Patent Application: US-53097302-A

Abstract:
a functional model for a set of experimental data has k independent parameters . the parameters are to be estimated from an experimental data - set of n data points , comprising “ inlier ” data points representative of the model and “ outlier ” data points which are not representative of the model . multiple subsets of the data points are defined , and each used to estimate the parameters of the model . the various estimates of the parameters are plotted in the parameter space to identify the peak parameters in the parameter space . data points which are not described by the model using the said peak parameters are judged to be outliers . the method makes it possible to identify up to n − k ′− 3 outliers .

Description:
suppose the experimental data - set comprises n input data points . each input data point is any quantity or vector denoted as x , x can be a vector of coordinates , gray level related quantities if the data originates from images , etc . x is called the feature vector of the input data point . in the embodiment , the model has k independent parameters p j ( j = 1 , . . . , k ) and is usually a function of x . the model is denoted as mod ( x ) given by : mod ( x )= p 1 . base 1 + p 2 . base 2 + . . . + p k . base k ( 1 ) where base j ( j = 1 , . . . k ) are known functions of the feature vector , x and the symbol “.” represents multiplication . a determination of the model is thus equivalent to the task of identifying the k parameters p 1 , . . . , p k using the experimental data - set . for each data point with feature vector x i , a corresponding model value mod ( x i ) can be calculated , where i = 1 , . . . , n . for inlier data points , x i and mod ( x i ) are related by equation ( 1 ), possibly with a noise , whereas outlier data points are not related by equation ( 1 ). the method proceeds by the steps shown in fig1 . in step 1 a number of subsets of the input data - set is generated . each subset is composed of at least k ′ ( k ′ is the number by which the k parameters will be uniquely determined in the subset containing any k ′ data points ) of the n input data points . the number of subsets with k ′ data points which can be formed in this way is c n k ′ =( n . ( n − 1 ). ( n − 2 ) . . . 2 )/( k ′. ( k ′− 1 ). ( k ′− 2 ). . . 2 ). note that in some applications all of these subsets may be generated , while in other applications only a portion of the total number of subsets may be generated . denote the total number of ways to form the subsets as m . in step 2 , for each of the subsets the parameters { p 1 , . . . , p k } are estimated either by least square mean estimation or by solving the k ′ linear equations . thus , each subset yields a respective point in the k - dimensional parameter space . hence in the k - dimensional parameter space , m parameter points are obtained from the estimation , with each parameter point denoted p i =( p 1 ( i ), p 2 ( i ), . . . , p k ( i )) t . here t stands for transpose . each subset of input data points will have a corresponding parameter point in the parameter space . in step 3 , count the number of occurrence of a parameter point ( histogram ), and plot the histogram in the parameter space to show , for each of the m parameter points , the number of subsets of input data points with the parameters close to the parameter point . for some applications , the parameters may need to be digitised with any digitisation method ( for example , an orientation of both 1 . 0 ° and 1 . 02 ° may both be digitised to 1 . 0 °). as the parameters derived from each subset of input data points will be distributed in the k - dimensional parameter space , a preferable way to get the histogram from the distribution is to specify the sizes of neighborhood in each coordinate of the parameter space . the neighborhood sizes can be specified by users or by any means . below a way to calculate the neighborhood sizes is illustrated . for the j - th ( j = 1 , 2 , . . . , k ) coordinate of all the m parameter points of the estimated parameters , arrange them in ascending order and still denote them as p j ( 1 ), p j ( 2 ), . . . , p j ( m ) for simplicity of denotation . the difference between p j ( t + 1 ) and p j ( t ) ( t = 1 , 2 , . . . , m − 1 ) is denoted dif ( p j , t ). the neighborhood size for the jth coordinate can be the median of dif ( p j , t ) for all t ranging from 0 to m − 1 , or the average of dif ( p j , t ), or any percent of the distribution of dif ( p j , t ) ( 100 percent will correspond to the maximum of dif ( p j , t ) while 0 percent will be 0 , and 10 percent corresponds to the neighborhood size so that the number of difference dif ( p j , t ) being smaller than the neighborhood size will be no more than 0 . 1 *( m − 1 )). having decided the neighborhood size for each coordinate of the parameters , namely , the j - th coordinate &# 39 ; s neighborhood size being δ j , the number of points for a given parameter point p i ( i = 1 , 2 , . . . , m ) in the parameter space is the number of parameter points p =( p 1 , p 2 , . . . p k ) t falling in the neighborhood this number of points is also called the number of occurrence of the subsets of input data with the parameters specified by the parameter point p i . in step 4 , we find the peak of the histograms found in step 3 . the k parameters corresponding to the peak of the histogram are called candidate peak parameters . if the number of occurrence of the histogram peak is greater than a predetermined threshold , e . g . 3 , and there is only one peak , then we may take the peak as a good estimate of the true parameters of the model , and the candidate peak parameters are called peak parameters . note that such a peak will generally be found when at least 3 of the subsets consists exclusively of inlier data points . this is bound to occur when there are at least k ′+ 3 inliers ( so that at least 3 subsets are composed entirely of inliers ), and thus the present method can cope even in the case that there are n − k ′− 3 outliers . in the case of multiple peaks exhibited in the histogram , depending on the nature of the original problem , one way is to take the candidate peak parameters with the maximum number of occurrence as the peak parameters . alternatively , one can pick up the candidate peak parameters with the maximum integration as the peak parameters . in step 5 we determine which input data points are such that they follow equation ( 1 ) with parameters equal to or very close to the peak parameters . such input points are judged to be inlier input data points . all other input points are judged to be outlier input points . in step 6 we determine a best estimate for the parameters using only the inliers . this can be done by a conventional method , such as a least square fit of the inliers . we now consider one specific example of the method , namely to derive the midsagittal plane ( msp ) from magnetic resonance ( mr ) brain images . determination of midsagittal plane of the human brain is 1 ) a prerequisite for talairach framework [ 7 ]; 2 ) the first step in spatial normalisation or anatomical standardisation of brain images ; 3 ) a first step in intra - subject , inter / intra - modality image registration ; 4 ) helpful to detection of brain asymmetry due to tumors as well as any mass effects for diagnosis . according to the patent application [ 5 ] entitled “ method and apparatus for determining symmetry in 2d and 3d images ” ( international application number pct / sg 02 / 00006 ), around 16 fissure line segments are extracted from 16 parallel planes of the volume ( axial slices ). due to the pathology or ubiquitous asymmetry presented in axial slices , some of the extracted fissure line segments deviate greatly from the expected fissure that should be removed in order to get a precise plane equation of the msp . there are two kinds of outliers to remove , i . e ., orientation outliers and plane outliers . as all extracted fissure line segments are from different parallel axial slices and they are supposed to form a plane ( the msp ), they should have the same orientation . those extracted fissure line segments deviating from the expected orientation are taken as orientation outliers and the rest of extracted fissure line segments as orientation inliers . for all the orientation inliers , some extracted fissure line segments may deviate from an expected plane , and are judged as plane outliers with the rest of orientation inliers judged as plane inliers . the plane equation of the msp is calculated by the least square error fit of all the plane inliers . both the expected orientation and expected plane are derived from the proposed invention described below . fig2 shows the steps to derive plane equation of the msp from the 16 extracted fissure line segments . in step 100 , orientation outliers are removed . in step 200 , plane outliers are removed . following this the plane equation of the msp is estimated . reference [ 5 ] includes a detailed description of the orientation outlier removal , but reference [ 5 ] can only deal with the orientation outlier removal based on empirical trial instead of a systematic framework while the current invention tends to provide a solution for the outlier removal of all kinds of models . for removal of plane outliers , the model is a three - dimensional plane , i . e ., where ( x , y , z ) are the coordinates in the three - dimensional image volume . in order to facilitate histogramming , it is supposed that there are 3 independent parameters for the model . each subset of data will contain two orientation inliers ( 4 three - dimensional points in three - dimensional image volume ). suppose there are n ′ ( n ′& lt ;= 16 ) orientation inliers . refer to fig3 for the steps to remove plane outliers and to calculate the plane equation of the msp : 1 ) from n ′ orientation inliers pick up any 2 orientations to form all the subsets ( step 201 ). there are altogether n ′ ( n ′− 1 )/ 2 different subsets . 2 ) calculate the least square fit plane equation of each subset ( step 202 ); 3 ) calculate the histogram of p 1 , p 2 , p 3 and p 4 by specifying the neighborhood sizes of p 1 being 0 . 1 , p 2 0 . 1 , p 3 0 . 1 , and p 4 1 . 0 ( step 203 ); 4 ) find the maximum peak of the histogram ( step 204 ) and denote the parameters corresponding to this peak as p 1 *, p 2 *, p 3 *, and p 4 * 5 ) judge those subsets as outlier subsets if their plane parameters ( p 1 , p 2 , p 3 , p 4 ) satisfying at least one of the following inequalities : the rest of the subsets are considered inlier subsets . those orientation inliers included in any of the inlier subset are judged as plane inliers . the rest of the orientation inliers are judged as plane outliers ( step 205 ). 6 ) finally the plane equation of the msp is the least square fit of the plane inliers ( step 206 ). efficient outlier removal is a key factor to deal with both normal and pathological images in medical imaging . in the case of extraction of the msp , the method proposed by liu et al [ 1 ] uses the robust standard deviation , but still the inliers may have a scattered orientation instead of the dominant one which corresponds to the maximum peak of the histogram . the next example will illustrate this . the method proposed by prima et al [ 4 ] uses the least trimmed squares estimation which can tackle at most 50 % of outliers while the embodiment can yield an outlier removal rate ( 3 plane inliers — 13 plane outliers out of 16 data ) 81 %. note that in this example , it is supposed that at least 3 strongly correlated subsets are available when at least k ′+ 3 ( k ′= 1 ) inliers are present ( the occurrence of the peak orientation will be no less than 3 ). in other words , the present method can function satisfactorily even when there are n − k ′− 3 outliers . in the next example , the difference between the embodiment of this invention and the result based on robust standard deviation as used by liu et al [ 1 ] is illustrated . suppose the orientations of 11 extracted fissure line segments are 50 °, 35 °, 30 °, 23 °, 17 °, 13 °, 11 °, 11 °, 11 °, 11 °, 9 ° respectively . the median of the angle is 13 °, and the robust standard deviation is 4 . 45 °. according to [ 1 ], only three angles ( 50 °, 35 °, 30 °) will be judged as outliers . the weighted estimation of orientation will be 15 . 8 °, and the average of the inlier orientation is 13 . 25 °. by the method disclosed in this invention , the peak parameter of the orientation is 11 ° by specifying the neighborhood size being 1 °, which is the dominant orientation . note the number of outliers is 6 which is beyond the limit of existing outlier removal methods , so it is understandable the existing methods will not able to remove all the outliers . the embodiment takes 11 ° as the inliers from the histogram and the number of outliers is 7 . the disclosure of the following references is incorporated herein in its entirety : liu y , collins r . t . and rothfus w . e ., “ robust midsagittal plane extraction from normal and pathological 3 - d neuroradiology images ,” 2001 , ieee transactions on medical imaging , 20 ( 3 ), p173 - 192 . 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