     μ   4       )     ·                 (       m     i   ⁢           ⁢   4       +         (       I   i   original     -       I   _     i       )     ·     m     i   ⁢           ⁢   4           μ   4         )               if             I   _     i     &lt;     I   i   original                       (   16   )               
where
 
                   I   _     i     =       ∑     k   =   1     4     ⁢       m   ik     ⁢     μ   k           ,         
and μ k  and m ik  have been defined before as the mean parameter of material type k in the image and the fraction of material type k in voxel i respectively In equation (16), index  1  refers to the segmentation of colonic air and index  4  refers to the segmentation of the bone/TM.
 
     Using equation (16), the image density values of the voxels in regions II and III are altered. The voxels with altered density values may then be classified in step  815 . In region III, the altered image density values of those voxels containing mixtures of colon tissues and tagged materials are restored to form the mucosa layer, and other voxels are classified as colon lumen voxels. In region II, the voxels on each side (or at each boundary) of the interface layer are altered by equation (16), but may not be completely restored due to the presence of more than two tissue types in some voxels. Referring to  FIGS. 6B and 6E , some small spots may be retained along the horizontal direction in region II  600  after processing by equation (16). 
     To eliminate this patch-like effect, the present exemplary embodiment introduces an equation to restore the image density for each voxel in region II. First, a subtraction volume SUB may be defined, which is generated by iteration: 
                         SUB   i     (   0   )       =     I   i   new       ⁢     
     ⁢     SUB   i     (   n   )       =       SUB   i     (     n   -   1     )       +       ∑     n   ∈     neighbor   ⁢           ⁢     (   i   )           ⁢             ⅇ     -       (       ∇   n     ⁢     SUB     (     n   -   1     )         )     2         ·                      ∇   n     ⁢     SUB     (     n   -   1     )                          ⁢                   (   17   )               
where ∇ represents the first-order derivative at the given position and the neighborhood of this voxel includes that voxel itself. In the present exemplary embodiment, after three iterations of equation (17), the image density at the interface layer may be further adjusted by I i   newII =I i   new −SUB i , i ∈ interface layer. The image density values in the interface layer may then be reduced to a reasonable level I i   newII  toward that of air level. Those voxels which exhibit modified density values, having a value close to that of air, are then classified as the colon lumen space in step  835 . These operations remove the patch-like effect and improve the image density restoration at both ends of region II, as shown in  FIG. 6C  and  FIG. 6F .
 
     The converted air voxels in regions I, II, and III make up the colon lumen space, from which a virtual colon model may be constructed for VC examination. Within the colon lumen space, a centerline or fly-path may be determined and a potential field may be constructed to facilitate the VC navigation through the entire lumen space to facilitate looking for polyps, as known in the art. The geometric information of the lumen border can be analyzed by a surface-based CAD technique to detect the polyps, as known in the art. The restored image density values of the mucosa layer, which may be comprised of five to ten voxels beyond the lumen border, can be extracted for texture analysis for improved CAD performance, according to the art.