A.W. Paeth, "A fast algorithm for general raster rotation," In Proceedings of Graphics Interface, pp. 77-81, 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... of the method, which is two dimensional, grows rapidly with the order L=n 1 of the model (typically, OL( per pixel) Fortunately, for the simplest transformations (scaling and rotations) there are ways to make the problem separable through a clever factorization of the transformation matrix [58]. This technique was used in [105] to design a high quality spline based procedure allowing to rotate images using 1D convolutions only; it was extended in [86] to allow for affine transformations in 2D and 3D as well. For image reductions, it is preferable to use a least squares approximation to ....

A.W. Paeth, "A fast algorithm for general raster rotation," in Proc. Graphics Interface '86 - Vision Interface '86, 1986, 77-81.

....1991] for a massively parallel volume rendering algorithm on the Connection Machine. They use a shearing technique to transform and resample the volume such that the view rays are in effect parallel to one of the coordinate axes. The particular shearing technique used is based on pure shears [Paeth, 1986; Tanaka et al. 1986] as opposed to shear scale transforms [Hanrahan, 1990; Smith, 1987] The 4 decomposition of a single axis rotation into pure shear transformations is given by 0 B cos 0 sin sin cos 1 C A = 0 B 1 0 tan a 2 0 1 1 C A 0 B 1 0 sin 1 ....

Paeth, A. W. A Fast Algorithm for General Raster Rotation. Proceedings Graphics Interface, pages 77--81, May 1986.

....computes products for any associative (not necessarily commutative) operator , 5] Compositing ( is associative. Numerical integration is also associative. In the next section we further discuss the mapping and our generalizations from [15] 2. 1 Processor Assignment by Permutation Warp Paeth [8], Tanaka et al. 11] and Schroeder et al. 9] have used pure shear matrix decomposition of rotation to create efficient resampling algorithms. The technique is a refinement of multipass filtering where the transform is restricted to rotation. By not actually resampling data, and using the shears ....

....samples being calculated are unique, but processors may receive more than one message because of virtualization. The density of messages across the network is the same if the slice and dice virtualization is used and communication remains efficient. 4 Time Quality Trade offs Multipass shears, [8][9] 11] 12] and direct warping, Section 2 [15] are not equivalent. Because each resampling discards the prior data, a shear filtering approach has more resolution error and interpolation error than a comparable direct filter. After a shear, all that is stored is the new samples. We used two test ....

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Paeth, Alan W. A Fast Algorithm For General Raster Rotation. In Proceedings Graphics Interface (May 1986), 77-81.

....4 6 we present our results. 2. Background An image or a volume is usually in the form of a regular rectilinear grid or a mesh of sampled function values termed pixels (image) or voxels (volume) When 2D images are subjected to affine transformations (e.g. translation, scaling, rotation [5] 11][27]) or when they are subjected to non affine grid deformation (perspective, texture mapping [12] warping [2] 37] the function value in the form of pixel intensity has to be resampled on the target grid, commonly called the resampling grid. Similarly, reconstruction is also needed when a 3D ....

Paeth A., "A Fast Algorithm for General Raster Rotation", Proceedings of Graphics Interface `86, pp. 77-81, May 1986.

....t s s s s s s s v u v u v v u v v u v = In this case, one needs to compute 1 1 h , such that = s s v u h v = 2 1 1 1 2 1 s s s s s t s v u v v u v v v u h v v u v ) t v u . 1 Some transformations may involve more than two passes [16]. In this work, we are interested in two pass decompositions. Relief Textures Oliveira and Bishop 6 One difficulty associated with serial warps is to find closed form solutions for 1 1 h . Sometimes, they do not exist at all [1] In these cases and also when there are multiple such ....

Paeth, A. A Fast Algorithm for General Raster Rotations. Proceedings of Graphics Interface'86, May, 1986, pp. 77-81.

....scaling not only complicates the resampling, but also may cause a situation (called bottleneck) where a beam (a volume row) is first shrunk and then magnified so that the original beam can not be recovered. A three shear decomposition of a 2D image rotation was introduced independently by Paeth [8] and Tanaka et al. 13] expressed as: R 2D (ff) cos ff Gamma sin ff sin ff cos ff = 1 0 Gamma tan ff 2 1 1 sin ff 0 1 1 0 Gamma tan ff 2 1 (1) A 1D shear operation does not suffer from any bottleneck problems and its resampling is also much simpler. A straightforward extension ....

A. W. Paeth. A fast algorithm for general raster rotation. In Proceedings of Graphics Interface '86, pages 77--81, May 1986.

....expressions parallel warp and serial warp which refer to the original 2 D map and to the composition of 1 D transforms that accomplishes a similar result, respectively. Assuming the row pass takes place first, a two pass serial warp 6 is accomplished 6 Some approaches use more than two passes [Paeth90]. Figure 2 1. Two pass affine transformation: 45 degrees rotation by applying two shear operations along the rows and columns of the image. Horizontal shear Vertical shear Rotation 16 by a horizontal shear followed by a vertical shear operation applied to the image. The horizontal pass ....

Paeth, A. "A Fast Algorithm for General Raster Rotations". Graphics Gems, Andrew Glassner, Editor. Academic Press, 1990, pp. 179-195.

....3 ) are chosen to match the desired translation vector s. Multiplying out Eq. 6] we find s = S (0) 3 S (1) 1 2 6 4 0 ffi 2 0 3 7 5 S (0) 3 2 6 4 ffi 1 0 0 3 7 5 2 6 4 0 0 ffi 3 3 7 5 = 2 6 4 ffi 1 ff 1 ffi 2 ffi 2 ffi 3 ff 0 ffi 1 (ff 0 ff 1 fi 0 )ffi 2 3 7 5 : [7] Solving Eq. 7] yields ffi 1 = s 1 Gamma ff 1 s 2 , ffi 2 = s 2 , and ffi 3 = s 3 Gamma ff 0 s 1 Gamma fi 0 s 2 . ....

....to match the desired translation vector s. Multiplying out Eq. 6] we find s = S (0) 3 S (1) 1 2 6 4 0 ffi 2 0 3 7 5 S (0) 3 2 6 4 ffi 1 0 0 3 7 5 2 6 4 0 0 ffi 3 3 7 5 = 2 6 4 ffi 1 ff 1 ffi 2 ffi 2 ffi 3 ff 0 ffi 1 (ff 0 ff 1 fi 0 )ffi 2 3 7 5 : 7] Solving Eq. [7] yields ffi 1 = s 1 Gamma ff 1 s 2 , ffi 2 = s 2 , and ffi 3 = s 3 Gamma ff 0 s 1 Gamma fi 0 s 2 . ....

Paeth AW. A fast algorithm for general raster rotation, in "Proc. Graphics Interface '86, Canadian Information Processing Society, Vancouver, 1986," pp. 77--81.

....we present our results. 2B ACKGROUND An image or a volume is usually in the form of a regular rectilinear grid or a mesh of sampled function values termed pixels (image) or voxels (volume) When 2D images are subjected to affine transformations (e.g. translation, scaling, rotation [5] 12] [28]) or when they are subjected to nonaffine grid deformation (perspective, texture mapping [13] warping [2] 38] the function value in the form of pixel intensity has to be reconstructed on the target grid, commonly called the resampling grid. Similarly, resampling is also needed when a 3D ....

A. Paeth, "A Fast Algorithm for General Raster Rotation," Proc. Graphics Interface `86, pp. 77-81, May 1986.

....] J x y , B I B J T p p OS SS I J I B I B J 31 To calculate the discrete samples of requires reconstruction of at points , which do not in general lie at s samples. Quality versus cost trade offs have resulted in many approximations of reconstruction [BARR81] BENN84] CATM80] FRASE85] [PAET86] [SCHR91] SMIT87] TANA86] WEIM80] WEIN90] Image warping is done in a one pass transforms, direct warps, and multiple pass transforms, multipass warps. Direct warps and multipass warps may be performed by serial or parallel algorithms. Image warping has many applications in computer graphics ....

.... Affine: Forward MCCMF (EREWF) Separable: SMIT87] Rotation: Rotation: TANA86] Orthogonal: WEST90] direct multipass General poly fit: YOKO86] General: WOLB89] Texture Map: FEIB80] General: GOSH89] General: WEIN90] direct direct direct General: Backwards MCCMB (CREWB) WEIM80][PAET86][SCHR91] CATM80] DREB88] HANR90] linear scanline adjacent convolve nth order polyfit nth order convolve bezier patch 32 The first factor, the transform direction, is the data flow of the program. An algorithm is forward mapping if data are passed by to the output [CATM80] DREB88] HANR90] ....

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A.W. Paeth, "A Fast Algorithm For General Raster Rotation," Proceedings Graphics Interface 1986 Vision Interface 1986 26--30 May 1986 Canadian Information Processing Society Vancouver, BC.

....of sites (see Fig. 4) Note that such shears are data blind (the contents of each site is moved without ever being examined) and invertible (each shear can be undone by a shear in the opposite direction) How to use three consecutive shears to rotate a 2 D image, as illustrated below, is well known[9]. R R R R S 1 Gamma S 2 Gamma S 3 Gamma We have developed an algorithm for doing 3 D rotations as well using only three shears[19] This algorithm (Fig. 4) is well matched to memory organization and access constraints of common storage media (dram or disk) and is ideally suited to ....

Paeth, Alan, "A fast algorithm for general raster rotation, " Proceedings, Graphics Interface '86, Canadian Information Processing Society, Vancouver, 77--81. Reprinted with corrections and additions in Andrew Glassner (ed.), Graphics Gems, Academic Press (1990), 179--195.

....function at each point, the program looks up the nearest precalculated value. A plot of the quantized cubic interpolation kernel is given in Figure 6.1. The figure shows the entire kernel for the 4 pixel region of support. 6.3. 1 Multiple pass Rotations Rotation by the three pass method of shears [104, 105] was also studied to determine its utility in parallel and sequential tomographic reconstruction. In the continuous domain, the three pass method realizes rotations exactly by shifting each row independently, followed by a shift of each column, and completed by another shift of each row. Equation ....

....errors is distinctly different from more standard rotation methods. Rotation accuracy degrades with angle. With any angle, the square pixels are warped into parallelograms with large angles, the area of the parallelogram ceases to equal the area of the square pixel we are using to represent it [104]. Also consider that with larger angles, the first shift results in more columns. More columns must be shifted differently, introducing more errors. Errors are minimized in our implementations by rotating arbitrary angles as a combination of perfect Sigma90 0 rotations and rotations between ....

A. W. Paeth, "A fast algorithm for general raster rotation," in Graphics Interface, pp. 77--81, 1986.

....processes the rows of an image and the second pass processes the columns. Multi pass methods have been used for both texture mapping [Catmull Smith80] Smith87] and for image processing [Wolberg88] to perform geometric correction of images [Fraser SchowengerdtBriggs85 ] or affine image warps [Paeth86] These methods work particularly well for affine and projective mappings, where the 1 D resampling warps for each pass are affine and projective (rational linear) respectively. Bilinear mappings are possible as well, but they require rational quadratic resampling warps [Catmull Smith80] ....

Alan W. Paeth, "A Fast Algorithm for General Raster Rotation", Graphics Interface '86, May 1986, pp. 77-81.

.... ecompose such mappings into a composition of two shear and scale passes: one horizontal and the i other vertical [Cat80] In a simpler variation discovered by Paeth, a rotational mapping is decomposed nto three passes of shears: the first horizontal, the second vertical, and the third horizontal [Pae86] s Filtering for this three pass rotate is particularly simple because resampling the scanlines involves no caling. Perspective Projection A naive method for texture mapping in perspective is to linearly interpolate the texture coordinates u g [ and v along the sides of the polygon and across ....

Alan W. Paeth, "A Fast Algorithm for General Raster Rotation", Graphics Interface '86,

....= N 1 p p T p T p ( p T 1 p ( I J I x y , J x y , B I B J 2 To calculate requires a reconstruction of at point , because does not, in general, lie at one of s samples. Quality versus cost trade offs have resulted in many approximations of reconstruction [1] 3] 8] [14] [16] 17] 20] 21] A resampling step, or pass is a filtering operation. Image warping is performed by a single resampling, direct warps, and multiple resamplings, multipass warps. Direct warps and multipass warps may be implemented as serial or parallel algorithms. Image warping has many ....

....allows dramatic improvements in algorithms such as coordinate calculations by differencing, efficient partitioning, and job assignment. Fig. 2 Warping Classification, with new algorithms: Backwards MCCMB and Forwards MCCMF Restricting 2D transforms to rotation allowed researchers [14][16] 17] to optimize by decomposing into multiple passes of two or three matrices. A nonscaling sequence of shears developed by [14] 17] is, EQ 1) A shear is a transform that operates on only one coordinate. Shears may scale (stretch or shrink axes) or not scale (distances are preserved) ....

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A.W. Paeth, "A Fast Algorithm For General Raster Rotation," in Proceedings Graphics Interface 1986 Vision Interface 1986, Vancouver, B.C., 26--30 May 1986, pp. 77-81.

....PISTON automatically performs the data redistribution for the operator. Two instances of the Rotate operator are used to rotate the DEM and image data, one for each data set. Rotate takes an image as input, distributed with a horizontal scan line K tiling, and rotates it using Paeth s algorithm [4]. The operator outputs the rotated image in a scan line K tiling. Internally, the algorithm used to perform the rotation requires three redistributions of the data, which are accomplished by specifyP2 H 4 ing the K tilings for the distributions required and calling the PMF functions to perform ....

Alan Paeth, "A Fast Algorithm for General Raster Rotation" Proceedings of Graphics Interface '86, pp 77-81.

....computes products for any associative (not necessarily commutative) operator , 16] 25] Compositing ( is associative. Numerical integration is also associative. FIGURE 4 Transforms and Communications in Permutation Warping for a Single Voxel 2. 1 Processor Assignment by Permutation Warp Paeth [24], Tanaka et al. 31] Schr der et al. 27] and we [35] have used pure shear matrix decomposition of rotation to create efficient resampling algorithms. A pure shear is a non scaling transform of a single coordinate. The technique is a refinement of multipass filtering where the transform is ....

....= det T ( 1 = T 12 . EQ 1) The other four equations are found by setting (EQ 2) The solution using s as given is , and . A special case is rotation, where , EQ 3) and by insertion and reduction by the half angle formula, and . This derivation shows how to calculate the result given in [24] [31] The same approach is used for three dimensional equiareal transforms solving a system of ten equations with nine unknowns in (EQ 4) and (EQ 5) EQ 4) EQ 5) The system appears to be over constrained, but can in fact be solved. The symbolic solution from Mathematica (TM) 39] is, ....

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Paeth, A. W. A Fast Algorithm For General Raster Rotation. Proc. Graphics Interface. Canadian Information Processing Society, 1986, pp. 77-81.

....and avoid other sampling artifacts this transformation must be based on an accurate re sampling method. The topic of accurate and efficient 2D raster transformations was dealt with extensively in the past. The interested reader is referred to the reference list [Hersch 1985, Heckbert 1986, Paeth 1986, Mitchel and Netravali 1988, Foley et al. 1990] 3: Template Based Discrete Rays We first make a note concerning the desired characteristics of the line algorithm that governs the steps taken by the ray. Unless these steps are determined solely by the line s slope, regardless of its length, we ....

Paeth, A. W., "A Fast Algorithm for General Raster Rotation", Graphics Interface'86, 77-81, 1986.

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A.W. Paeth, "A fast algorithm for general raster rotation," In Proceedings of Graphics Interface, pp. 77-81, 1986.

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Alan W. Paeth, "A Fast Algorithm for General Raster Rotation", Graphics Interface '86,

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