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- Intensity,
- Structures,
- Multichannel,
- Images,
- Tissues,
- Rendering,
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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 165 Fig. 3. Plots of normalized p resp**on**ses of **local** structure filters for corresp**on**ding **local** models, S fh …0; r †; f g, where r is c**on**tinuously varied and f ˆ i s i1 ( 1 ˆ 1, s ˆ 2 , and i ˆ 1; 2; 3; 4). See Appendix A for the theoretical derivati**on**s of the resp**on**se curves shown in (b)-(d). (a) Resp**on**se of the edge filter for the edge model ( ˆ edge). (b) Resp**on**se of the line filter for the line model ( ˆ line). (c) Resp**on**se of the blob filter for the blob model ( ˆ blob). (d) Resp**on**se of the sheet filter for the sheet model ( ˆ sheet). where 2 2fsheet; line; blobg. S int can also be extended so as to be combined with Gaussian blur, in which case we represent it as S int ff; f g. 3.3 Implementati**on** Our **3D** **local** structure filtering methods described above assume that volume data with isotropic voxels are used as input data. However, voxels in medical volume data are usually anisotropic since they generally have lower resoluti**on** al**on**g the third directi**on**Ði.e., the directi**on** orthog**on**al to the slice planeÐthan within slices. Rotati**on**al invariant feature extracti**on** becomes more intuitive in a space where the sample distances are uniform. That is, structures of a particular size can be detected **on** the same scale independent of the directi**on** when the signal sampling is isotropic. We therefore introduce a preprocessing procedure for **3D** **local** structure filtering in which we perform interpolati**on** to make each voxel isotropic. Linear and spline-**on** methods are often used, but blurring is inherently involved in these approaches. Because, as noted above, the original volume data is inherently blurrier in the third directi**on**, further degradati**on** of the data in that directi**on** should be avoided. For this reas**on**, we opted to employ sinc interpolati**on** so as not to introduce any additi**on**al blurring. After Gaussian-shaped slopes are added at the beginning and end of each profile in the third directi**on** to avoid unwanted Gibbs ringing (see Appendix B), sinc interpolati**on** is performed by zero-filled expansi**on** in the frequency domain [22], [23]. The sinc interpolati**on** and **3D** **local** structure filtering were implemented **on** a Sun Enterprise server with multi- CPUs using multithreaded programming. With eight CPUs (168 MHz), interpolati**on** and single-scale **3D** filtering for a 256 256 128 volume were performed in about five minutes. Separable implementati**on** was used for efficient **3D** Gaussian derivative c**on**voluti**on**. For example, the computati**on** of the sec**on**d derivatives of Gaussian in the Hessian matrix was implemented using three separate c**on**voluti**on**s with **on**e-dimensi**on**al kernels as represented by f x i y j z k…x; @ 2 f†ˆ @x i @y j @z G…x; f† f…x† k ˆ di dx i G…x; d j f† dy j G…y; d k f† dz k G…z; f†f…x† ; …20† where i, j, and k are n**on**negative integers satisfying i ‡ j ‡ k ˆ 2 and 4 f was used as the radius of the kernel [10]. Using this decompositi**on**, the amount of computati**on** needed can be reduced from O…n 3 † to O…3n†, where n is the kernel diameter. Details of the implementati**on** of eigenvalue computati**on** for the Hessian matrix are described in [10].

166 **IEEE** TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. 6, NO. 2, APRIL-JUNE 2000 Fig. 4. Multichannel tissue **on****on**s. Discrete **on****on**. The resultant multidimensi**on**al opacity functi**on** …p† (right) is obtained by the min of the discrete **on****on**tinuous **on**e-dimensi**on**al (1D) opacity functi**on** 0 …S int † (lower left). (b) Example using the min (AND) 4 MULTICHANNEL TISSUE CLASSIFICATION BASED ON LOCAL STRUCTURES 4.1 Representati**on** of Classificati**on** Strategy A multidimensi**on**al feature space can be defined whose axes corresp**on**d to the measures of similarity to different **local** structures selected from S 0;1 , S 2 , and M 2 , where 0;1 2fint; edgeg and 2 2fsheet; line; blobg. Let p ˆ …p 1 ;p 2 ; ...;p m † (m is the number of features) be a feature vector in the multidimensi**on**al feature space, which c**on**sists of multichannel values of the measures of similarity to different **local** structures at each voxel. The opacity functi**on** …p† and color functi**on** c…p† ˆ…R…p†;G…p†;B…p†† for volume rendering are given as functi**on**s of the multidimensi**on**al variable p. We c**on**sider the following problem: Given tissue classes t ˆ…t 1 ;t 2 ; ...;t n † (n is the number of tissue classes), how can we determine the tissue opacity functi**on** j …p† and tissue color c j …p† for tissue t j ? We use a 1D weight functi**on** …p i †, whose range is ‰0; 1Š, of feature p i as a primitive functi**on** to specify the opacity functi**on** j …p† for tissue t j . We c**on**sider a generalized combinati**on** rule of the 1D weight functi**on**s to obtain the opacity functi**on** …p† formally given by …p† ˆmax min 1;i…p i †; min 2;i…p i †; ...; min ri…p i † ; 1im 1im 1im …21† where r is the number of terms min 1im ki …p i † and ki …p i † denotes the weight functi**on** for feature p i in the kth term. The max and min operati**on**s in (21) respectively corresp**on**d to uni**on** and intersecti**on** operati**on**s in fuzzy theory [24]. Suppose ki …p i † has a box-shaped functi**on** given by where ki …p i †ˆ…p i ; L ki ;H ki †; …x; L; H† ˆ 1; L x

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