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Tissue classification based on 3D local intensity ... - IEEE Xplore

Tissue classification based on 3D local intensity ... - IEEE Xplore

SATO ET AL.: TISSUE

SATO ET AL.: TISSUE CLASSIFICATION BASED ON 3D LOCAL INTENSITY STRUCTURES FOR VOLUME RENDERING 163 Fig. 2. Schematic diagrams of measures of similarity to local structures. The roles of weight functions in representing the basic conditions of a local structure are shown. (a) Line measure. The structure becomes sheet-like and the weight function approaches zero with deviation from the condition 3 ' 2 , blob-like and the weight function ! approaches zero with transition from the condition 2 1 ' 0 to 2 ' 1 0, and stenosislike and the weight function ! approaches zero with transition from the condition 2 1 ' 0 to 1 0. (b) Blob measure. The structure becomes sheet-like with deviation from condition 3 ' 2 , and line-like with deviation from the condition 2 ' 1 . (c) Sheet measure. The structure becomes blob-like, groove-like, line-like, or pit-like with transition from 3 1 ' 0 to 3 ' 1 0, 3 1 ' 0 to 1 0, 3 2 ' 0 to 3 ' 2 0, or 3 2 ' 0 to 2 0, respectively. in which 0 < 1 (Fig. 1b). is introduced in order to give !… s ; t † an asymmetrical characteristic in the negative and positive regions of s . Fig. 2a shows the roles of weight functions in representing the basic conditions of the line case. In (6), j 3 j represents the condition 3 0, … 2 ; 3 † represents the condition 3 ' 2 and decreases with deviation from the condition 3 ' 2 , and !… 1 ; 2 † represents the condition 2 1 ' 0 and decreases with deviation from the condition 1 ' 0 which is normalized by 2 . By multiplying j 3 j, … 2 ; 3 †, and !… 1 ; 2 †, we represent the condition for a line shown in Table 1. For the line case, the asymmetric characteristic of ! is ong>basedong> on the following observations: . When 1 is negative, the local structure should be regarded as having a blob-like shape when j 1 j becomes large (lower right in Fig. 2a). . When 1 is positive, the local structure should be regarded as being stenotic in shape (i.e., part of a vessel is narrowed), or it may be indicative of signal loss arising from the partial volume effect (lower left in Fig. 2a). Therefore, when 1 is positive, we make the decrease with the deviation from the 1 ' 0 condition less sharp in order to still give a high response to a stenosis-like shape. We typically used ˆ 0:25 and ˆ 0:5 (or 1) in our experiments. Extensive analysis of the line measure, including the effects of parameters and , can be found in [10]. The specific shape given in (7) is ong>basedong> on the need to p generalize two line measures, 3 2 and min… 3 ; 2 †ˆ j 2 j (where 3 < 2 < 0), suggested in earlier work [13]. These measures take into account the conditions 3 0 and p 3 ' 2 . j 3 j … 2 ; 3 † in (6) is equal to 3 2 and j 2 j when ˆ 0:5 and ˆ 1, respectively. In this formulation [10], the same type of function shape as that in (7) is used for (8) to add the condition 2 1 ' 0. We can extend the line measure to the blob and sheet cases. In the blob case, the condition 3 ' 2 ' 1 0 can be decomposed into 3 0 and 3 ' 2 and 2 ' 1 .By representing the condition t ' s using … s ; t †, we can derive a blob filter given by S blob ffg ˆ j 3j … 2 ; 3 † … 1 ; 2 † 3 2 1 < 0 0; otherwise: …9† In the sheet case, the condition 3 2 ' 1 ' 0 can be decomposed into 3 0 and 3 2 ' 0 and 3 1 ' 0. By representing the condition t s ' 0 using !… s ; t †, we can derive a sheet filter given by S sheet ffg ˆ j 3j!… 2 ; 3 †!… 1 ; 3 † 3 < 0 …10† 0; otherwise: Fig. 2b and Fig. 2c show the relationships between the eigenvalue conditions and weight functions in the blob and sheet measures.

164 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. 6, NO. 2, APRIL-JUNE 2000 3.2 Multiscale Computation and Integration of Filter Responses Local structures can exist at various scales. For example, vessels and bone cortices can, respectively, be regarded as line and sheet structures with various widths. In order to make filter responses tunable to a width of interest, the derivative computation for the gradient vector and the Hessian matrix is combined with Gaussian convolution. By adjusting the standard deviation of Gaussian convolution, local structures with a specific range of widths can be enhanced. The Gaussian function is known as a unique distribution optimizing localization in both the spatial and frequency domains [19]. Thus, convolution operations can be applied within local support (due to spatial localization) with minimum aliasing errors (due to frequency localization). We denote the local structure filtering for a volume blurred by Gaussian convolution with a standard deviation f as S ff; f g; …11† where 2fint; edge; sheet; line; blobg. The filter responses decrease as f in the Gaussian convolution increases unless appropriate normalization is performed [20], [21]. In order to determine the normalization factor, we consider a Gaussian-shaped model of edge, sheet, line, and blob with variable scales. An ideal step edge is described as 1; if x>0 h edge …x† ˆ …12† 0; otherwise; where x ˆ…x; y; z†. By combining Gaussian blur with (12), a blurred edge is modeled as ! h edge …x; r †ˆ 1 exp jxj2 h 2 3 2 3 2 2 edge …x†; …13† r r where represents the convolution and r is the standard deviation of the Gaussian function to control the degree of blurring. Sheet, line, and blob structures with variable widths are modeled as h sheet …x; r †ˆexp x2 2 2 ; …14† r h line …x; r †ˆexp x2 ‡ y 2 ; …15† 2 2 r and h blob …x; r †ˆexp jxj2 2 2 r ! ; …16† respectively, where r controls the width of the structures. We determine the normalization factor so that S fh …x; r †; f g satisfies the following condition: . max r S fh …0; r †; f g is constant, irrespective of f , where 0 ˆ…0; 0; 0†. The above condition can be satisfied when the Gaussian first and second derivatives are computed by multiplying by f or 2 f , respectively, as the normalization factor. That is, the normalized Gaussian derivatives are given by f xp y q…x; f†ˆf p‡q f @ p‡q @x p @y G…x; f†g f…x†; q …17† where p and q are nonnegative integer values satisfying p ‡ q 2, and G…x; † is an isotropic 3D Gaussian function with a standard deviation (see Appendix A for the derivation of the normalization factor for secondorder local structures). Fig. 3 shows the normalized response of S fh …0; r †; f g (where f ˆ i s i1 , 1 ˆ 1, p s ˆ 2 , and i ˆ 1; 2; 3; 4) for 2fedge; sheet; line; blobg when r is varied. In the edge case, the maximum of the normalized response of S edge fh edge …0; r †; f g is p 1 ( 0:399) when 2 r ˆ 0, that is, the case of the ideal edge without blurring. By increasing f , S edge fh edge …0; r †; f g gives a higher response to blurred edges with a larger r , while the response to the ideal edge remains constant (Fig. 3a). In the line case, the maximum of the normalized response S line fh line …0; r †; f g is 1 4 …ˆ 0:25† when r ˆ f [10]. That is, S line ff; f g is regarded as being tuned to line structures with a width r ˆ f . A line filter with a single scale gives a high response in only a narrow range of widths. We call the curves shown in Fig. 3b, Fig. 3c, and Fig. 3d width response curves, which represent filter characteristics like frequency response curves. The width response curve of the line filter can be adjusted and widened using multiscale integration of filter responses given by M line ff; 1 ;s;ngˆmax 1in S lineff; i g; …18† where i ˆ s i1 1 , in which 1 is the smallest scale, s is a scale factor, and n is the number of scales [10]. The width response curve of multiscale integration using the four scales consists of the maximum values among the four single-scale width response curves and gives nearly uniform responses in the width range between r ˆ 1 and p r ˆ 4 when s ˆ 2 (Fig. 3b). While the width response curve can be perfectly uniform if continuous variation values are used for f , the deviation from the continuous case is less than 3 percent using discrete values for f with p s ˆ 2 [10]. Similarly, in the cases of Ssheet fh sheet …0; r †; f g and S blob fh blob …0; r †; f g, the maximum of the normalized 2 response is p … 3 † 3 … 0:385† when r ˆ p f q q 2 (Fig. 3c), and 2 3 … 3 5† 5 3 … 0:186† when r ˆ 2 f (Fig. 3d), respectively (see Appendix A for the derivation of the above relationships). For the second-order cases, the width response curve can be adjusted and widened using the multiscale integration method given by M 2 ff; 1 ;s;ngˆmax 1in S 2 ff; i g; …19†

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