GVF-Based Transfer Functions for Volume Rendering

Transcription

1 GVF-Based Transfer Functions for Volume Rendering Shaorong Wang 1,2 and Hua Li 1 1 National Research Center for Intelligent Computing Systems, Institute of Computing Technology, Chinese Academy of Sciences, China, Graduate School of the Chinese Academy of Sciences, Beijing, China, {shrwang, lihua}@ict.ac.cn Abstract. Transfer function is very important for volume rendering. One common approach is to map the gradient magnitude to opacity transfer functions. However, it catches too many small details. Gradient vector flow (GVF) vectors have large magnitudes in the immediate vicinity of the edges, where the GVF vectors keep coordinate with the vectors of the gradient of the edge map. While in homogeneous regions where the intensity is nearly constant, the magnitudes of gradient vectors are nearly zero and GVF diffuses the edge gradient. Because of these aspects, we extend GVF to color space and apply it for opacity transfer functions. Experiments show that our method enhances edge features and makes a visual effect of diffusing along the edges. 1 Introduction Several data sets become available from projects such as the Visible Human Project at the National Library of Medicine, the Whole Frog Project at Lawrence Berkeley, and the Chinese Digital Human Project at the Institute of Computing Technology of Chinese Academy of Sciences. It s a big challenge to get realistic volume visualization of these photographic volume data sets. The currently dominant techniques for volume visualization consist of surfacebased rendering and direct volume rendering [1]. Generally, the first step of surfacebased rendering technique is to reconstruct the surface by 2D contour reconstructing method [2] or iso-surface extraction method such as the Marching Cubes algorithm [3] or Morse theory [4]. And then surface is rendered after being reconstructed. Usually the reconstruction step needs a segmentation performance, which can be very difficult. On the contrary, direct volume rendering method maps voxel directly into screen space without using geometric primitives as an intermediate representation. Ray-Casting method [5], Splatting method [6] and Shear-Warp method [7] are the most common direct volume rendering methods. All these algorithms perform the following processes. First, assign color value and opacity value to each voxel. Then project the voxel into the image plane. Finally compose the projected samples. Transfer functions are used to map image properties to visualization characteristics, such as color, opacity, and texture. A review of transfer function techniques is given in [8]. Many previous studies on transfer function brought out approach by mapping gradient magnitude to opacity value. For color image processing is nonlinear, it is important to choose an appropriate color space [9]. In [10], Ebert and etc study color H.-P. Seidel, T. Nishita, and Q. Peng (Eds.): CGI 2006, LNCS 4035, pp , Springer-Verlag Berlin Heidelberg 2006

2 728 S. Wang and H. Li spaces for volume rendering, and give some transfer functions by mapping the voxel s color gradient magnitude to opacity value according to the color distance defined in RGB color space and CIE LUV color space. The results show that this method can generate high quality image which display multi-organs such as fattiness, muscles and bones. Our work of deriving opacity values is different from traditional transfer function approaches in the following ways: We map gradient vector flow magnitude to opacity transfer function instead of gradient magnitude. Compared with other transfer function, GVF transfer function has two main advantages: First, it captures the strong edges information and gets rid of the weak edges which may be noise and disturb the whole visual effect. Second, GVF transfer function also catches the details in homogeneous regions. Finally, the opacity of the voxel is small when the voxel is near away from the edges, which makes a special visual effect. We extend GVF model to vector data. Tradition GVF model only deal with scalar data and the edge map must be gradient magnitude if it is applied to vector data. We extend GVF model using an auxiliary image, which is usually a component standing for image luminance in the color space. 2 Extended GVF Gradient vector flow is firstly used to describe external force of snake model [11]. Snake model or active contour model [12] is dynamic force model, which draw the closed curve or surface to the object edge or surface. The driving forces consist of 1 internal forces and extern forces: E = E 0 int + Eextds, where E int are internal forces that hold the curve together and keep the curve from bending too much. And E ext are external forces that attract the curve toward the object boundaries. There are several kinds of external forces, in which the most common are: E = Ixy (, ),E = Ixy (, ),E = G ( xy, )* Ixy (, ). ext ext ext The basic idea of snake model is to design the external forces to move the active contour toward the edges by difference of intensity or intensity gradient. Gradient fields have some important properties listed below: First, gradient vectors are normal to the edge; Second the magnitudes get the large value in the immediate vicinity of the edges. Third, the intensity is nearly constant and the gradient magnitude is nearly zero in homogeneous regions. The last two properties result in that capture range of the gradient is small and the active contour can not converge to the concave boundary such as U-shaped object. GVF field is an extern force field that has much larger capture range than any other field based gradient has. It diffuses the gradient vectors of an edge map computed from the image. For a 2D image, GVF vector V(x,y)=(u(x,y),v(x,y)) is defined by minimizing the energy functional, E = μ V+ f V f dxdy (1) σ

3 GVF-Based Transfer Functions for Volume Rendering 729 Where edge map f(x, y) derived from image and has the similar style as the common extern forces of active contour model. To minimize the energy functional, V is set to be f when f is large for the second term dominates the functional. The functional is dominated by the first term when f is small and V can be slowly-varying vector to get lower energy. GVF vector is nearly equal to the gradient of edge map when it is near the edge and then diffuses to the homogeneous regions away from the edges. As shown in the definition of GVF, edge map f(x, y) is a scalar field. In order to calculate GVF vector of color data, the GVF vector can be solved by using the calculus of variations: uxt = uxt uxt f f + f vxt vxt vxt f f f (,) μ (,) ((,) x)( x y ) (,) = μ (,) ((,) y)( x + y ) (2) In Equation(2), only the partial derivatives of the edge map f are used. So the GVF model can be extended to vector data field if the partial derivatives of the edge map are given. Considering that most common color spaces have three components, we suppose the edge map is f = ( f1, f2, f3). We define the distance of two vectors in the color space as in [10]: f( X) f( Y) = ( f ( X) f ( Y)) + ( f ( X) f ( Y)) + ( f ( X) f ( Y)), so f( x+ h, y) f( x, y) fx = lim( ) =± ( f ) + ( f ) + ( f ) h 0 h x x x f( x, y+ h) f( x, y) fy = lim( ) =± ( f ) + ( f ) + ( f ) h 0 h y y y , (3) In equation (3), the sign of partial derivatives need to be confirmed. In this paper we introduce the partial derivatives of a reference scalar image fa as an auxiliary data, and assign the sign of fa to that of f x and f y. We take the gray component of RGB, I component of HSI and L component of CIELUV as the auxiliary image respectively. This approach is feasible because the auxiliary images stand for luminary in their respective color space. 3 GVF Transfer Function The design of the transfer function is to classify each voxel into different types. For scalar data, the most common method to classify the voxel is to segment data according to the data intensity. But for vector data, especially color image data of human organize, there are no automatic and reliable approach to segment or classify the all the materials. One of the most useful classification methods is by human interaction, marking and editing the materials little by little according to the professional knowledge of the corresponding field. But in fact it is a time-consuming and tedious work. Design of transfer function which maps the image color or intensity to opacity value, is a substitute way and has been proofed to be efficient.

4 730 S. Wang and H. Li The choice of the color space is very important for transfer function. In [10], the color vector distance and gradient are defined to design transfer function in RGB and CIELUV color space. The general opacity in the rending process is as opacity = ( vo * s) exp, where s is a factor, exp is an exponent coefficient, and vo is gradient magnitude. The basic idea of our approach is to take the magnitude of the GVF vector as the desired transfer function. We first choose a proper color space. And then produce an edge map. Finally we map the GVF vector derived from the edge map to opacity transfer function. As discussed in section 3, GVF transfer function has the following properties. First, it captures boundary of the image where gets a small opacity value. The much stronger the boundary is (it means that the gradient magnitude is large), the bigger opacity value is. Second, GVF transfer function diffuses the edge gradient to the slow-varying region. From equation (1) we know that each GVF vector is dependent to the vectors in the region nearby, and GVF transfer function is a global function and robust to the noise. On the other hand, the gradient function is local function, and each gradient vector is calculated only by the data in its neighborhood. 4 Results and Discussion We introduce RGB, CIELUV and HSI color space in this paper. As opacity is some dependent to luminance and the human s ocular system is more sensitive to luminance than color s thickness, L component of LUV and I component of HIS are fit for designing the transfer function. We take some notations to denote the different kinds of transfer functions. X-G represents gradient magnitude of X component. For instant, L-G is gradient magnitude of L component. X-GVF stands for magnitude of GVF vector whose edge map is gradient magnitude of X component, and X-GVF2 denotes magnitude of GVF vector whose edge map is X component. X-GVF and X-GVF2 differ from their edge maps. In HSI color space, I [0, 255] and HS, [0,1]. I component is the principal component and H and S component can be neglected. The transfer function pairs such as (HSI-G, I-G), (HSI-GVF, I-GVF) have the similar visual effects. In the following experiments we firstly render some regular geometry volume objects. Then we apply our approach to human organ volume data. 4.1 Simple Object Our first experimental materials are some regular volume objects. They are circle, rectangle and textured rectangle annulations, as shown in Fig. 1. The gradient effect only exists in the region several voxels around the edge and gradient transfer function only captures the edge, as shown in Fig. 2a. On the contrary, GVF diffuses edge gradient to the homogeneous region away from the edges. The opacity is large in the immediate vicinity of edges, and the visual effect is much better, as shown in Fig. 2b.

5 GVF-Based Transfer Functions for Volume Rendering 731 a. circle b. rectangle c. textured a. I-G b. I-GVF2 c. I-GVF Fig. 1. Source images Fig. 2. Circle X-GVF transfer function is gradient magnitude is anisotropic. Gradient operator is anisotropic in numerical algorithm and calculating the gradient is an important step in the GVF. When the edge map is processed by gradient operator, the anisotropy is accumulated. The rendering results show the unhomogeneity of X-GVF transfer function, as in Fig. 2c, while X-GVF2 transfer is much better (Shown in Fig. 2b). GVF weakens the edges whose gradient magnitude is small. As discussed in section 3, E is mainly dominated by the smooth term when is small. We apply a transform f ( f ) coeff out = to the edge map to enhance the weak edge. When in coeff < 1, the weak edge is enhanced. Shown in Fig. 3, Fig. 3a is the map of I-GVF and Fig. 3b-d are the figures of I-GVF whose transform coefficients are 1, 1.5, 0.5 respectively. Shown in Fig. 3d, the weak edges in the bottom of the rectangle are enhanced and majority of texture details are maintained. f a. I-G b..0 c. 1.5 d. 0.5 Fig. 3. Enhance edges The final rendering results are shown in Fig. 4. circle LUV-G LUV-GVF LUV-GVF2 rectangle LUV-G LUV-GVF LUV-GVF2 textured LUV-G LUV-GVF LUV-GVF2 Fig. 4. Rendering results

6 732 S. Wang and H. Li 4.2 Volume Data We take takes 128 slices from the thorax section of the Male data set of VHP(slice 1300 to slice 1427) to form a volume data with size 587*341*128. a. source b. L component c. L-G d. L-GVF Fig. 5. VHP slice To explore the effectiveness of our transfer functions, we choose a slice (slice 1300, shown in Fig 5-a) to test our approach. Fig. 5b is L component of the slice and Fig. 5c is the gradient of L component. Comparing gradient vector map with GVF vector map (Fig. 5d), gradient snaps more weak edges, while GVF catches the strong edges and diffuses the edges gradient. The edges of GVF vector map look much thicker. Observing the close-up figures of L-G and L-GVF in Fig. 6, gradient vectors are much disordered while GVF vectors look more regular and uniform. a.. close-up of L-G b. close-up of L-GVF Fig. 6. Close-up We first give some volume rendering results without any special transfer function, as shown in Fig. 7a. And the rendering results with transfer function L-G are shown in Fig. 7b. The information in gradient transfer function includes the most of stronger edges and lots of weak edges, which makes the large edges inconspicuous and the little edges indistinct because of too many edges and little discrimination. Compared with gradient transfer function, GVF transfer function captures the large edges and diffuses the edges gradient, which makes a special visual effect. The GVF transfer functions, especially those whose edge maps are from color intensity, have good uniformity because of diffusion effect. As discussed in section 4.1, a transform can be taken to the input edge maps. In Fig. 7c-f, GVF transfer functions in the same row are the same types, but their edge maps have been transformed with different coefficients. The transform coefficient in the first column is less than that in the second column. Comparing the rendering results, we can find that X-GVF2 with larger transform coefficient has better visual

7 GVF-Based Transfer Functions for Volume Rendering a. original 733 b.l-g c. I-GVF2(0.5) d.i-gvf2(1.5) e. RGB-GVF(0.80) f. RGB-GVF(1.2) g. L-GVF2 h.rgb-gvf2 i. L-GVF2 j.i-gvf2 Fig. 7. Rendering Results of VHP slice effect, while smaller pre-transform coefficient produces better X-GVF transfer function. From the results, we also know X-GVF2 transfer function is better than XGVF. Fig 7g-j shows other rendering results of our VHP data.

8 734 S. Wang and H. Li 5 Conclusion In this paper, we design opacity transfer function based on GVF vector magnitude. The experiment shows that choosing the approximate color space, component, and pre-transform coefficient, GVF transfer function is better than that based on gradient magnitude. GVF transfer function captures and enhances the large edges and diffuses them, which make the edges look thick. Acknowledgment. This work was funded by National Key Basic Research Plan (grant No: 2004CB318006) and National Natural Sciences Plan (grant No: , ). All the research work is conducted on Dawn 4000A Server Platform of Institute of Computing Technology, Chinese Academy of Sciences. And Mrs. Jingcai Shi gives us a lot of useful help. References 1. Peter Shirley and Allan Tuchman. Volume visualization methods for scientific computing. 2. Bernhard Geiger. Three-dimensional modeling of human organs and its application to diagnosis and surgical planning. PhD thesis, INRIA, France, William E. Lorenson and Harvey Cline. Marching cubes: A high-resolution 3D surface construction algorithm. In Proceedings of SIGGRAPH 1987, pages , Milnor, J. W. Morse Theory. Princeton, NJ: Princeton University Press, M. Levoy. Display of surfaces from volume data. IEEE Computer Graphics and Applications, vol. 8, no. 5, pp: , May L. Westover, Footprint evaluation for volume rendering. Computer graphics, vol. 24, no. 4, T. Todd Elvins. A Survey of Algorithms for Volume Visualization. Computer Graphics, Volume 26, Number 3, August 1992, pp Pfister, H., Lorensen, B., Bajaj, C., Kindlmann, G., Schroeder, W., Avila, L. S., Martin, K., Machiraju, R., Lee, J.: The transfer function bake-off. IEEE Comput. Graphics Appl. 21,3 (2001), G.Sapiro and D.L.Ringach, Anisotropic Diffusion of Multivalued Images with Applications to Color Filtering, IEEE Trans. Image Processing, vol. 5, no. 11, pp , D.S. Ebert, C.J. Morris, P. Rheingans, and T.S. Yoo. Designing effective transfer functions for volume rendering from photographic volumes. IEEE Transactions on Visualization and Computer Graphics, 8(2): , June C. Xu and J.L. Prince, Snakes, Shapes, and Gradient Vector Flow, IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 7, no. 3, pp , Mar M. Kass, A. Witkin, and D. Terzopoulos, Snakes: Active contour models, Int l J. Computer Vision, vol. 1, pp , 1987.