Patent Application: US-43807807-A

Abstract:
a method is proposed for segmenting a 2 - or 3 - d space , spanned by a set of medical data comprising intensity values at locations within the space , to estimate the position of an object of medical significance . the method using a level set function having a level set which provides a model of boundary of the object . the level set function is iteratively updated by a force defined based on the medical data . for computational efficiency , only the force in the narrowband of the model boundary needs to be calculated . minimization of an energy function , related to the force , provides a termination condition for the iteration . high level knowledge can be incorporated in several ways , such as by an explicit force term which takes over from the force based on the medical data when prior knowledge of the object is about to be violated by the model .

Description:
an embodiment of the invention will now be described with reference to fig1 to 3 . this embodiment is used for the segmentation of a medical dataset which is a set of intensity values , such as an mr image . the intensity values are associated with respective points in a two - or three - dimensional space . the segmentation identifies an object within the image . specific examples are given of the use of the embodiment for segmenting orbital muscles within mr images , but application of the embodiment to other segmentations tasks is straightforward . the embodiment employs an initial model which is defined in the space of the image . if the space of the image is 2d , the initial model is a contour represented polygon , or , if the space of the image is 3d , the model is a surface represented by triangle meshes . the contour or surface forms one or more closed inside regions which indicate object itself . the model is chosen manually based on the prior - knowledge of the object &# 39 ; s shape and size , and initially located according the location of the object in the image . this model is then allowed to evolve , according to equations defined below , within the space of the image to form a final model which is the estimate of the position of the object . positions in the space are denoted by a vector x . the initial model is expressed as a zero level set of a higher dimensional level set function φ ( x , t ), defined as follows . the initial value of φ ( x , t ) at time t = 0 , that is φ ( x , 0 ), is defined as the signed euclidian distance map ( maurer 2003 ) of the model : with negative value inside the model , and positive outide the model . the evolution of the function φ ( x , t ) is governed by the following equation : where , φ t ( x , t ) denotes the time derivative of φ ( x , t ), f ( x , t ) denotes the total force which is a function of the model at time t and of the image dataset ( explicit expressions for f ( x , t ) are given below ), and n ( x , t ) is the unit normal of the level set , i . e . the unit vector which is defined by : the numerical solution of equation ( 1 ) is obtained by approximating the real time variable t by a series of discrete time t i = i · δt , where i = 0 , 1 , 2 , . . . and the time step is denoted as δt . the functions φ t ( x , t ), n ( x , t ) and f ( x , t ) can be approximated as φ ( x , t i ), n ( x , t i ) and f ( x , t i ) respectively . the time derivative φ t ( x , t ) is approximated as the difference between the respective values of φ ( x , t i ) at two successive time steps , divided by δt . the spatial derivative on the right hand side of ( 2 ) is treated by upwind numerical schemes ( sethian , 1999 ), which gives : φ ( x , t t + 1 )= φ ( x , t i )− δ t · n ( x , t i )· f ( x , t i )( i = 0 , 1 , 2 , . . . ). ( 3 ) to describe the evaluation state of the deformable model at a given time t , we introduce several notations : that is , the set of all locations x such that φ ( x , t ) is at least zero . ∂( t ): the boundary of the region ω ( t ), i . e . the set of all locations such that φ ( x , t )= 0 . b r ( t ): the “ narrowband ” of the boundary ∂( t ), which is defined as the set of points with euclidean distance to ∂( t ) less than a given threshold r . #( s ): the cardinal number which is equal to the number of discrete points in a given set s . we now turn to the problem of defining f ( x , t i ). to do so , we first define a novel function called the “ gradient flux ”. to begin with , according to the object we are trying to model , we define a “ gradient characteristic number ” g , which describes the gradient direction at a boundary of the object . in general terms , we select g according to whether the desired object is a valley of the intensity ( g = 1 ), or a peak of the intensity ( g =− 1 ). if the desired object is neither of these things , such as a saddle - point of the intensity , we select g = 0 . more specifically , if the object is such that all gradients on the object boundary point to the outside , g is set to 1 . if all gradients on the objection boundary point to the inside , g is set to − 1 . otherwise , g is set to 0 . for example , in a t1 mr orbital image , g is 1 for orbital muscles ; in t1 mr human brain images , g is 1 for the hippocampus , − 1 for ventricles , and 0 for caudate nuclei . now we define the “ gradient flux ” as follows . this is intended to produce the result that the nearer the boundary of the model is to that of the object in the image , the larger the average gradient flux on the boundary of the model . referring to fig1 , we denote the boundary ∂( t ) of the model at time t by reference numeral 1 . thus , the model at this time is the region ω ( r ) to the left of the model boundary 1 . the ( unknown ) boundary of the object which is to be identified by the segmentation technique is denoted by reference numeral 3 . consider a point 5 on the boundary 1 . the point 5 has coordinate x , and the normal direction to the boundary at this point is shown on fig1 as n ( x , t ). we will define the gradient flux at point 5 such that only the corresponding point in the image contributes to the gradient flux at point x . the corresponding point on the image is defined in such a way that it requires little computing cost to find it . specifically , it is defined as a point on the line segment which has its centre at point x and extends in direction n ( x , t ) to a predefined length d outside of the boundary ∂( t ), and in direction − n ( x , t ) to a predetermined length d inside the boundary . thus , to find this corresponding point we only have to search along this line segment , which is relatively computationally cheap . any point on the line segment has coordinates x + r · n ( x , t ), and denoting the intensity value of the image dataset at this point on the line segment as i ( x + r · n ( x , t ), the gradient vector of the image intensity at this point can be written as ∇ i ( x + r · n ( x , t )). it is expected that the point 7 at which the line segment intersects with the object boundary 3 is such that ∇ i ( x + r · n ( x , t )) is a local maximum . furthermore , ∇ i ( x + r · n ( x , t )) should be consistent with the expected direction : if the gradient characteristic number g of the object is 1 , then the gradient vector ∇ i ( x + r · n ( x , t )) should point to the outside of the model ; and if the gradient characteristic number g is − 1 , then the gradient vector ∇ i ( x + r · n ( x , t )) should point to the inside of the model . to give a numerical value to the gradient flux , suppose that a point charge were located at point x + r · n ( x , t ). thus , the electric flux density at point x would decrease with the distance r according to the function 3 /( 4πr 2 ). by analogy , we will define the gradient flux density at any point along the line segment such that the gradient flux decreases with the distance to the boundary in the same way as the electric flux . in order to avoid the gradient flux becoming infinite for r = 0 , we define the gradient flux using the function 3 /( 4πr 2 + 1 ) instead of 3 /( 4πr 2 ), so that gradient flux is defined at r = 0 . according to these heuristic rules , the embodiment calculates the gradient flux density e ( x , n ( x , t )) at any model boundary point 5 having location x as : where n ( x , t ) denotes the outside normal of the model boundary 1 at point x ; ∇ i ( x + r · n ( x , t )) is the gradient of the image at x + r · n ( x , t ); d is the predefined maximum search length ; the function ξ indicates the consistency requirement concerning the gradient direction , and is defined as : having now defined the gradient flux , we can now define the force f ( x , t ) used in eqn . ( 1 ). unlike traditional level set techniques , in which the force f ( x , t ) is directly calculated at the point x , we first calculate the force on the model boundary , and then expand the force to a narrowband of the boundary . specifically , for a point x on the boundary ∂( t ), we define its neighborhood n r ( x , t ) as the set of points that are on the boundary ∂( t ) and which have distance to x less than a given threshold r . as in deformable surface models ( liu 2005 , shen 2001 ), a boundary point x of the model is considered as the vertex of its boundary patch n r ( x , t ), on which the force at point x is calculated . the calculation combines the model constraint ( s ) with image features . this is done by defining the total force f ( x , t ) using ( i ) a gradient flux - based force f a ( x , t ) obtained using the gradient flux as defined above , which in turn is based on the intensity values , and ( ii ) an intensity based force f b ( x , t ) which ensures that model obeys the model constraints . first we obtain the gradient flux - based force . as a first step , the total gradient flux q 0 ( x , t ) on the boundary patch n r ( x , t ) is calculated as : then , the gradient fluxes q + ( x , t ) and q − ( x , t ) are calculated . these correspond respectively to the patch n r ( x , t ) being shifted to the outside of the model region ω ( r ) with unit translation n ( x , t ), and inside the model region ω ( t ) with unit translation − n ( x , t )). q + ⁡ ( x , t ) = ∑ x ′ ∈ n ′ ⁡ ( x , t ) ⁢ e ⁡ ( x ′ + n ⁡ ( x ′ , t ) , n ⁡ ( x ′ , t ) ) ( 9 ) q - ⁡ ( x , t ) = ∑ x ′ ∈ n ′ ⁡ ( x , t ) ⁢ e ⁡ ( x ′ - n ⁡ ( x ′ , t ) , n ⁡ ( x ′ , t ) ) ( 10 ) where , x ′+ n ( x ′, t ) and x ′− n ( x ′, t ) are points obtained by shifting the point x ′ with translation n ( x ′, t ) and − n ( x ′, t ) respectively . when the q + ( x , t ) is higher than q − ( x , t ) and q 0 ( x , t ), the model boundary ∂( t ) is attracted to expand at point x ; when q − ( x , t ) is higher than q + ( x , t ) and q 0 ( x , t ), the boundary ∂( t ) is urged inward at point x ; in other cases , the boundary should remain stationary . based on these heuristic rules , the gradient flux - based force f a ( x , t ) is calculated as : f a ⁡ ( x , t ) = { q + ⁡ ( x , t ) · n ⁡ ( x , t ) q + ⁡ ( x , t ) & gt ; q 0 ⁡ ( x , t ) , q + ⁡ ( x , t ) & gt ; q - ⁡ ( x , t ) - q - ⁡ ( x , t ) · n ⁡ ( x , t ) q - ⁡ ( x , t ) & gt ; q 0 ⁡ ( x , t ) , q - ⁡ ( x , t ) & gt ; q + ⁡ ( x , t ) q 0 ⁡ ( x , t ) · n ∞ otherwise ( 11 ) where , n − is the unit vector n ( x , t ) with an additional flag to indicate that the force is used to keep the boundary ∂( r ) fixed at point x . now we calculate the intensity - based force . let μ ( t ) and σ ( t ) denote the mean and variation of the image intensity within the region , and let and l ( x , t ) denote the mean of the image intensity on the boundary patch n r ( x , t ), i . e . : in the case that the gradient characteristic number g of an object is 1 , i . e . the intensity inside the object is less than that outside the object , if the intensity l ( x , t ) at the boundary point x is greater than μ ( t )+ σ ( t ), it is expected that it will be necessary to remove point x from region ω ( t ). thus , we generate an intensity - based force at x which pushes the boundary ∂( t ) toward the inside of the region ω ( t ). conversely , otherwise , if l ( x , t )& lt ; μ ( t )+ σ ( t ), then we generate an intensity - based force to push x outwards . the cases for g =− 1 and 0 are similar to g = 1 . specifically , motivated by these heuristic rules , the embodiment generates an intensity - based force f b ( x , t ) at a boundary point x which is calculated as : f b ⁡ ( x , t ) = { n ⁡ ( x , t ) · φ ⁡ ( 1 - g · [ i ⁡ ( x ) - μ ⁡ ( t ) ] / σ ⁡ ( t ) ) ( g = 1 , - 1 ) n ⁡ ( x , t ) · φ ⁡ ( 1 -  i ⁡ ( x ) - μ ⁡ ( t )  / σ ⁡ ( t ) ) ( g = 0 ) ( 15 ) and accordingly limits the force f b ( x , t ) within the interval [− 1 , 1 ]. let f a ( t ) and f max ( t ) denote the average and maximum magnitude of forces f a ( x , t ) on the boundary ∂( t ), i . e . : the total force f ( x , t ) is finally estimated from the gradient flux - based force f a ( x , t ) and the intensity - based force f b ( x , t ) with the force f a ( x , t ) as the priority : we have now fully defined all elements of eqn . ( 1 ) for a point on the boundary . the algorithm implements the level set evolution by a fast narrowband approach ( sethian 1999 ). we expand the boundary force f ( x , t ) into a narrowband b r ( t ) of the boundary ∂( t ). in the 3d case , for any voxel x we use n 26 ( x ) to denote its 26 - neighborhood ( i . e . its 26 nearest neighbours , which are directly adjacent ). at any given time t , we define a set of neighbourhoods { ω ( k ) }, for k = 0 , 1 , 2 , . . . , with corresponding boundaries {∂ ( k ) }. ω ( 0 ) is defined to be equal to ω ( t ), and its boundary ∂ ( 0 ) is initially set to ∂( t ). for k = 0 , 1 , 2 , etc . we define the region ω ( k + 1 ) and its boundary ∂ ( k + 1 ) iteratively from ω ( k ) and ∂ ( k ) : ∂ ( k + 1 ) =[∪ xε ∂ ( k ) n 26 ( x )]− ω ( k ) , ω ( k + 1 ) = ω ( k ) ∪∂ ( k + 1 ) ( 20 ) the force f ( x , t ) on the set of points ∂ ( 0 ) is first calculated according to eqn . ( 19 ). the embodiment then uses the force on each set of points ∂ ( k ) to obtain the force on the set of points ∂ ( k + 1 ) as follows . for any voxel which is ∂ ( k + 1 ) and has position x , the force f ( x , t ) is a weighted average of the force on those of its 26 nearest - neigbors which are in the set ∂ ( k ) , i . e . : f ⁡ ( x , t ) = ∑ x ′ ∈ n 26 ⁡ ( x ) ⋂ ∂ ( k ) ⁢ f ⁡ ( x ′ , t ) / d ⁡ ( x , x ′ ) / # ⁢ ( n 26 ⁡ ( x ) ⋂ ∂ ( k ) ) ( 21 ) where , d ( x , x ′) denotes the euclidean distance of two points x and x ′. the force expansion in the 2d case is similar to that in the 3d case . it differs just by changing the 26 - neighborhood in the 3d case into a 8 - neighborhood in the 2d case . the embodiment makes use of an “ energy function ” to determine when its iterations should terminate . the energy ē ( t ) of the model is estimated as the negative of the average gradient flux density on the model boundary ∂( t ): we are now in a position to define the complete algorithm . this will be explained with reference to fig3 . the algorithm used by the embodiment is based on the topology preserved narrowband method presented in ( han 2003 ). specifically , we propose the following algorithm to update the value φ ( x , t i + 1 ) from φ ( x , t i ): 1 . initialisation ( step 10 ). set i = 0 and the initial value φ ( x , t 0 ) to be the signed euclidean distance map of the initial model ( i . e . it is zero for each point on the contour ( 2d case ) or surface ( 3d case ) which is the initial estimate of the object position , and at all other points is the euclidian distance from the contour / surface multiplied by a sign − 1 or + 1 according whether the point is inside or outside the region enclosed by the contour or surface ). initialize the sign function s ( x , t 0 ), and the narrowband b k ( t 0 ), where k is a manually - set parameter to specify the width of the narrowband b k ( t ) which is defined as the region ω ( k ) obtained according to eqn 20 . 2 . calculate force f ( x , t i ) on the boundary ∂( t i ) of the model region ω ( t i ) ( step 20 ). 3 . expand the force f ( x , t ) from boundary ∂( t i ) to its narrowband b k ( t i ) ( step 30 ). 4 . for each point x in b k ( t i ), calculate the new value φ ( x , t i + 1 ) ( step 40 ) a ) detect whether x is a simple point of the point set ω ( t i ) using the method presented in ( bertrand , 1994 ), where , a simple point means its addition or removal from ω ( t i ) will not change the topology of ω ( t i ). b ) if x is a simple point , update φ ( x , t t + 1 ) according to eqn . ( 3 ), and then update s ( x , t i + 1 ) according to eqn . ( 4 ). c ) otherwise , i . e ., x is not a simple point , set φ ( x , t i + 1 )= ε * sign ( φ ( x , t i )), where , ε is a small positive number , so that φ ( x , t i + 1 ) takes a small value while keep the same sign with the old value φ ( x , t i ). in this way , the topology of ω ( t i ) is preserved . 5 . calculate the total energy e ( t i + 1 ) ( step 50 ). 6 . determine whether e ( t i + 1 )& gt ; e ( t i ) ( step 60 ), and if so return φ ( x , t i ) as the final evolution result ( step 70 ); otherwise , go to the next step to begin the next iteration . alternatively , the termination criterion may be based on the number of iterations reaching a threshold . these two approaches may be combined , so that the algorithm terminates if the number of iterations reaches a threshold without a minimal energy having been reached . 7 . reinitialization : determine if the zero level set of φ ( x , t i + 1 ) reaches to the boundary of the current narrowband b k ( t i ) ( step 80 ). if so , reinitialize φ ( x , t i + 1 ) to be the signed distance function of its zero level set ( step 90 ). the embodiment was used to segment orbital muscles from mr images . fig2 shows the segmentation of five muscles with five models in one 2d slice . fig2 ( a ) shows the intensity values of the data set . fig2 ( b ) shows the initial contours . fig2 ( c ) shows the result of the embodiment . it can be seen that the models are deformed to fit the original data quite accurately . 1 . kass , m ., witkin , a ., and terzopoulos , d . 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