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

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 functi**on**s in representing the basic c**on**diti**on**s of a **local** structure are shown. (a) Line measure. The structure becomes sheet-like and the weight functi**on** approaches zero with deviati**on** from the c**on**diti**on** 3 ' 2 , blob-like and the weight functi**on** ! approaches zero with transiti**on** from the c**on**diti**on** 2 1 ' 0 to 2 ' 1 0, and stenosislike and the weight functi**on** ! approaches zero with transiti**on** from the c**on**diti**on** 2 1 ' 0 to 1 0. (b) Blob measure. The structure becomes sheet-like with deviati**on** from c**on**diti**on** 3 ' 2 , and line-like with deviati**on** from the c**on**diti**on** 2 ' 1 . (c) Sheet measure. The structure becomes blob-like, groove-like, line-like, or pit-like with transiti**on** 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 regi**on**s of s . Fig. 2a shows the roles of weight functi**on**s in representing the basic c**on**diti**on**s of the line case. In (6), j 3 j represents the c**on**diti**on** 3 0, … 2 ; 3 † represents the c**on**diti**on** 3 ' 2 and decreases with deviati**on** from the c**on**diti**on** 3 ' 2 , and !… 1 ; 2 † represents the c**on**diti**on** 2 1 ' 0 and decreases with deviati**on** from the c**on**diti**on** 1 ' 0 which is normalized by 2 . By multiplying j 3 j, … 2 ; 3 †, and !… 1 ; 2 †, we represent the c**on**diti**on** for a line shown in Table 1. For the line case, the asymmetric characteristic of ! is **on** the following observati**on**s: . 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 deviati**on** from the 1 ' 0 c**on**diti**on** less sharp in order to still give a high resp**on**se 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 **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 c**on**diti**on**s 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 formulati**on** [10], the same type of functi**on** shape as that in (7) is used for (8) to add the c**on**diti**on** 2 1 ' 0. We can extend the line measure to the blob and sheet cases. In the blob case, the c**on**diti**on** 3 ' 2 ' 1 0 can be decomposed into 3 0 and 3 ' 2 and 2 ' 1 .By representing the c**on**diti**on** 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 c**on**diti**on** 3 2 ' 1 ' 0 can be decomposed into 3 0 and 3 2 ' 0 and 3 1 ' 0. By representing the c**on**diti**on** 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 relati**on**ships between the eigenvalue c**on**diti**on**s and weight functi**on**s in the blob and sheet measures.

164 **IEEE** TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. 6, NO. 2, APRIL-JUNE 2000 3.2 Multiscale Computati**on** and Integrati**on** of Filter Resp**on**ses Local structures can exist at various scales. For example, vessels and b**on**e cortices can, respectively, be regarded as line and sheet structures with various widths. In order to make filter resp**on**ses tunable to a width of interest, the derivative computati**on** for the gradient vector and the Hessian matrix is combined with Gaussian c**on**voluti**on**. By adjusting the standard deviati**on** of Gaussian c**on**voluti**on**, **local** structures with a specific range of widths can be enhanced. The Gaussian functi**on** is known as a unique distributi**on** optimizing **local**izati**on** in both the spatial and frequency domains [19]. Thus, c**on**voluti**on** operati**on**s can be applied within **local** support (due to spatial **local**izati**on**) with minimum aliasing errors (due to frequency **local**izati**on**). We denote the **local** structure filtering for a volume blurred by Gaussian c**on**voluti**on** with a standard deviati**on** f as S ff; f g; …11† where 2fint; edge; sheet; line; blobg. The filter resp**on**ses decrease as f in the Gaussian c**on**voluti**on** increases unless appropriate normalizati**on** is performed [20], [21]. In order to determine the normalizati**on** factor, we c**on**sider 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 c**on**voluti**on** and r is the standard deviati**on** of the Gaussian functi**on** to c**on**trol 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 c**on**trols the width of the structures. We determine the normalizati**on** factor so that S fh …x; r †; f g satisfies the following c**on**diti**on**: . max r S fh …0; r †; f g is c**on**stant, irrespective of f , where 0 ˆ…0; 0; 0†. The above c**on**diti**on** can be satisfied when the Gaussian first and sec**on**d derivatives are computed by multiplying by f or 2 f , respectively, as the normalizati**on** 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 n**on**negative integer values satisfying p ‡ q 2, and G…x; † is an isotropic **3D** Gaussian functi**on** with a standard deviati**on** (see Appendix A for the derivati**on** of the normalizati**on** factor for sec**on**dorder **local** structures). Fig. 3 shows the normalized resp**on**se 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 resp**on**se 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 resp**on**se to blurred edges with a larger r , while the resp**on**se to the ideal edge remains c**on**stant (Fig. 3a). In the line case, the maximum of the normalized resp**on**se 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 resp**on**se in **on**ly a narrow range of widths. We call the curves shown in Fig. 3b, Fig. 3c, and Fig. 3d width resp**on**se curves, which represent filter characteristics like frequency resp**on**se curves. The width resp**on**se curve of the line filter can be adjusted and widened using multiscale integrati**on** of filter resp**on**ses 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 resp**on**se curve of multiscale integrati**on** using the four scales c**on**sists of the maximum values am**on**g the four single-scale width resp**on**se curves and gives nearly uniform resp**on**ses in the width range between r ˆ 1 and p r ˆ 4 when s ˆ 2 (Fig. 3b). While the width resp**on**se curve can be perfectly uniform if c**on**tinuous variati**on** values are used for f , the deviati**on** from the c**on**tinuous 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 resp**on**se 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 derivati**on** of the above relati**on**ships). For the sec**on**d-order cases, the width resp**on**se curve can be adjusted and widened using the multiscale integrati**on** method given by M 2 ff; 1 ;s;ngˆmax 1in S 2 ff; i g; …19†

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