P. Heckbert. Fundamentals of Texture Mapping and Image Warping. M.S. Thesis, UCB/CSD 89/516, CS Division, U.C. Berkeley, June 1989.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....a few [Glas95] 7 Figure 2. 1: Complex Natural Surfaces The use of texture mapping, in its various forms, has the capacity to enrich the visual appearance of objects without requiring the considerable computations that would be necessary for a truly physically based simulation of the surface [Heck89]. Work on Texture mapping began with the early work of Blinn and Newel, altering surface color [Blin76] Texturing has also been applied to a wider range of surface attributes such as: surface normal [Blin78] transparency [Gard85] and surface displacement [Cook84] among others. Figure 2.2 ....

Paul S. Heckbert. Fundamentals of Texture Mapping and Image Warping. Technical Report No. UCB/CSD 89/516, California Institute of Technology, 1989.

....kernel and averaged. The values of the filter kernel can be precalculated and stored into a lookup table, therefore allowing any radially symmetric filter to be used. However, reconstruction of the texture samples is not considered. In his classic work on texture mapping and filtering, Heckbert [26] derives the resampling formula, which explicitly states the process of reconstructing, mapping, filtering and sampling a signal consisting of a discrete set of samples. Based on this, he revises the EWA filter to include reconstruction of texture samples. To express the resulting filter in a ....

....avoid aliasing, and finally sampling the bandlimited representation to produce a discrete output signal. In the context of splatting, the input space is the object space where the splats are defined, and the output space is the screen space. The resampling process has been described by Heckbert [26], and is as follows: discrete input samples f(u k ) u k U reconstructed input f c (u) f(u k ) # r(u) f(u k )r(u u k ) mapping g c (x) f c (M 1 (x) continuous output g # c (x) g c (x) # h(x) n g c (t)h(x discrete output samples g(x o ) g # c (x o ) x o X where ....

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Paul S. Heckbert. Fundamentals of texture mapping and image warping. Master's thesis, The University of California, June 1989.

....function is expressed as f c (u) w k r k (u u k ) 1) where u k are the local coordinates of the point P k . The value f c (u) gives the color of the point Q. To render a point based object, the texture function f c is mapped to the screen space with Heckbert s resampling framework [6]. It involves the following conceptual steps: first, the continuous texture function f c is reconstructed from sample points using Equation (1) second, f c u 0 Q Q s neighbourhood support of the basis function r k 3D object space 2D parametrization local parametrization Figure 2. ....

....in a closed form as a single elliptical Gaussian. An elliptical Gaussian in 2D with the variance matrix V is (x) 2# # V 2 x V 1 x . Matrix V 1 is so called conic matrix and x V 1 x = const are the isocontours of the Gaussian , that are ellipses iff V is positive definite [6]. The variance matrices for basis function r k and the prefilter h are denoted V k and respectively. Usually V = I (the identity matrix) With Gaussians, Equation (3) becomes # k (x) k I (x k ) 4) where J k is the Jacobian of the object to screen mapping m evaluated at u k . ....

P. Heckbert. Fundamentals of texture mapping and image warping. Master's thesis, University of California, 1989.

.... systems [12, 15] differential points [29] spectral processing of point sampled surfaces [40] multiresolution point based modeling [33] and the MLSsurfaces [2] EWA surface splatting of Zwicker et al. 67] combines the ideas of Levoy and Whitted [32] with Heckbert s resampling framework [27] to produce a high quality splatting technique that features anisotropic texture filtering, edge antialiasing and order independent transparency. Zwicker et al. also extended the EWA splatting to volume rendering [66] Hybrid polygon point rendering systems [12, 15] definitely leave the idea of ....

....[7, 6, 8] The particular case of alias in the context of splatting is explained by Zwicker in [24] EWA surface splatting EWA surface splatting [67] uses fuzzy elliptical splats and band limits them to avoid aliasing. It is a formal framework for splatting that is based on Heckbert s EWA filter [27] used for texture filtering in polygonal rendering. The definition of the texture function on the surface of a point based object is illustrated in Figure 14. The point based object is represented as a set of irregularly spaced points p , each associated with a basis function r k and ....

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Paul S. Heckbert. Fundamentals of texture mapping and image warping. Master's thesis, Dept. of Electrical Engineering and Computer Science, University of California, Berkeley, June 1989. 3, 14, 15, 34, 35

....as fc(u) wr(u u) 5) where uk are the local coordinates of the point Pk. The value fo(u) gives the color of the point Q. To render a point based object whose texture is defined by Equation (5) the texture function fo has to be mapped to the screen space. Heckbert s resampling framework [7] is used for this purpose. It involves the following conceptual steps: first, the continuous texture function fo in objectspace is reconstructed from sample points using Equation (5) second, fo is warped to screen space using the affine approximation of the object to screen mapping m, third, the ....

....kernel in a closed form as a single elliptical Gaussian. An elliptical Gaussian in 2D with 1 xTV x the viance matrix V is Gv(x) 2 where IVl is the determinant of V. Matrix V is so called conic matrix and xTv x c e the isocontos of the Gaussian Gv, that e ellipses iff V is positive definite [7]. The viance matrices for basis function r and the pre filter h e denoted V and V n respectively. Usually V n I (the identity matrix) With Gaussians, Equation (7) becomes p(x) laGvz i(x m(u) 8) Neglecting the occlusion we can express the depth blurred continuous screen space signal dof ....

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P.S. Heckbert. Fundamentals of texture mapping and image warping. Master's thesis, Dept. of Electrical Engineering and Computer Science, University of California, Berkeley, June 1989.

....space. First, the texture has to be reconstructed from the samples stored in the texture map using a reconstruction filter. After it has subsequently been warped to screen space according to the viewing projection, a prefilter has to be applied, before finally sampling onto the output pixel grid [4]. Naturally, in practice certain trade offs have to be #Hadwiger,Hauser# VRVis.at, http: www.VRVis.at vis theussl cg.tuwien.ac.at, http: www.cg.tuwien.ac.at home made for performance reasons. Where in theory the filtering process is the same for both magnification and minification, i.e. ....

P. Heckbert. Fundamentals of Texture Mapping and Image Warping. Master's thesis, University of California at Berkeley Department of Electrical Engineering and Computer Science, 1989.

....values will be a function of both the pose x p and the texture T . Predicting the measurement can then be reformulated as asking the question: given x p and T , what is the value I(p; x p ; of each pixel in the image of the patch This is exactly the problem addressed by texture mapping [7, 12]. The most basic form of texture mapping simply inverts the mapping m between (homogeneous) texture coordinates (s; t; u) and image coordinates (x; y; w) In this scheme, each pixel is inverse mapped to its preimage in texture space, and assigned the value of the nearest integer texture ....

....according to the mapping m, and then pre filtering it with an ideal low pass filter that cuts off undesired high frequencies. By combining these two filters the predicted value for each pixel p can be obtained by convolving the texture image with a resampling filter ae centered around m (p) [7] I(p; x p ; T ) T (k) ae(m (p) k) 3) In practice we use a Gaussian low pass filter for both the reconstruction filter and the prefilter. In the planar case, this has the convenient property that the combined filter is again a (warped and space variant) Gaussian filter in texture space ....

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P. S. Heckbert. Fundamentals of texture mapping and image warping. Master's thesis, University of California, Berkeley, 1989.

....quality texture filtering. In contrast to previous point rendering approaches, surface splatting uses a novel screen space formulation of the Elliptical Weighted Average (EWA) filter [3] the best anisotropic texture filtering algorithm for interactive systems. Extending the framework of Heckbert [6], we derive a screen space form of the EWA filter for irregularly spaced point samples without global texture parameterization. This makes surface splatting applicable to high resolution laser range scans, terrain with high texture detail, or point sampled geometric objects (see Figure 1) A ....

....properties onto the surfaces [5] If these texture functions are inappropriately bandlimited, texture aliasing may occur during projection to raster images. For a general discussion of this problem see [21] Although we develop our contributions along similar lines to the seminal work of Heckbert [6], our approach is fundamentally different from conventional texture mapping. We present the first systematic analysis for representing and rendering texture functions on irregularly point sampled surfaces. The concept of representing objects as a set of points and using these as rendering ....

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P. Heckbert. Fundamentals of Texture Mapping and Image Warping. Master's thesis, University of California at Berkeley, Department of Electrical Engineering and Computer Science, June 17 1989.

....[6] 7] 8] Most of these approaches are based on a splatting approach similar to splatting in volume rendering. In this paper, we present a framework for high quality splatting. Our derivation proceeds along similar lines as Heckbert s elliptical weighted average (EWA) texture filter [9], therefore M. Zwicker and M. Gross are with the Computer Graphics Lab, Swiss Federal Institute of Technology (ETH) Zurich, Switzerland. E mail: #zwicker, gross# inf.ethz.ch H. Pfister and J. van Baar are with MERL, Cambridge MA. E mail: #pfister, jeroen# merl.com c #2002 IEEE. Personal use of ....

....texture functions on point sampled objects and avoiding aliasing during rendering in [23] The surface splatting technique can replace the heuristics used in previous methods and provide superior texture quality. We develop EWA splatting along similar lines to the seminal work of Heckbert [9], who introduced EWA filtering to avoid aliasing of surface textures. We recently extended his framework to represent and render texture functions on irregularly point sampled surfaces [23] and to volume splatting [24] Section 6.4 will show the connection between EWA volume and surface ....

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P. Heckbert, "Fundamentals of texture mapping and image warping," M.S. thesis, University of California at Berkeley, Department of Electrical Engineering and Computer Science, June 17 1989.

....# # # # # # # # # # # # # # # # # # a b c d e f g h # # # # # # # # # # = # # # # # # # # # # # # # # # # # # # # Gaussian elimination is used to calculate a h. Calibrating a projective transform like this is explained by Heckbert [11]. A similar method has been used by Sukthankar [29] using a camera to detect the projected calibration points. The Escritoire could use a camera in a similar way, but the user would still be required to select a number of projected points to calibrate the digitizer so there would be no saving in ....

HECKBERT, P. S. Fundamentals of Texture Mapping and Image Warping. Master's thesis, University of California, Berkeley, 1989. See pages 17--21 for projective mappings between homogeneous co-ordinate spaces.

....back to front into the final image. We introduce a new footprint function for volume splatting algorithms integrating an elliptical Gaussian reconstruction kernel and a low pass filter. Our derivation proceeds along similar lines as Heckbert s elliptical weighted average (EWA) texture filter [4], therefore we call our algorithm EWA volume splatting. EWA volume rendering is attractive because it prevents aliasing artifacts in the output image while avoiding excessive blurring. Moreover, it works with arbitrary elliptical Gaussian reconstruction kernels and efficiently supports ....

....elliptical footprints very similar to Westover [19] As pointed out in Section 4, our technique can be used with irregular grids to efficiently and accurately project and rasterize the elliptical splat kernels. We develop EWA volume splatting along similar lines to the seminal work of Heckbert [4], who introduced EWA filtering to avoid aliasing of surface textures. We recently extended his framework to represent and render texture functions on irregularly pointsampled surfaces [21] Section 5 will show the connection between EWA volume and surface splatting. 3 Preliminaries 3.1 The ....

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P. Heckbert. Fundamentals of Texture Mapping and Image Warping. Master's thesis, University of California at Berkeley, Department of Electrical Engineering and Computer Science, June 17 1989.

....the 2D projective transformation, H,has eight degrees of freedom, h 1 ,h 2 , h 9 . Considering the inhomogeneous solution and parameterizing the entries of H by a vector q, image points of two images of a plane are related by: q 1 x q 2 y q 3 q 4 x y q 6 (3. 12) As suggested in [14], instead of representing the projective transformation by the eight coe#cients of the homography, the positional o#sets of the four corners of some image region is used to define the various motion models. This is illustrated in Figure 3.4, where four examples of motion sampling vectors #q i are ....

P. Heckbert. Fundamentals of texture mapping and image warping. In Masters Thesis, U.C. Berkeley, 1989.

....the concept of environment mapping, which uses a spherical image to model the environment surrounding an object. Such a model is useful to simulate highly specular reflections on mirror like objects, a crude first approach at a global illumination model. Recent texture mapping techniques [48] use mipmapping for efficient storage and processing of pre filtered images [100] They also employ more traditional image processing algorithms [35] and a set of algorithms called image warping algorithms [101] All those algorithms allow the transformation of images, so that they can be ....

Paul S. Heckbert. Fundamentals of texture mapping and image warping. Master's thesis, University of California at Berkeley, June 1989. 162

....hand drawn, or be a photograph. The pattern is represented as an image made up of pixels called texels. During the rendering process, these texels are conceptually mapped onto the polygon and then viewed through the pin hole camera and viewed on the screen. This can be expressed in matrices as [1] 2 4 x i w i y i w i w i 3 5 = P 3x4 # G 4x3 # T 3x3 # 2 4 x t y t 1 3 5 The 3 by 3 T matrix scales and translates and rotates the texture to put it on the two dimensional polygon; this is an affine map. The 4 by 3 matrix G takes the two dimensional polygon and puts it into 3d ....

P. Heckbert. Fundamentals of texture mapping and image warping. Master's thesis, The University of California at Berkeley, June 1989.

.... morphing, or to produce the effects of nonstandard projections for applications such as environment mapping[7] Image warping is also applied to correction of lens distortion[3] and aligning images taken from different viewpoints[4] Further applications of image warping are described by Heckbert[8]. Catmull and Smith[2] deform textures using 3D affine transformations. The transformations are decomposed into separate horizontal and vertical stages to simplify scan line implementation in hardware. They suggest caching intermediate values in a special framebuffer to avoid recalculating ....

....in pose can be catered for by warping the image. Seitz and Dyer[14] similarly use image morphing to produce shape transformations that look 3D using only 2D images. McMillan and Bishop[11] show the derivation of a suitable transformation to use warping to achieve image based rendering. Heckbert[8] provides a thorough study of the image warping that can be achieved for texture mapping. Methods of filtering to reduce aliasing are also covered. Resampling of images during correction of camera distortion is described by Chiang and Boult[4] 3 Implementation 3.1 Reference Implementation A ....

Paul Heckbert, Fundamentals of Texture Mapping and Image Warping, Master's Thesis, Department of Electrical Engineering and Computer Science, University of California, Berkeley, CA, June 1989.

....Both provide anisotropic filtering when the compression is along one axis of the texture, but perform as poorly as MIP mapping along the texture diagonal. Heckbert s elliptically weighted average (EWA) method offers a better solution by using an elliptical Gaussian filter in texture space [8]. The ellipse approximates the projection of a circular Gaussian pixel filter from screen space to texture space. The elliptical filter can be constructed by integrating raw texels, or by combining several MIP map samples within the filter footprint (a process called footprint assembly) ....

HECKBERT, P. S. Fundamentals of texture mapping and image warping. Master's thesis, Department of Electrical Engineering and Computer Science, University of California, Berkeley, June 1989.

....and image understanding have naturally led to the same direction. In computer graphics, the progress towards image based rendering was first motivated by the desire to increase the visual realism of the approximate geometric descriptions by mapping images onto their surfaces, i.e. texture mapping [Heck89]. Besides TIP, other image based rendering methods in computer graphics include viewpoint interpolation [CW93, MB95] view morphing [SD96] and interpolation from dense samples [LH96, Gort96] In viewpoint interpolation, new views were synthesized from two cylindrical panoramic views created by ....

P. Heckbert. Fundamentals of Texture Mapping and Image Warping. Master's thesis, UCB/CSD 89/516, CS Division, U.C. Berkeley, June 1989, 86 pages.

....on paper, and we used cylindrical panoramas exclusively rather than planar photographs. We have added the capability to use scanned drawings and photographs as textures applied to perspective rectangles. Texture mapping in this fashion is achieved with projective 2D mappings and image re sampling [4]. We use transparency channels with texture to alleviate the limitations of a rectilinear shape. For example, a painting of a tree can be scanned and a binary mask created using off the shelf image editing software. The mask defines the tree as opaque and the rest of the rectangle transparent. The ....

Paul S. Heckbert. Fundamentals of Texture Mapping and Image Warping. Master's thesis, UCB/CSD 89/516, CS Division, EECS Dept, UC Berkeley, May 1989.

....is only a quick run through the field. 2.1 Textures and Mappings Starting with the pioneering work of Catmull, Blinn and Newell [blin76] many techniques have been developed to map images unto object surfaces. An excellent survey of the first ten years or so has been written by Heckbert [heck89]. A step closer to IBR was to use environment mapping, that is to use real images mapped on a virtual environment to affect reflection or illumination in the scene [gree86] The basic concept of mapping images on geometric objects has been applied also very early in animation, notably with movie ....

P. S. Heckbert. "Fundamentals of texture mapping and image warping". M.Sc. thesis, Dept of EECS, UCB, June 1989.

....again. In fact, since the time transformation might introduce high frequencies, it is advisable to filter some high frequencies before resampling. This is a wellknown operation called sampling rate conversion in the area of signal processing, and resampling in the context of warping and morphing [10]. Since the human auditory perception is related to the oscillations of the air pressure, it is essential to the analysis 2 of the spectral effects of the transformation. From Fourier Analysis we have: f(at) # 1 a # f( # a ) where # f(#) F f(t) #) Therefore, it becomes clear ....

P. Heckbert. Fundamentals of texture mapping and image warping. Master thesis (technical report no. ucb/csd 89/516), University of California, Berkeley, 1989.

....on paper, and we used cylindrical panoramas exclusively rather than planar photographs. We have added the capability to use scanned drawings and photographs as textures applied to perspective rectangles. Texture mapping in this fashion is achieved with projective 2D mappings and image re sampling [4]. We use transparency channels with texture to alleviate the limitations of a rectilinear shape. For example, a painting of a tree can be scanned and a binary mask created using off the shelf image editing software. The mask defines the tree as opaque and the rest of the rectangle transparent. The ....

Paul S. Heckbert. Fundamentals of Texture Mapping and Image Warping. Master's thesis, UCB/CSD 89/516, CS Division, EECS Dept, UC Berkeley, May 1989.

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P. Heckbert. Fundamentals of Texture Mapping and Image Warping. M.S. Thesis, UCB/CSD 89/516, CS Division, U.C. Berkeley, June 1989.

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P. Heckbert. Fundamentals of Texture Mapping and Image Warping. Master's thesis, University of California at Berkeley, Department of Electrical Engineering and Computer Science, June 17 1989.

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P. S. Heckbert. Fundamentals of texture mapping and image warping. MS thesis, Department of Electrical Engineering and Computer Science, University of California, Berkeley, June 1989.

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P. S. Heckbert. Fundamentals of texture mapping and image warping. Master's Thesis, Department of Electrical Engineering and Computer Science, University of California, Berkeley, 1989.

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P. S. Heckbert, "Fundamentals of texture mapping and image warping," Master's thesis, Dept. of Electrical Engineering and Computer Science, University of California, Berkeley, June 1989.

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P. Heckbert, "Fundamentals of Texture Mapping and Image Warping," MS thesis, CS Division, Univ. of California, Berkeley, 1989.

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P.S. Heckbert, "Fundamentals of Texture Mapping and Image Warping," master's thesis, Univ. of California, Berkeley, 1989.

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P.S. Heckbert, Fundamentals of texture mapping and image warping. Master's thesis, Computer Science Division, UC Berkeley (1989).

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Paul Heckbert. Fundamentals of texture mapping and image warping. MA Thesis, Berkeley, 1989.

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P. Heckbert, "Fundamentals of texture mapping and image warping," M.S. thesis, CS Division, U.C. Berkeley, USA, 1989.

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P.S. Heckbert, Fundamentals of texture mapping and image warping. Master's thesis, Computer Science Division, UC Berkeley (1989).

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Paul Heckbert. Fundamentals of texture mapping and image warping. MA Thesis, Berkeley, 1989.

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P. S. Heckbert, Fundamentals of Texture Mapping and Image Warping, Master's thesis, University of California at Berkeley, June 1989.

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P. S. Heckbert, "Fundamentals of texture mapping and image warping," Master's thesis, Dept. of Electrical Engineering and Computer Science, University of California, Berkeley, June 1989.

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P. Heckbert. Fundamentals of Texture Mapping and Image Warping. Technical Report ucb/csd 89/516, University of California, Berkeley, 1989.

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P. Heckbert, "Fundamentals of texture mapping and image warping," M.S. thesis, University of California at Berkeley, Department of Electrical Engineering and Computer Science, June 17 1989.

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P. Heckbert. Fundamentals of texture mapping and image warping. M.sc. thesis, University of California, Berkeley, June 1989.

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P. Heckbert. Fundamentals of Texture Mapping and Image Warping. Master's thesis, University of California at Berkeley, Department of Electrical Engineering and Computer Science, June 17 1989.

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Paul S Heckbert, "Fundamentals of Texture mapping and Image Warping", Master's thesis June 1989, Computer Science division, University of California, Berkeley, CA.

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P. Heckbert. Fundamentals of Texture Mapping and Image Warping. Technical Report ucb/csd 89/516, University of California, Berkeley, 1989.

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Paul Heckbert, \Fundamentals of texture mapping and image warping," M.S. thesis, University of California at Berkeley, 1989.

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P. Heckbert. Fundamentals of texture mapping and image warping. Master's thesis, University of California at Berkeley, Department of Electrical Engineering and Computer Science, June 17 1989.

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Heckbert, P. S., Fundamentals of Texture Mapping and Image Warping, Masters Thesis, Department of Electrical Engineering and Computer Science, The University of California at Berkeley, Technical Report Number UCB/CSD 89/516, June 1989.

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P. S. Heckbert. "Fundamentals of Texture Mapping and Image Warping". M.Sc. Thesis, Department of Electrical Engineering and Computer Science, University of California, Berkeley, June 1989.

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P. S. Heckbert. Fundamentals of Texture Mapping and Image Warping. Master's thesis, University of California at Berkley, 1989.

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Paul S. Heckbert, "Fundamentals of Texture Mapping and Image Warping," Master's Thesis, (Technical Report No. UCB/CSD 89/516), University of California, Berkeley, 1989. (www.cs.cmu.edu/ ph)

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P. Heckbert. Fundamentals of Texture Mapping and Image Warping. Master's thesis, University of California at Berkeley, Department of Electrical Engineering and Computer Science, June 17 1989. 1, 2, 5

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P.S. Heckbert, "Fundamentals of texture mapping and image warping, "Masters Thesis, Dept. of EECS, UCB, Technical Report No. UCB/CSD 89/516, June 1989.

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P.S. Heckbert, Fundamentals of texture mapping and image warping. Master's thesis, Computer Science Division, UC Berkeley (1989).

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