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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 161 modalities and anatomical structures, we dem**on**strate that multidimensi**on**al opacity functi**on**s **on** **3D** **local** **intensity** structures can be highly effective for characterizing tissues and that the quality of the resultant volumerendered images is significantly improved. The organizati**on** of the paper is as follows: In Secti**on** 2, we characterize the novelty of our approach through a review of related work. In Secti**on** 3, we formulate **3D** filters for the enhancement of specific **local** **intensity** structures with variable scales. In Secti**on** 4, we describe multichannel classifiers **on** **local** **intensity** structures. In Secti**on** 5, we present experimental results using both synthesized volumes and real CT (computed tomography) and MR (magnetic res**on**ance) data. In Secti**on** 6, we discuss the work and indicate the directi**on**s of future research. 2 RELATED WORK 2.1 Multidimensi**on**al Opacity Functi**on** Using Derivative Features The earliest menti**on** of a multidimensi**on**al opacity functi**on** was by Levoy [1], who assigned opacity as a twodimensi**on**al (2D) functi**on** of **intensity** and gradient magnitude in order to highlight sharp changes in volume data, that is, object surfaces. More recently, Kindlmann and Durkin analyzed a **3D** histogram whose axes corresp**on**d to the original **intensity**, gradient magnitude, and sec**on**d derivative al**on**g the gradient vector [14]. While their main emphasis is **on** the semiautomated generati**on** of an opacity functi**on** **on** inspecti**on** of the **3D** histogram, they dem**on**strate that surfaces with different c**on**trasts can be discriminated using opacity assignment by a 2D functi**on** of **intensity** and gradient magnitude. The differences between the work of Kindlmann and Durkin and ours can be summarized as follows: . Although Kindlmann and Durkin **on**ly use gradient magnitude as an additi**on**al variable of an opacity functi**on**, we use several types of feature measurements **on** first and sec**on**d derivatives, including gradient magnitude and sec**on**d-order structures such as sheet, line, and blob. . Even if we c**on**sider **on**ly the use of gradient magnitude, our approach focuses **on** its utility for avoiding mis**on****on****on**e of the main issues in tissue **on****on** obtained from vector-valued MR data was effectively used by Laidlaw et al. to resolve the problem [15]. Although we address the same problem, our approach is different in that we dem**on**strate the utility of multichannel informati**on** obtained from scalar data rather than vector-valued data. 2.3 Image Analysis for Extracting Local Intensity Structures Feature extracti**on** is **on**e of main research topics in the image processing field. In the 2D domain, various filtering techniques have been developed to enhance specific types of features and a unified approach for classifying **local** **intensity** structures using the Hessian matrix and gradient vector of the **intensity** functi**on** has been proposed [16]. Although there have been relatively few studies in the **3D** domain, successful results have been reported in the enhancement of line structures such as blood vessels [9], [10], planer structures such as narrow articular spaces [17], and blob structures such as nodules [18]. Am**on**g these reports, **on**e unified framework is **on** a 3 3 tensor representing **local** orientati**on** [11], [12], [17], which can be used to guide adaptive filtering for orientati**on** sensitive smoothing. Another framework is **on** a 3 3 Hessian matrix [9], [10], [13], which is similar to the tensor model, but **on** directi**on**al sec**on**d derivatives. The possibility of using the Hessian for line and sheet detecti**on** in volume data was suggested by Koller et al. [13]. Further, a line measure **on** the Hessian and its multiscale integrati**on** was formulated and intensively analyzed by Sato et al. [9], [10]. The approach described in this paper has two novel features as compared with these previous studies. . The multiscale line measure is extended and generalized to different types of **local** **intensity** structures, including sheet and blob. . A multidimensi**on**al feature space whose axes corresp**on**d to different types of **local** **intensity** structures, as well as the original **intensity** values, is used to characterize each tissue. 3 MULTICALE **3D** FILTERS FOR ENHANCEMENT OF LOCAL STRUCTURES We design **3D** filters resp**on**ding to specific **3D** **local** structures such as edge, sheet, line, and blob. The derived filters are **on** the gradient vector and the Hessian matrix of the **intensity** functi**on** combined with normalized Gaussian derivatives. The filter characteristics are analyzed using Gaussian **local** structure models. The formulati**on** of this secti**on** is the generalizati**on** of a line measure [10] to different sec**on**d-order structures. 3.1 Measures of Similarity to Local Structures Let f…x† be an **intensity** functi**on** of a volume, where x ˆ…x; y; z†. The sec**on**d-order approximati**on** of f…x† around x 0 is given by f II …x† ˆf…x 0 †‡…x x 0 † > rf 0 ‡ 1 2 …x x 0† > r 2 f 0 …x x 0 †; where rf 0 and r 2 f 0 denote the gradient vector and the Hessian matrix at x 0 , respectively. Thus, the sec**on**d-order structures of **local** **intensity** variati**on**s around each point of a volume can be described by the original **intensity**, the gradient vector, and the Hessian matrix. The gradient vector is defined as rf ˆ…f x ;f y ;f z †; where partial derivatives of volume f…x† are represented as f x ˆ @ @x f, f y ˆ @ @y f, and f z ˆ @ @z f. Gradient magnitude is given by …1† …2†

162 **IEEE** TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. 6, NO. 2, APRIL-JUNE 2000 TABLE 1 Basic C**on**diti**on**s for Each Local Structure and Representative Anatomical Structures Each structure is assumed to be brighter than the surrounding regi**on**. q jrfj ˆ fx 2 ‡ f2 y ‡ f2 z : The gradient magnitude has been widely used as a measure of similarity to a **3D** edge structure. The Hessian matrix is given by 2 f xx f xy 3 f xz r 2 f ˆ 4 f yx f yy f yz 5; …3† f zx f zy f zz where partial sec**on**d derivatives of f…x† are represented as f xx ˆ @2 @x f, f 2 yz ˆ @2 @y@z f, and so **on**. Let the eigenvalues of r 2 f be 1 , 2 , 3 ( 1 2 3 ), and their corresp**on**ding eigenvectors be e 1 , e 2 , e 3 , respectively. The eigenvector e 1 …x†, corresp**on**ding to the largest eigenvalue 1 …x†, represents the directi**on** al**on**g which the sec**on**d derivative is maximum, and 1 …x† gives the maximum sec**on**dderivative value. Similarly, 3 …x† and e 3 …x† give the minimum directi**on**al sec**on**d-derivative value and its directi**on**, and 2 …x† and e 2 …x† the minimum directi**on**al sec**on**d-derivative value orthog**on**al to e 3 …x† and its directi**on**, respectively. 2 …x† and e 2 …x† also give the maximum directi**on**al sec**on**d-derivative value orthog**on**al to e 1 …x† and its directi**on**. f, jrfj, 1 , 2 , and 3 are invariant under orth**on**ormal transformati**on**s. A multidimensi**on**al space can be defined whose axes corresp**on**d to these invariant measurements. The basic idea of the proposed method is to use the multidimensi**on**al space for tissue **on****on**trast, **on**ly f is usually used in the c**on**venti**on**al method. f and jrfj can be regarded as the **intensity** itself and the edge strength, respectively, which are intuitive feature measurements of **local** **intensity** structures. Thus, we define the S int filter, which takes an original scalar volume f into a scalar volume of **intensity**, as S int ffg ˆf; and the S edge filter, which takes an original scalar volume into a scalar volume of a measure of similarity to an edge, as S edge ffg ˆjrfj: 1 , 2 , and 3 are combined and associated with the intuitive measures of similarity to **local** structures. Three types of sec**on**d-order **local** structuresÐsheet, line, and blobÐcan be classified using these eigenvalues. The basic c**on**diti**on**s of these **local** structures and examples of anatomical structures that they represent are summarized in Table 1, which shows the c**on**diti**on**s for the case where structures are bright in c**on**trast with surrounding regi**on**s. C**on**diti**on**s can be similarly specified for the case where the c**on**trast is reversed. Based **on** these c**on**diti**on**s, measures of similarity to these **local** structures can be derived. With respect to the case of a line, we have already proposed a line filter that takes an original volume f into a volume of a line measure [10] given by S line ffg ˆ j 3j … 2 ; 3 †!… 1 ; 2 † 3 2 < 0 …6† 0; otherwise; where is a weight functi**on** written as … s ; t †ˆ …s t † ; t s < 0 …7† 0; otherwise; in which c**on**trols the sharpness of selectivity for the c**on**diti**on**s of each **local** structure (Fig. 1a), and ! is written as 8 …1 ‡ s >< j t j † t s 0 !… s ; t †ˆ …1 s j tj † j tj > s > 0 …8† >: 0; otherwise; …4† …5† Fig. 1. Weight functi**on**s in measures of similarity to **local** structures. (a) … s ; t †, representing the c**on**diti**on** t ' s , where t s . … s ; t †ˆ1 when t ˆ s . … s ; t †ˆ0 when s ˆ 0. (b) !… s ; t †, representing the c**on**diti**on** t s ' 0. !… s ; t †ˆ1 when s ˆ 0: !… s ; t †ˆ0 when t ˆ s 0 or s … jtj †0.

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