Efficient filter computation with symmetric matrix kernels

Transcription

1 Efficient filter computation with symmetric matrix ernels Manfred opp nstitute of Computer raphics, Visualization and Animation roup Technical University of Vienna, arlsplatz 3/86-2, A-040 Wien Abstract This paper presents an algorithm for filter calculations using symmetric matrix ernels. This algorithm outperforms traditional methods for ernels larger than or equal to 5x5 on machines based on RSC designs, where the time needed to calculate an addition equals the time needed for a multiplication. The algorithm is based on a decomposition of the ernel matrix into several ernel matrices of decreasing size, which can be computed very fast because of spatial coherence. A comparison with traditional methods shows the efficiency of the presented approach. eywords: image processing, anti-aliasing, filtering, symmetric matrix ernels, spatial coherence ntroduction iltering using matrix ernels is a common technique in image processing [Russ92][abe89] and anti-aliasing [ole90][watt92]. The filtered pixel is the weighted sum of the original and its neighbouring pixels within a rectangle defined by a maximum distance in x- and y- direction which is determined by the ernel size. The weights form a (M x ) matrix called the ernel matrix. The dimensions M and of the ernel matrix are usually odd, which guarantees that the maximum distance of accounted neighbouring pixels from the original pixel in one dimension is the same in positive and in negative direction. Equation () describes the filter computation using a ernel matrix: M p p m M + + xy, = ij, x+ i m, y+ j n; = ; n = ; () 2 2 i= j= p xy,... filtered pixel at position (x,y) p xy,... original pixel at position (x,y)... (M x ) ernel matrix According to Equation (), a straight forward implementation of a general ernel filter would require (M ) multiplications and (M -) additions for each filtered pixel. Since a picture usually contains several hundreds of thousands to several millions of pixels, efficient filtering is important to get the results in resonable time.

2 n this paper we want to cope with ernel filters, which are symmetric with respect to the x- axis and the y-axis. iven a (M x ) ernel matrix, is symmetric with respect to the x- axis, iff ij, = M + i, j; i M; j (2a) and symmetric with respect to the y-axis, iff ij, = i, + j ; i M; j (2b) Most of the frequently used filter ernels are symmetric with respect to the x- and y-axis, including auss ernels [Russ92], Box ernels [Watt92], and Bartlett ernels [Crow8]. A simple improvement for symmetric ernels to the general ernel filter algorithm could mae use of the symmetry to reduce the number of multiplications: The four symmetric pixels are added and the common weight ij, = M + i, j= M + i, + j = i, + j is multiplied afterwards. This improved method needs only /4 (M ) multiplications if M and are even and a little bit more, if M or are odd, but still (M -) additions. 2 Basic algorithm 2. Concept of our proposed algorithm Our goal is to reduce the number of operations, consisting of additions and multiplications, needed in the filtering. This goal will be reached through the reduction of additions, but the number of multiplications will be larger than or equal to those in traditional methods. Therefore we are assuming, that the calculation time for a multiplication equals the calculation time for an addition, which is usually true on RSC based machines. The efficiency of our algorithm is based on spatial coherence [röl93], the use of information gained from the filtering of neighbouring pixels for the actual filtering. To achieve the goal we will recursively decompose the ernel matrix into the sum of a smaller ernel matrix and two less computational expensive ernel matrices of the same size. Equation (3) shows this decomposition of the ernel into three simpler to compute ernels. rom () it can be derived, that the sum of the filtering with the ernels i, i=,2,...,anz, is equal to the filtering with ernel Chapter 2.2 describes our decomposition scheme, chapter 2.3 explains the filter algorithm based on the decomposition and appendix B shows an example of the proposed algorithm with a 5x5 ernel filter. anz i i=.

3 , 2, L,, 2, 22, 2, 2, + = M L M = A + B+ = () 3 M M M, 2,, M, M, M, 2 L M, M, A, A2, L A A,, S 0 L 0 S 0 0 L 0 0 A2, A22, A2, A + + 2, , L, 2 0 M L M + M... M + M M L M M + + AM AM AM A, 2,, M, M 2, L M 2 2 0, AM, AM, 2 L AM, AM, S S 0 0 L Matrix decomposition The first step in the algorithm is a decomposition of the (M x ) ernel matrix in a scalar value S, a (M x ) matrix A and a ((M-2)x(-2)) matrix 2. S, A and 2 are computed according to the following equations with =: S =, (4) A, = AM 2 + 2, = A, = A M 2 + 2, = (5a) Ai, = Ai, = i, ; < i < M (5b) A, j AMj,, j j 2 2 (5c) Aij, = Ai, A, j; < i< M 2 + 2; < j< (5d) + ij, = i+, j+ Ai+, j+ ; i M 2 ; j 2 (6) urthermore A can be written as the product of a vertical vector v and a horizontal vector h: h = A, j; v = Ai, ; A= v h; (7) i M 2 + 2; j With this recursive calculation scheme, S, v, h and + are calculated from,, until one of the dimensions of the matrix + reaches one or two. The result of this recursive decomposition are q scalar values S, q vertical vectors v, and q horizontal vectors h and the remaining ernel matrix q+ MM P with q = M P min, M P Q. 2 2 M QP 2.3 ilter computations with the components irst we can sum up the pixels affected by the scalar values S i, i q. This leads to the first intermediate sum:

4 q i $ ii, $ M + i, i $ M + i, + i $ i, + i i= sum = S cp + p + p + p h (8) p$ ij, = px+ i my, + i n; i M; j The next step is to compute the ernel filtering with the ernel A, q and add these filterings to the intermediate sum. But instead of applying the whole matrix A on the window of neighbouring pixels, we apply the horizontal filter h on each line of this window, followed by the vertical filter v applied on the result of the horizontal filtering. rom () and (7) it can be proved, that this method produces the same results as applying A itself. ote that because of the symmetry of v and h, only one multiplication has to be performed in the filtering for each pair of opposite pixels. The ernel calculations with the horizontal filters can be reused in filtering of the neighbouring vertical pixels. Therefore only one new horizontal filter has to be applied for the calculation of one ernel A on a new pixel window except in the calculation of the first lines of the picture. inally the ernel matrix filtering with q+ is computed and added to the previous sum using the conventional method with symmetry enhancements described in chapter. ote that the size of the remaining ernel q+ is less than or equal to ((M-+2) x 2) or (2 x ( - M + 2), respectively. or square ernels the remaining ernel q+ is (x) or (2x2), respectively. 3 Optimizations One obvious optimization is to stop the recursive decomposition, if the last computed intermediate ernel equals the null matrix O. ote that the Box ernel and the Bartlett ernel need only one decomposition step, because these ernels can directly be written as the product of a vertical and a horizontal vector with integer values. Also obvious, scalar values S can be ignored, if they are 0. Our algorithm outperforms traditional ones, if the size of the ernel is big and is beaten, if the size is small. Therefore one optimization is to alter the decomposition scheme to stop decomposing, if the remaining ernel + can be computed faster with a traditional algorithm. Often integer arithmetic is used for the calculation of the filtering with integer values in the ernel and the result is divided by a scaling value. This is because the frame buffer only deals with integer color values and the filtering is made for direct display. Therefore floating point arithmetic would cause some additional conversion overhead. f floating point arithmetic is used because of the needed precision of the result, the decomposition of the ernel matrix can be modified to get rid of most of the scalar values S :

5 (, = 0) DO calculate decomposition according to (4),(5),(6),(7) ELSE DO S =0 (9) A, = AM 2 + 2, = A, = AM 2 + 2, =, (0) calculate rest of matrix A and + according to (5b)-(5d), (6) Ai, h = A, j; v = ; i M 2 + 2; A, j () With this decomposition scheme, S can only be - or 0, therefore most additions in (8) can be sipped, because S =0. Therefore (8) can be replaced with: $ ii, $ M + i, i $ M + i, + i $ i, + i S 0 ; i q sum = cp + p + p + p h (2) i 4 Results Table shows a comparison of operations needed for the computation of one filtered pixel using square matrix ernels with odd dimensions which are symmetric with respect to the x- and y-axis. We used odd square matrix ernels for our comparison, because these are the most frequently used ernels in image processing and anti-aliasing. Standard is the brute force implementation of general ernel matrix filtering. Symmetric refers to the standard method improved by reducing multiplications through adding up the symmetric pixels before the multiplication. Base is the algorithm described in chapter 2 and Optimized uses our basic algorithm enhanced with the optimization to stop decomposing, if the remaining ernel + can be computed faster with the Symmetric algorithm. The decomposition threshold for this algorithm with odd square matrix ernels is 3x3. To calculate the operations needed for the Base algorithm with a x ernel, we need: the operations needed for a (-2)x(-2) ernel 4 additions and multiplication for the addition of the four corner points multiplied with S - additions and (-)/2 multiplications for the filtering with the horizontal vector - additions and (-)/2 multiplications for the filtering with the vertical vector addition to add the result of the vector filtering to the rest. Because the first and the last weights in the horizontal and vertical vector are, only (-)/2 and not (+)/2 multiplications are needed for the vector filtering. Therefore to calculate the operations needed for a x ernel with odd we need the operations needed for a (-2)x(-2) ernel filter plus 2 +3 additions plus multiplications. The only difference in the calculation of operations for the Optimized algorithm is that it starts from a 3x3 ernel filtering computed with the Symmetric algorithm.

6 Standard Symmetric (Sym) Base Optimized Add Mul Op Add Mul Op Add Mul Op Add Mul Op 3x Sym Sym Sym 5x x x x x x Table - Comparison of algorithms with ernel symmetric with respect to the x- and y-axis Usually square ernel filters, which are symmetric with respect to the x- and y-axis are also symmetric with respect to the diagonals. Therefore the Symmetric algorithm can be extended to cope with the diagonal symmetry, which reduces the number of multiplications needed for a filter computation to be smaller than that of our algorithm. But the number of operations is still lower in our proposed algorithm, therefore it beats the Symmetric algorithm on RSC machines, where the time needed for an addition equals the time needed for a multiplication. Table 2 shows the actual numbers. Symmetric (Sym) Optimized general case Optimized Box ernels Optimized Bartlett ernels Add Mul Op Add Mul Op Add Mul Op Add Mul Op 3x3 8 3 Sym Sym Sym x x x x x x Table 2 - Comparison of algorithms with ernel symmetric with respect to the x- and y-axis and the diagonals and optimized algorithms for Box and Bartlett ernels Table 2 also contains the number of operations needed for a highly optimized filtering with Bartlett ernels and Box ernels, described in chapter 3. The Bartlett ernels itself are shown in appendix A. The Optimized Box ernel computation does not need a multiplication, because all values in the ernel matrix are. ote that in these optimizations the number of operations needed for the filtering of one pixel is only linear to the horizontal or vertical dimension of the ernel.

7 5 Conclusions This paper presented an algorithm for efficient computation of matrix ernels filtering with symmetry with respect to the x- and y-axis. Our proposed algorithm clearly beats the brute force method and also the enhanced method using the symmetry to reduce the number of multiplications for ernels larger than or equal to 5x5. n the case of square matrix ernels, which are also symmetric with respect to the diagonals, the proposed algorithm uses less operations, but more multiplications with large ernels than the standard algorithm improved for exploiting these symmetries. Therefore the proposed algorithm also performs better than the symmetry enhanced standard algorithm on RSC machines, where the calculation time for an addition equals the calculation time for a multiplication. Also optimizations of the basic algorithm using special properties of some specific ernels were exploited. Examples for these include the Box and the Bartlett ernels. Acnowledgements want to than Martin eda, Eduard röller and Werner Purgathofer for their helpful comments. ote This paper is also available online as a Technical Report from the World-Wide Web server of the nstitute of Computer raphics at the Technical University of Vienna at or via TP at ftp://ftp.cg.tuwien.ac.at/pub/tr/94/tr ps.gz References [Crow8]. C. Crow, A Comparison of Anti-aliasing Techniques, EEE Computer raphics and Applications (), 40-48, an 98. [ole90] ames D. oley, Andries van Dam, Steven. einer and ohn. ughes, Computer raphics - Principles and Practice, Second Edition, Addison-Wesley Publishing Company 990, pp [röl93] Eduard röller, Coherence in Computer raphics, PhD Thesis at the Technical University of Vienna, Verband der wissenschaftlichen esellschaften Österreichs (VWÖ) 993. [abe89] Peter aberäcer, Digitale Bildverarbeitung - rundlagen und Anwendungen, Third Edition, Carl anser Verlag 989, pp [Russ92] ohn C. Russ, The mage Processing andboo, CRC Press 992.

8 [Watt92] Alan Watt and Mar Watt, Advanced Animation and Rendering Techniques, Addison-Wesley and ACM Press 992, pp -54. A Bartlett ernels B 2 B = B = B = B = = B Example of the algorithm with a 5x5 ernel filter iven a 5x5 ernel matrix and the 5x5 matrix W containing the original and neighbouring pixels: = W= To chec the results of our algorithm, we compute the ernel filtering directly: p direct = =79.

9 According to (4), S = 2- =, A can be computed from (5a)-(5d) and 2 from (6): A = rom A h and vcan be computed via (7): = = = h= b g v= ow we enter the second level of the recursion and compute S, A,, h and v: S 2 = 5- = A = = b 5 24g= b 29g h= b 6 g v= = Since the dimension of 3 is one, the decomposition is halted. Let us loo at the filter computations with the components: irst we compute the sum of the pixels affected by the scalar values S and S 2 using (8): sum = (+3+0+2) + 4 ( )= 22 The next step to do is to compute the ernel filters h and v on W, 2 h and 2 v on the inner 3x3 window of W and add it to the sum v hbwg v = = v 0 = = v h 0 v 6 0 v = + + = = + + = sum 2 = sum = 79 inally we have to add the x ernel filter 3 on the inner point of W to the sum: 3 p = sum 2+ ( 0) = 79+ ( 29) 0= 79 This yields to the same result as with the direct ernel filter computation. 4