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....stores the corresponding volume block s mean and standard deviation. At run time, a user supplied error tolerance is provided and an octree is fit to the pyramid. Based on the local error measurements, different regions of the volume can be rendered in different resolutions. Danskin and Hanrahan [2] proposed an importance sampling method, where the sampling rate along a ray is changed according to the local and accumulated opacities. A special type of pyramid, called Range pyramid, is used to facilitate the adaptive sampling. Wilhelms and Van Gelder proposed a selective octree traversal ....

J. Danskin and P. Hanrahan. Fast algorithms for volume ray tracing. In Proceedings of 1992 Workshop on Volume Visualization, pages 91--99. ACM SIGGRAPH, 1992.

....the virtual angioscopy system point picking: interactive point picking, implemented by casting a ray that starts from the 2D mouse position, makes it possible to interactively drag rulers to measure morphological features. To detect walls during ray casting, we use an accumulated opacity algorithm [5] that traces the ray from front to back and stops when the accumulated opacity is larger than a user settable threshold. As the algorithm is used for boundary detection, during ray tracing the opacity of voxels which are smaller than a user defined threshold are set to zero. 3 Implementation and ....

DANSKIN, J., AND HANRAHAN, P. Fast algorithms for volume ray tracing. 1992.

....solving the single scattering illumination integral given above. This technique requires casting rays from a viewpoint through a volume for each pixel in the output image. Various sampling and illumination techniques are employed to approximate and extend equation (1) see [Upson88] Levoy89] or [Danskin92]) In our implementation we use two types of sampling along a ray: one in which the samples are evenly spaced and one where the samples occur only on the faces of the rectilinear data cells. In this latter cases the samples are unevenly spaced. This approach to volume rendering makes it easy to ....

Danskin, John and Pat Hanrahan, Fast Algorithms for Volume Ray Tracing. Proceedings of the 1992 Workshop on Volume Visualization (October 1992), ACM SIGGRAPH New York. pp. 91-98.

....so that computation can be omitted in transparent regions of the volume. These data structures are built during a preprocessing step from a classified volume: a volume to which an opacity transfer function has been applied. Such spatial data structures include octrees and pyramids [13] 12] 8] [3], k d trees [18] and distance transforms [23] Although this type of optimization is data dependent, researchers have reported that in typical classified volumes 70 95 of the voxels are transparent [12] 18] Algorithms that use spatial data structures can be divided into two categories ....

....of at least five faster than previous algorithms running on comparable hardware. Other than a slight loss due to the two pass resampling, our algorithm does not trade off quality for speed. This is in contrast to algorithms that subsample the data set and can therefore miss small features [10] [3]. Section 2 of this paper describes the shear warp factorization and its important mathematical properties. We also describe a new extension of the factorization for perspective projections. Section 3 describes three variants of our volume rendering algorithm. The first algorithm renders ....

Danskin, John and Pat Hanrahan. Fast algorithms for volume ray tracing. In

....images in seconds [34] Because of the high computational demands of volume rendering many acceleration techniques have been considered. Foremost amongst these are hierarchical data structures which encode the fact that large subregions of the volume are either empty or homogeneous [35] 36] [15]. This knowledge can be used to adjust the step size in the evaluation of the integral. These techniques, although not posed as wavelet transforms correspond to the use of a Haar transform of the original dataset. More recently Muraki [44] used a Battle Lemarie [3] wavelet decomposition for ....

DANSKIN,J.,AND HANRAHAN, P. Fast Algorithms for Volume Ray Tracing. In Proceedings of the ACM Volume Visualization Symposium, 91--98, 1992.

....images, but are very expensive because of the redundant data structure traversal. Many acceleration techniques have been proposed and they mostly rely on spatial data structures such as pyramid and octree, that subdivide 3D space into volume hierarchies to assist rays to locate voxels of interest [4, 8, 15, 24, 13, 16]. The use of such data structures adds overhead in computing the intersections between the rays and objects and this offsets the savings substantially. Yagel and Kaufman [26] developed an algorithm that determines the sequence of voxels to be intersected by a discrete ray using a pre computed ....

J. Danskin and Pat Hanrahan. Fast algorithms for volume ray tracing. In Proc.

....entire volume dataset through semi transparent images. This allows the viewers to peer inside the volume space for better comprehension and exploration of the dataset. Volume rendering algorithms can be roughly classified into two categories: the image space approach such as raycasting algorithms[Lev90, DH92, YK92], and the object space approach such as splatting and cell projection[UK88, Wes90, WG91] Hybrid algorithms that combine the advantages of both approaches have also been developed[LL94, SFH97] Volume graphics is a new 3D graphics paradigm proposed by Kaufman, Cohen and Yagel[KCY93] It employs ....

J. Danskin and P. Hanrahan. Fast algorithms for volume ray tracing. In Proc.

....National University of Singapore, Heng Mui Keng Terrace, Kent Ridge, Singapore 119597. 1.2 Related Work There are a large number of publications on the topic of volume rendering. They can be roughly classified into two categories: image space approaches such as raycasting based algorithms[2, 13, 25], and object space approaches[11, 21, 22] Using hardware assisted 3D texture mapping for volume rendering has also been reported by several researchers[1, 6, 19, 23] and it usually works in the object space. Although it seems natural to use 3D texture mapping to deform the volumes, not much has ....

J. Danskin and Pat Hanrahan. Fast algorithms for volume ray tracing. In Proc.

....and visualization, and a variety of software hardware solutions have been developed in rendering of polygonal and volumetric models. A good method for the visibility problem needs to cull hidden elements efficiently as well as compute the correct visibility. The early ray termination technique [10, 2] offers an effective solution for the visibility problem in volume visualization. Although it is very natural for image order algorithms, such as ray casting, this method is not easily combined with object order algorithms like splatting. During rendering, a single occlusion test can often resolve ....

J. Danskin and P. Hanrahan. Fast algorithms for volume ray tracing. In Proceedings of the

....by a transfer function that maps data values to color (emission) and opacity (absorbtion) These quantities are then used to synthesize virtual images. We focus our interest on scalar data fields sampled on a uniform rectilinear grid, like e.g. tomographic data. The popular ray casting approach [7] approximates the physical equation of light transfer by resampling the data field and integrating the interpolated discreet emission and absorbtion values along the rays of sight. The most significant contribution to the computational cost of volume rendering is caused by the huge number of ....

J. Danskin and P. Hanrahan. Fast Algorithms for Volume Ray Tracing. In Worksh. of Vol. Vis. ACM, 1992.

....a fixed iso surface, the extraction of which is not optimized in this approach. First steps towards iso surface generation on reduced tetrahedral grids were made by Cignoni et al. 6] Traditional ray casting has been accelerated by adapting the integration step size to the variance in the data [7] and by efficiently stepping through the volume [31] Other approaches again use hierarchical [17] or compressed [28, 11] representations of the volume. 3 THE FINITE ELEMENT APPROACH Finite element analysis is a numerical method for solving partial differential equations which is widely used in ....

....using the trapezoidal rule. For the two dimensional case this is explained in Fig. 4. One can clearly see, that sample points become denser in regions of smaller tetrahedra, which corresponds to regions of high variance in the data. This sampling technique was also used by Max et al. 21] Danskin [7] introduced the idea of importance sampling based on octrees. The variance of an octree cell is used to determine the next step size. In our case the step size information is extracted from the data structure itself Sample Point Figure 4: Sampling points for ray integration in a 2D mesh Since ....

J. Danskin and P. Hanrahan. Fast Algorithms for Volume Ray Tracing. In Transactions 1992 workshop on Volume Visualization, pages 91--98, 1992.

....volume ray casting 1 INTRODUCTION The integration acceleration technique followed in this work is based on the idea of importance sampling. In volume areas of high variance the integration step size should be shorter than in areas of low variance. This technique has been described by Danskin in [DH92] This way the relation of approximation error and number of sample points is optimized. To change the step size during integration Danskin uses a pyramidal data structure and the variance value at a certain position to determine the next step size. In contrast our approach is based on an ....

J. Danskin and P. Hanrahan. Fast algorithms for volume ray tracing. In Transactions 1992 workshop on Volume Visualization, pages 91-- 98, 1992.

....transfer have been developed. They differ considerably in the physical phenomena they account for and in the way the emerging numerical models are solved. The expensive basic ray casting approach [20] has been accelerated in various ways. Among other strategies this comprises adaptive sampling [8, 7], exploiting coherence [19] using special purpose architectures [17, 23] and taking advantage of hardware in graphics workstations [3] Planes Parallel to Image Plane 3D Texture Mapping Rendered Image Trilinear Interpolation Compositing (Blending) Figure 1: Volume rendering with ....

J. Danskin and P. Hanrahan. Fast Algorithms for Volume Ray Tracing. In Worksh. of Vol. Vis., pages 91--98. ACM, 1992.

....very well by volume rendering techniques. In this section the used volume visualization algorithms are described briefly. We used a standard ray casting approach for the visualization process with different transfer functions. For a detailed presentation of volume rendering techniques we refer to [5, 7, 11, 14, 17, 18, 29]. The use of transfer functions for generating iso surfaces without polygonal extraction are discussed in detail for example in Levoy [17] The idea behind volume ray casting is that rays are traced back from each pixel of the desired image into the volume. The integral form of the equation of ....

J. Danskin and P. Hanrahan. Fast Algorithms for Volume Ray Tracing. Transactions Workshop on Volume Visualization, pages 91--98, 1992.

....ray tracing idea [27] is to shoot a ray of sight through every pixel into the volume, reconstructing the function value at appropriately chosen sample points along the ray and blending the mapped color and opacity values. Acceleration of this expensive technique is achieved by adaptive sampling [11,9], by exploiting coherence [25] by parallelizing in image and object space, and by exploiting hardware in graphics workstations [3] or in special purpose architectures [43] 2.2 Hierarchical Approaches The first hierarchical approaches applied in volume visualization were based on octrees. The ....

J. Danskin and P. Hanrahan. Fast Algorithms for Volume Ray Tracing. In 1992 Symposium on Volume Visualization, pages 91--98. ACM SIGGRAPH, 1992.

....to the final pixel intensity. Image space methods, i.e. ray casting, are known to produce superior quality images for the cost of repeated re sampling of the data. Although several acceleration techniques have been proposed such as adapting the integration step size to the variance in the data [4, 14] or efficiently stepping through the volume [25] none of these methods allows for interactive image generation rates. Since voxel based projection methods [13, 23, 24] in general avoid the numerically complex re sampling of the original domain, the expected frame rates are much higher than those ....

J. Danskin and P. Hanrahan. Fast Algorithms for Volume Ray Tracing. In Transactions 1992 workshop on Volume Visualization, pages 91--98, 1992.

....pixel is displayed if the difference of the current function value and the iso value changes sign. In this case, the surface normal needed for the lighting computation coincides with the direction of the function gradient. Acceleration of this expensive technique is achieved by adaptive sampling [9, 7], by exploiting coherence [17] by parallelizing in image and object space, and by exploiting hardware in graphics workstations [5] or in special purpose architectures [20] Related to our work are approaches where volume rendering is performed on compressed data sets. Wavelet based techniques ....

J. Danskin and P. Hanrahan. Fast Algorithms for Volume Ray Tracing. Transactions Workshop on Volume Visualization, pages 91--98, 1992.

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J. Danskin. and P. Hanrahan, Fast algorithms for volume ray tracing, 1992.

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J. Danskin & P. Hanrahan "Fast algorithms for volume ray tracing", Workshop on Volume Visualization, pp. 91-98, 1992.

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J. Danskin and P. Hanrahan. Fast algorithms for volume ray tracing. Proceedings of the 1992.

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J. Danskin and P. Hanrahan. Fast algorithms for volume ray tracing. In Proc. 1992.

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J. Danskin and P. Hanrahan. Fast algorithms for volume ray tracing. In Proceedings of 1992 Workshop on Volume Visualization, pages 91--98. ACM SIGGRAPH, 1992.

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J. Danskin and Pat Hanrahan. Fast algorithms for volume ray tracing. In Proc.

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J. Danskin and P. Hanrahan. Fast algorithms for volume ray tracing. In

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J. Danskin and P. Hanrahan. Fast algorithms for volume ray tracing. In

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