Real-Time Volume Rendering for Virtual Colonoscopy

Transcription

1 Real-Time Volume Rendering for Virtual Colonoscopy Wei Li, Arie Kaufman, and Kevin Kreeger Department of Computer Science State University of New York at Stony Brook Stony Brook, NY , USA Abstract. We present a volume rendering system that is capable of generating high-quality images of large volumetric data (e.g., ) in real time (30 frames or more per second). The system is particularly suitable for applications that generate densely occluded scenes of large data sets, such as virtual colonoscopy. The central idea is to divide the volume into sets of axis-aligned slabs. The union of the slabs approximates the shape of a colon. We render sub-volumes enclosed by the slabs and blend the slab images. We use the slab structure to accelerate volume rendering in various aspects. First, empty voxels outside the slabs are skipped. Second, fast view-volume clipping and occlusion culling are applied based on the slabs. Third, slab images are reused for nearby viewpoints. In addition, the slabs can be created very efficiently and they can be used to approximate perspective rendering with parallel projection, so that our system can benefit from fast parallel projection hardware and algorithms. We use image-warping to reduce the artifacts due to the approximation. 1 Introduction Virtual colonoscopy [14, 9] is a non-invasive alternative to optical colonoscopy. The system takes a spiral CT scan of the patient s abdomen after the entire colon is cleansed. Several hundred high-resolution CT images are rapidly acquired during a single breathhold of about seconds, forming a volumetric abdomen data set. A model of the real colon is then segmented from the abdomen data set. To support interactive virtual navigation and detection inside the human colon, it is critical to render internal colon views in real-time. For years, volume rendering has been notorious as a hog of computational power and memory. The large sizes of colon volumes (approaching ) make the situation worse. One trend of acceleration techniques is the so-called indirect volume rendering, that transform the volume into an intermediate format, such as iso-surface meshes. Surfacebased virtual colonoscopy system providing near real-time exploration has been developed based on high-end graphics hardware [9]. However, it is shown that direct volume rendering technique provides more realistic colonic images, flexible visualization of interior structures for polyps and other abnormalities, and shorter preprocessing time [26, 24].

2 During the last decade, many acceleration methods for direct volume rendering have been proposed, such as early ray termination [13], empty space skipping [5], rendering with parallel super computer [18, 11], using dedicated hardware [21] and image-based rendering [17]. Virtual colonoscopy systems exploiting some of these methods have been reported [26, 24]. However, these systems demand expensive hardware, such as SGI power challenge. In this paper, we present a rendering system that exploits both the tortuous feature of colon and the inter-frame coherence of the navigation. Our method is able to render high quality images with an average frame rate of 50 Hz. We first create volume-aligned bounding box sets for all the voxels that may contribute to views during navigation. All the boxes are slab-shaped and are with the same thickness. Since generally less than 20% of voxels are inside a colon, our algorithm skips large amount of external voxels. Further more, because of the winding nature of colon, only a small percentage of the internal voxels is visible for a given viewpoint. In our system, any two adjacent boxes share a rectangular-shaped window, called portal, which is the bounding rectangle of the colon cross-section on box surfaces. Since rays can only pass through portals, we calculate the intersection of view-frustum with the portals to determine the subset of visible slabs. All the slabs and portals are volume-aligned, therefore, the clipping simply becomes a comparison. Rendered images of slabs are then alpha-blended to generate the final image. Warping and reusing the slab images greatly accelerates the rendering. The main contribution of this paper is that our slab structure is carefully designed so that it is can be constructed efficiently as well as integrates multiple functions, including empty space skipping, visibility culling, approximating perspective rendering with parallel projection and image-based rendering. The rest of the paper is organized as follows. First, we review related work and focus on direct volume rendering approaches applicable to interactive or real-time volume visualization. In Section 3 through 6, we present our slab-based volume rendering techniques that enable real-time navigation of a virtual colon. In Section 7, we give the experimental results and we conclude the paper in Section 8. 2 Related Work In previous work, our group developed ray-casting-based virtual colonoscopy systems [26, 24] on a high-end 16-processor SGI Power Challenge. Besides using multi-processors, the rendering are further accelerate by techniques such as polygon assisted ray casting (PARC) [1] and space leaping [5] respectively. Obviously, it is difficult to port such systems to low-cost systems. Parker et al. [20] present a brute-force interactive volume ray tracing system on shared-memory multiprocessor machine that has high intrinsic cost. In their system, a volume is divided into small units called bricks to improve data locality while our slabs are used for various other purposes. Shear warp [12] is credited to be the fastest volume software rendering method so far. However, our experiments show that it is not sufficient for interactive rendering to simply use the method for data as big as Another popular acceleration approach of volume rendering is to exploit texture mapping hardware [4, 25]. Trilinear interpolation of the 3D texture mapping hardware

3 are exploited to resample the density [4] and possibly the gradient volume [6]. Both Westermann et al. [25] and Meißner et al. [16] apply color matrix onto the extracted gradient textures to compute directional lighting. They also achieve iso-surface rendering and enable semi-transparent transfer function. Salama et al. [22] enhanced 2Dtexture-based volume rendering in both quality and speed with the OpenGL extension of multi-texture and multi-stage rasterization. Although the results are promising, 3D texture mapping hardware is still expensive hence has limited availability while using 2D texture mapping hardware alone does not satisfy either the speed or the quality requirements of virtual colonoscopy. Researchers have attempted image-based techniques to speedup direct volume rendering. In many image-based surface-rendering approaches, the pixel values in the reference images are reprojected to obtain novel views. Due to the facts that volume rendering usually need to handle partially transparent voxels, which adds more viewdependent features, reprojection is carried out in segment, i.e. partial ray, level [8], rather than reusing the values of the pixels, which are actually the composite of whole rays. Alternatively, segments can be grouped into layers and the per-layer reprojection is applied which can take the advantage of texture mapping hardware [3]. Mueller et al s IBR assisted volume rendering [17] works in a similar fashion by dividing the volumes into slabs and reuse slab images for nearby views. They also use a coarse geometry to reduce the gaps among the slab images. Similarly, our system first perform volume rendering in slab level, then reuse warped slab images as much as possible. However, our slab structure also approximates the shape of a colon, hence is efficient for empty space skipping. Furthermore, portals associated with the slabs enables fast occlusion culling. In addition, our system supports self-occlusion and multi-resolution meshes. Bartz et al [2] proposed to utilize the VIZARD II architecture for interactive volume rendering in virtual endoscopy applications. In a preliminary version of PC-based virtual colonoscopy system [23], we utilized a Mitsubishi s VolumePro board [21]. The board can render a volume at 30 frames per second (FPS), but only with parallel projection. To provide perspective views that are necessary for navigating inside a colon, we adopted a multi-pass rendering method to approximate perspective projections with parallel projected slab images. The thickness of slabs has to be small enough to keep the approximation error less noticeable. In this paper, we use image-warping to reduce such approximation distortion. The previous system is able to deliver low-quality images (sample distance: 1 voxel) at about 5 frames per second while the system presented in this paper capable of generating higher quality images (sample distance: 0.5 voxel) in real-time. 3 The slab structure In this paper, a slab refers to a sub-volume bounded by an axis-aligned box. The size of the box in one dimension is significantly smaller than those in the other two dimensions. We call this smaller size 1 the thickness of the slab. All the slabs have the same thickness. As mentioned in the previous section, the slab structure is the basis of our volume 1 In rare cases, this smaller size is not the smallest in the three.

4 rendering system. First of all, the slabs serve as the bounding boxes of the possibly visible voxels during the navigation. Under the assumption that a virtual camera stays inside a colon during navigation and any voxel behind colon wall does not contribute to the image, all the voxels outside colon can be skipped during rendering. In other words, only voxels within the slabs need processing. Because these bounding boxes are slab-shaped, for each colon volume, we create three axis-aligned box sets, one for each major axis. During rendering, a set of slabs is selected depending on the current viewing direction. Figure 5 (also in color plate) shows one set of slabs. Apparently, the union of these slabs is an approximation of the colon. Each adjacent slab pair share a portal, which is the axis-aligned bounding rectangle of the cross-section of a colon on the shared face of the two slabs. All portals of the same slab set are parallel, as shown in Figure 1. Since rays can only pass through portals to traverse from one slab to its neighbor, we intersect the view-volume with the portals to determine visible slabs for a given viewpoint. Y p A B C Portals Colon X Fig. 1. 2D image of a set of slabs and portals. Portals are drawn as thick lines. 4 Creating the slabs To create the slabs, we construct a binary volume that differentiates voxels inside colon from others. Only the internal voxels may contribute to rendering for virtual colonoscopy. In the binary volume, the value of an interior voxel is one and all the others are zero. The creation of the slabs is done independently for each axis. Since the thickness of the slabs is all the same, for an axis, we cut the volume with a group of uniformly spaced parallel planes perpendicular to the axis. The spacing of the planes is just the thickness of slabs. Every slab is then bounded by two of such parallel planes. If an interior voxel v inside a slab is known, then the definition of the slab is the smallest axis-aligned bounding box enclosing the set of all interior voxels reachable from v without moving through any non-interior voxel and without moving out of the two bounding planes or out of the volume.

5 According to the definition, the computation can be done independently for the subvolume within each pair of parallel planes. We first compress the 3D sub-volume into a 2D slice by merging voxels lined up orthogonally to the planes. The merging is essentially an logical OR operation. That is, if any of the voxel along the line perpendicular to the planes line is one, the resulting pixel value is one too; otherwise it is zero. Figure 2 sketches in 2D such a compression. By replacing each sub-volume with the a slice, we obtain a compressed binary volume, whose size is just 1/d of the original volume in the direction orthogonal to the cutting planes, where d is the thickness of the slabs. Next, we find all the connected regions with 2D region-grow on the compressed volume with the following pseudocode. set all voxels of the compressed volume to be unflagged; for each voxel{ if it is unflagged && it is an internal voxel{ start 2D region grow; { if a interior voxel flag it; } obtain the bounding rectangle of the region; extrude the bounding rectangle to a box; } } In the algorithm, each voxel is inspected exactly once for whether to apply a region grow; and each internal voxel is flagged exactly once. Obviously the time complexity is linear to the number of total voxels in the compressed volume. 5 Slab-based volume rendering To render a single frame, the system first chooses the set of slabs depending on which major axis is closest to the current viewing direction. Next, slabs containing voxels (possibly) visible to the current virtual camera are determined. Each visible slab is rendered and the slab images are composited to create the final image. The rendering of the slabs can be done with arbitrary methods, ray casting, splatting, shear-warp or hardware. When the camera is looking from one end of a long tube and looking towards the other end, the system needs to render more slabs; while the camera is looking at the colon wall, only a few slabs need to be handled. One good thing is that the larger number Cutting Planes Sub-Volume Slice Fig. 2. Compress a sub-volume between a pair of cutting planes to a slice. Black voxels have value 1 and white voxels have value 0.

6 of slabs occurred only when the navigation direction is nearly parallel to the centerline; it happens to be the time that more slabs can be reused. The visibility detection is carried out by accumulatively intersecting the view-volume with the portals of the visible slabs. The slabs are intersected in the order from nearest to the furthest. If the accumulated intersection diminishes, the computation stops and all the slabs further are ignored. The idea is similar to that in [9] where they project the portals onto the image plane. Since their portals reside on arbitrary planes, the projection has to be computed separately for each portal and the projected shape of the portals can be arbitrary quadrilateral. To facilitate computation, bounding rectangles of those quadrilaterals aligned with the image window are used. This round-up increases the possibility of failing to cull invisible slabs and incur unnecessary work for rendering. Our method takes the advantage of the axis-alignment of the portals. We project the image window onto the first portal plane and calculate the bounding rectangle of the projection. The bounding rectangles of the projection on the following portal planes are obtained incrementally. In this fashion, only one projection and one round-up are necessary, consequently, our method is faster and more accurate. Figure 6 (also in color plate) shows a set of detected visible slabs overlapped on a slice image. Visible slabs are outlined with blue boxes. The visible parts of the slabs that approximate the shape of the view-volume are marked in yellow. Note in the figure, the slabs look wider than the cross-section of the colon, which is because only one slice is shown, whereas the bound boxes consider neighboring slices as well. When the visible slabs are determined, their images are obtained either by direct rendering or by reprojection from images of the same slabs rendered previously from nearby viewpoints (see next section). Then the images are alpha-blended with texture mapping hardware by defining textured rectangles in 3D space, as shown in Figure 3. Slab Images Camera Fig. 3. Blending of slab images. It is possible that the slab images are rendered with parallel projection while the slab images are perspectively blended. This approach is used in [23] to approximate perspective projection with parallel projection hardware, such as VolumePro. In their methods, slabs of arbitrary orientation are allowed, whereas VolumePro can only handle axis-aligned sub-volumes. Therefore, to get an image of a slab, an axis-aligned

7 bounding volume of the slab, which is usually much larger than the slab, has to be processed. There is no such extra work with axis-aligned slabs, as in the system presented in the paper. Furthermore, they choose the thickness of a slab according to the distance between the slab and the viewpoint, so that the approximation error is less than a predetermined threshold. Whereas in this paper, we adopt slabs of uniform thickness to facilitate reusing slab images. When using approximated perspective projection in applications like virtual colonoscopy, the slab surfaces which should be invisible in real perspective rendering, display the most notable distortion, as shown in Figure 7. Figure 7(a) is an actual image from a colon navigation while Figure 7(b) illustrates such distortion by simplifying the colon with three pipes segments. Obviously, the approximated perspective rendering can fit into our framework of slab-based rendering. Furthermore, we can utilize the portal information to reduce the distortion error or to relax the thickness limit of slabs. Rather than mapping a slab image to a plane as shown in Figure 3, we propose to warp the image to a simple geometric model constructed from the portals. Figure 4(a) shows in 2D such a model and Figure 4(b) shows the effects seen from the image plane. The front portal is enlarged while the back one is shrunken if they are perspectively projected. It is intuitive that this kind of warping is helpful in significantly reducing the distortions exhibited in Figure 7 with almost no extra cost. Figure 9 shows (also in color plate) a comparison of an actual colon scene with and without warping. Back Portal Back Portal Colon Wall Camera Front Portal (a) Front Portal (b) Fig. 4. Warping a slab image to the portal model. (a) 2D diagram of the portal model. The thick lines represent the surfaces to which the slab images are mapped. (b) The warping effects seen on the image plane if perspective projection is used. 6 Caching, Warping and Reusing of Slab Images During the navigation, the visibility of the slabs have strong coherency across frames. That is, it is highly possible that a slab stays visible through multiple frames. Our system exploits such coherence by reusing existing images rendered for previous frames, so long as the change of the viewing angle is within an error tolerance. Slab images can be prerendered and stored in a database. Each slab will be rendered from a discrete viewing

8 angle set with parallel projection. Then during navigation, slab images are queried by a slab index and a view angle. Alternatively, we dynamically cache the slab images for the current frame or insert slab images into the database. This strategy keeps the database smaller and/or avoids the lengthy preprocessing step for generating all the images. In virtual colonoscopy, the colon wall is classified as non-transparent. Therefore, we can utilize the image warping techniques developed for image-based-surface rendering before reusing the images, so that fewer sampling angles for each slab is essential. The basic idea of image warping is to reproject an image rendered for one viewpoint to another. One way of doing the reprojection is to associate each pixel with a depth value hence a shaded 3D point can be restored from each pixel. Next the 3D point is projected to the new view. However our algorithm accommodate the usage of VolumePro, from which, the per pixel depth information is unavailable. Moreover, to exploit the power of texture mapping hardware, we also want to avoid the per-pixel warping before texture mapping as in relief texture mapping [19]. Instead, we texture map to the geometric models of the slabs. The portal model discussed in the previous section serves the purpose, in addition to reduce the parallel-to-perspective distortion. We can also use a more accurate model. For each slab, we create two depth images during the preprocessing steps. The depth is defined as the orthogonal distance from the first non-transparent voxel to one of the slab faces containing portals. Then we build a regular triangular mesh from the depth information. Figure 8c shows such a mesh. Only quadrilaterals are displayed for clarity. The quadrilaterals are divided into triangles when being used. The warping is carried out by texture mapping a slab image to the triangular mesh and the textured triangles are projected one by one to the image plane. During the reprojection, self-occlusion may occur. Since the triangular patches may be translucent and should be composited in back-to-front order, we process the triangles in McMillan s occlusion compatible order [15]. In our implementation, we project vertices of the mesh to the plane on which the slab image is rendered to find the texture coordinates. This is equivalent to projective texture mapping. Since perspective distortion shrinks objects in the distance, it also shrinks the errors in the distance [10]. In our system, we build a multiresolution meshes and choose the mesh of the appropriate resolution, which we call mipmesh. Our method is similar to Mueller et al. s IBR assisted volume rendering [17], but they did not consider the self-occlusion problem and did not use mipmesh. Benefiting from the regularity of the mesh, occlusion compatible order is easy to apply and it is trivial to convert individual triangles into triangle strips. A simple but very effective strategy is that we keep a list of texture objects created from the slab images. Therefore, we can keep the reusable slab images in the texture memory of the texture mapping hardware, instead of transferring them from host memory to texture memory every frame. This trick makes the system ten times faster than without it. This is another reason that we don t pre-warping the image before texture mapping. Otherwise, the textures will change and need to be updated every frame. Our system can be considered as having two level caches of slab images. The first level holds the texture object in texture memory and the second level stores slab images in the host memory.

9 7 Experimental Results We implement our volume rendering system on a personal computer equipped with Pentium III 700 and GeForce 256 graphics board. The computer is also installed with a VolumePro board. If slab images are reused, the frames mainly depends on the time spent on warping and texture transferring, hence the frame rate is not much different between with and without VolumePro. Actually, we have tested our system using pure software ray-casting to render the slab images. The software ray casting is accelerated by space leaping. The difference in frame rates between using VolumePro and using the software ray-caster is even smaller the frame-rate variance during the navigation. The major disadvantage for a system without VolumePro is that it takes much longer during the preprocessing if all possible slab images are generated or exhibit longer delay when switching slab sets. All the following results come from a system with VolumePro. With our 2D region grow algorithm, the creation of three slab sets of for a data set only takes about 1 minute. We configure the rendering with 0.5 voxel ray spacing in all X, Y and Z axes. Figure 10 displays a couple of scenes from our virtual colonoscopy system. Table 1 and Table 2 shows the performance of our system. Without reusing slab images, although only a small portion of voxels is picked by the slab structure and VolumePro is exploited, the system only delivers half frame per second. By caching slab images in host memory, the frame rate reaches 10 Hz. Caching slab images in texture memory of the graphics board gives another order of magnitude improvement. Table 2 shows the dependence of the frame rate on image warping. 2 2 grid refers to a regular mesh whose distance between adjacent grid point is two voxels and 1 1 grid has similar meaning. In our implementation, the triangular patches are sent to graphics hardware as multiple triangle strips. We can see the warping takes significant percentage of the rendering time with higher resolution meshes. It implies we may optimize the warping by applying mesh simplification. All the images showed in this section are warped with 2 2 meshes in real-time (with frame rates over 30 Hz). Table 1. Performance of slab-image cache Method Frame Rate (FPS) No cache 0.5 Host memory 10 Host memory & texture memory 100+ Figure 8 and Figure 9 shows the effects of image warping on the quality of image. Even the slab images are composited with the same view direction as they are rendered, there are cracks between slabs without warping because of the parallel-to-perspective distortion. The frame rate of the virtual colonoscopy system varies significantly depending on several parameters. All the frame rates shown here are conservative estimate in that they

10 Table 2. Performance with image warping Method Frame Rate (FPS) No warping grid grid are closer to the lower end than from the higher end. Taking advantages of multiple acceleration techniques based on the slab structure, our system achieves real-time volume rendering for virtual colonoscopy. 8 Discussion and Future Work In this paper, we present a slab-based volume rendering system that achieves real-time rendering for virtual colonoscopy on low cost personal computers. The core of the system is the axis-aligned slab structure. The slab structure can be created efficiently and it integrates empty space skipping, visibility culling, approximated perspective projection and image-based rendering. However, images composited from axis-aligned slabs exhibit noticeable artifacts when the view direction is oblique and switching between different slab sets shows aliasing. We are going to study whether more slabs set can solve the problem with tolerable storage requirement. Our current system create a polygonal mesh for each slab independently. We will try to conserve connection information between slabs. Moreover, we will apply mesh simplification to further accelerate or to allow meshes of higher resolution. With the mesh simplification, we expect to face difficulties in using occlusion compatible order and triangle strips. Currently, we only use one image per slab for compositing. Blending from multiple images of the same slab, as in Debevec s work [7], will also be within our future investigation. There are noticeable jerky motions when the system switches to different slab set and the slab images have not been generated. We will try similar idea as in [3] to amortize the computation of rendering a whole new group of slabs by predicting future viewpoint and direction. Acknowledgments This work has been supported by grants from NIH #CA82402, Office of Naval Research under grant N , E-Z-EM Inc and Viatronix Inc. The patients data sets were provided by the University Hospital of the State University of New York at Stony Brook. The authors wish to thank Min Wan, Baoquan Chen, Klaus Mueller, Manuel Oliveira and others for their discussion and suggestion. We would also like to thank the anonymous reviewers for their comments. References 1. Rick Avila, Lisa Sobierajski, and Arie Kaufman. Towards a Comprehensive volume Visualization System. IEEE Visualization, pages 13 20, 1992.

11 2. Dirk Bartz and Michael Meiner. Translucent and opaque direct volume rendering for virtual endoscopy applications. Proceedings International Workshop on Volume Graphics 2001, Martin L. Brady, Kenneth Jung, HT Nguyen, and Thinh Nguyen. Two-phase perspective ray casting for interactive volume navigation. IEEE Visualization 97, pages , November Brian Cabral, Nancy Cam, and Jim Foran. Accelerated volume rendering and tomographic reconstruction using texture mapping hardware. Symposium on Volume Visualization, pages 91 98, October Daniel Cohen and Zvi Sheffer. Proximity clouds, an acceleration technique for 3d grid traversal. The Visual Computer, 11(1):27 38, Frank Dachille, Kevin Kreeger, Baoquan Chen, Ingmar Bitter, and Arie Kaufman. Highquality volume rendering using texture mapping hardware SIGGRAPH / Eurographics Workshop on Graphics Hardware, pages 69 76, August Paul E. Debevec, Yizhou Yu, and George D. Borshukov. Efficient view-dependent imagebased rendering with projective texture-mapping. Eurographics Rendering Workshop 1998, pages , June Taosong He and Arie Kaufman. Fast stereo volume rendering. IEEE Visualization 96, pages 49 56, October Lichan Hong, Shigeru Muraki, Arie Kaufman, Dirk Bartz, and Taosong He. Virtual voyage: Interactive navigation in the human colon. Proceedings of SIGGRAPH 97, pages 27 34, August Kevin Kreeger, Ingmar Bitter, Frank Dachille, Baoquan Chen, and Arie Kaufman. Adaptive perspective ray casting Volume Visualization Symposium, pages 55 62, October ISBN Philippe Lacroute. Analysis of a parallel volume rendering system based on the shear-warp factorization. IEEE Transactions on Visualization and Computer Graphics, 2(3), September Philippe Lacroute and Marc Levoy. Fast volume rendering using a shear-warp factorization of the viewing transformation. Proceedings of SIGGRAPH, pages , July Marc Levoy. Efficient ray tracing of volume data. ACM Transactions on Graphics, 9(3): , July W. Lorensen, F. Jolesz, and R. Kikinis. The exploration of cross-sectional data with a virtual endoscope. In In R. Satava and K. Morgan (eds.), Interactive Technology and the New Medical Paradigm for Health Care, pages , Leonardo McMillan. An Image-Based Approach to Three-Dimensional Computer Graphics. PhD thesis, Univertity of North Carolina, Computer Science Department, Michael Meißner, Ulrich Hoffmann, and Wolfgang Straßer. Enabling classification and shading for 3D texture mapping based volume rendering using OpenGL and extensions. IEEE Visualization, pages , October Klaus Mueller, Naeem Shareef, Jian Huang, and Roger Crawfis. Ibr-assisted volume rendering. Late Breaking Hot topics session at Visualization 99 Visualization, Jason Neih and Marc Levoy. Volume rendering on scalable shared-memory mimd architectures Workshop on Volume Visualization, pages 17 24, Manuel M. Oliveira, Gary Bishop, and David McAllister. Relief texture mapping. Proceedings of SIGGRAPH 2000, pages , July Steven Parker, Michael Parker, Yarden Livnat, Peter-Pike Sloan, Charles Hansen, and Peter Shirley. Interactive ray tracing for volume visualization. IEEE Transactions on Visualization and Computer Graphics, 5(3): , July - September 1999.

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13 Fig. 5. One set of slabs. The union of the slabs approximates the shape of a colon Fig. 8. Warping a slab image by a polygonal mesh. (a) Unwarped slab image; (b): Warped slab image; (c): Polygonal mesh Fig. 6. Visible slabs from a viewpoint. The slabs are outlined with darker box and the visible parts of the slabs are shown as lighter boxes. Fig. 9. Images showing the effects of warping. (a) Blending of slab images without warping; (b) After warping, the artifacts are much less noticeable. Fig. 7. Images showing the distortion of the approximated perspective volume rendering. Figure (a) is an actual image from colon navigation. Highlighted areas marked with white rectangles exhibit notable distortion. Figure (b) is the image of a three-segment pipe. The internal ring is the face of the back segment uncovered by the front segment. Fig. 10. Internal colon views rendered by the system.