Herman, Gabor T.

Liu, H. K.

English

Herman Gábor T.

Liu H. K.

University of Szeged

A simple shading for computer displayed surfaces t B y G . T . H E R M A N a n d H . K . L i u To the memory of Professor László Kalmár Introduction In order to achieve a visually realistic representation of the surface of a three-dimensional object on a cathode-ray tube, one has to solve the problems of remov-ing the hidden parts of the surface and shading its visible parts. A commonly used approach which allows efficient solutions to both these problems is to approximate the surface so that it is composed of planar polygons [2]. Lack of continuity in the shading across polygonal boundaries causes undesirable artifacts which can be Temoved by continuous shading procedures such as the one suggested by Gouraud [2]. In this note we discuss an extremely simple and efficient alternative method for removing the artifact mentioned above. In the application area which is our main concern our method is not only two orders of magnitude faster than that of Gouraud's , but it also produces superior displays. In this application area the sur-face is approximated by a composition of squares, each one of which is parallel to one of three mutually perpendicular planes. In the next section we describe the nature of our application area. Three-dimensional reconstruction from projections The problem of reconstructing a three-dimensional object f rom a set of its two-dimensional projected images has arisen in fields ranging f rom electron micro-scopy to holographic interferometry. For example, in medicine we use an x-ray source to project the body onto the two-dimensional surface of a film. From a sub-set of all possible projections of a body, reconstruction algorithms [1] produce a three-dimensional array of numbers in which each number is an estimate of the average density of the body in one of a set of equal, non-overlapping cubes, called voxels (volume elements). In order to see the shape of ,a particular organ, a three-dimensional boundary detection algorithm has been devised [3]! This detects the organ's boundary surface and generates the description of the surface in terms of the square faces of the voxels. 60 G. T. Herman and H. K. Liu An example For a demonstration of our ideas we use a plastic cast of an isolated canine left ventricle with some beads inserted on the surface. Figure 1 shows a television image of this cast, blurred so as to make the resolution similar to that obtained in the reconstruction process. A variant of the Algebraic Reconstruction Technique [1] has produced f rom 34 x-ray projections of our cast a 64 x 64 x 28 density array, with the edge o f each voxel being 1.04X 1.04X 1.68 mm. (Based on such a reconstruction variations in the surface smaller than a voxel could not possibly be displayed. For a fair evaluation of the efficacy of our display procedure its output should be compared to an image of the original which has been blurred as in Figure 1 to remove features. Fig. I A television image of the plastic cast of a canine left ventricle Fig. 2 Computer-generated display of the cast which was determined from 28 reconstructed cross-sectional levels of the cast A simple shading for computer displayed surfaces 61 smaller than the resolution of the reconstruction procedure.) A surface detection algorithm [3] has been applied to the reconstructed array and it produced an estimat-ed surface of the cast consisting of a connected collection of faces of voxels. This collection is displayed in Figure 2 using a shading which is uniform within each face and whose value depends on the angle the face makes with an assumed direc-tion of light and on the distance of the face f rom an assumed light source (War-nock's shading rule as given on page 624 of [2]). As can be seen, approximating the curved surfaces with small square surfaces generates an undesirable visual effect, caused mainly by having only three directions in which surface elements lie. Similar, though less disturbing artifacts are observable whenever a curved surface is displayed as a collection of planar polygons. Gouraud's shading procedure Gouraud [2] suggested a way to get rid of the artifact apparent in Figure 2. This approach is to modify the computation of the shading on each surface so that continuity exists across surface boundaries. This continuity can be achieved by assigning as normal at a vertex the average of the normals to each surface associated with this particular vertex. Each surface has a different shading for each of its vertices and the shading at any particular point inside the surface has to be com-puted as a continuous function of the shading a t the vertices of the surface. Fig. 3 Computer-generated display of the cast by use of Gouraud's shading procedure Due to the large number of the surfaces in the objects we are interested in (typically 10,000), this approach is prohibitively time consuming. Also, the results are far f rom acceptable, because the surfaces are either parallel or perpendicular to each other. Figure 3 shows a display produced by Gouraud 's method applied to the surface of the cast shown in Figure 2. 62 G. T. Herman and H. K. Liu A simple smoothing procedure In this section, we describe an alternative approach to this problem. Instead of smoothing the surfaces in object-space, we smooth the intensities in image-space. Let G j , . . . , G9 denote the shading intensities in a screen dot and its eight neigh-bors before smoothing, as indicated by the diagram below. After smoothing, the new intensity of the center dot will be Gj+ W* 2 f^Gi+W2* 2 f t * Gi /-'new t—2 i = 6 - r 9 • 1 + tv* 2 f i + w 2 * Z f i i=2 ¡ = 6 If the dot labelled by 1 is on an edge and does not have a neighbor labelled by i, then ft is assumed to be zero, otherwise equals one. W is a user adjustable weighting factor. Fig. 4 Computer-generated display of the cast by the use of the proposed shading procedure This simple smoothing method gives surprisingly good results. Figure 4 shows the display shown in Figure 2 after our smoothing procedure with w=0 .8 . The time required for this smoothing procedure is under eight seconds (CDC 3500) while our implementation of Gouraud's method needed 985 seconds to produce the smoothing shown in Figure 3. (Note that the time required by our procedure A simple shading for computer displayed surfaces 63 is dependent only on the number of picture elements in the display, while the time required by Gouraud 's method depends on the number of polygons f rom which the surface is made up.) Also, in our opinion, the true image (Figure 1) is better approximated by the display produced by our method (Figure 4) then it is by the display produced by Gouraud 's method (Figure 3). Fig. 5 Computer-generated display of a canine heart which was determined from 30 reconstructed cross-sectional levels of the heart Fig. 6 Computer-generated display of a part of a canine left lung which was determined from 64 reconstructed cross-sectional levels of the intact thorax of a dead dog 64 G. T. Herman and H. K. Liu: A simple shading for computer displayed surfaces Further examples We demonstrate our display method on reconstructions of two further bio-logical objects. In Figure 5, we display the surface of an isolated canine heart whose 6 4 X 6 4 X 3 0 reconstruction has been produced from 50 x-ray projections. In Figure 6, we display the surface of a par t of the left lung of a dog. Thirty-five x-ray projections of an intact dead dog have been taken and the thorax and contents of the dog were reconstructed as a 6 4 X 6 4 X 6 4 array. The input to the surface detection algorithm was a 28 X 64 X 64 subarray containing most but not the whole of the left lung. The surface detection algorithm [3] was applied to detect the surface of the lung in the thorax and the result is displayed in Figure 6. The imprints in the lung of the heart and of the major airway above it are clearly visible. Conclusion We have proposed, a display smoothing algorithm which is very efficient if the surface to be displayed consists of many polygonal surface elements. Efficiency is due to the smoothing being done on the image and not on the surface. In the display of organ surfaces obtained by three-dimensional reconstruction, our algo-rithm produces results visually superior to Gouraud 's smoothing procedure in a small fraction of the time required by that procedure. Acknowledgements. The authors wish to thank Dr. James F. Greenleaf for valuable discussions and Drs. Lowell D. Harris, Erik L. Ri tman and Richard A. Robb for providing test data. The authors are indebted to Ms. Jean F rank and her coworkers for secretarial and typing assistance. The research of the authors is supported by the United States Public Heal th Service, under grants H L 18968, H L 4664 and RR7. Abstract This note proposeses a simple way of removing the artifact caused by approximating curved surfaces with polygons in computer-generated three-dimensional display. The method is com-pared with Gouraud's continuous shading method. DEPT. UF COMPUTER SCIENCE STATE UNIVERSITY O F NEW YORK A T BUFFALLO 4226 RIDGE LEA R O A D AMHERST, N E W Y O R K 14226' References [1] GORDON, R., G. T. HERMAN and S. A. JOHNSON, Image reconstruction from projections, Sci-entific American, v. 233, 1975, pp. 56—68. [2] GOURAUD, H., Continuous shading of curved surfaces, IEEE Trans. Computers, v. C—20, 1971, pp. 623—629. [3] Liu, H. K., Two- and three-dimensional boundary detection, Computer Graphics and Image Processing, v. 6, 1977, pp. 123—134. (Received May 29, J978) 

A simple shading for computer displayed surfaces

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A simple shading for computer displayed surfaces

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