Efficient Eclidean Distance Transform Using Perpendiclar Bisector Segmentation Jn Wang and Ying Tan Key Laboratory of Machine Perception (Ministry of Edcation), Peking Uniersity Department of Machine Intelligence, School of EECS, Peking Uniersity, Beijing, P.R. China bedoins@pk.ed.cn ytan@pk.ed.cn Abstract In this paper, we propose an efficient algorithm for compting the Eclidean distance transform of two-dimensional binary image, called PBEDT (Perpendiclar Bisector Eclidean Distance Transform). PBEDT is a two-stage independent scan algorithm. In the first stage, PBEDT comptes the distance from each point to its closest featre point in the same colmn sing one time colmn-wise scan. In the second stage, PBEDT comptes the distance transform for each point by row with intermediate reslts of the preios stage. By sing the geometric properties of the perpendiclar bisector, PBEDT directly comptes the segmentation by featre points for each row and each segment corresponding to one featre point. Frthermore, by sing integer arithmetic to aoid time consming float operations, PBEDT still achiees exact reslts. All these methods redce the comptational complexity significantly. Conseqently, an efficient and exact linear time Eclidean distance transform algorithm is implemented. Detailed comparison with state-of-the-art linear time Eclidean distance transform algorithms shows that PBEDT is the fastest on most cases, and also the most stable one with respect to image contents.. Introdction Gien a binary image, whose elements hae only the ale backgrond (featre) pixels and featre (backgrond) pixels, its distance transform comptes the distance for each pixel between that pixel and the featre pixel closest to it []. Distance transform (DT) algorithms This work is spported by National Natral Science Fondation of China (NSFC), nder grant nmber 75 and 73, and partly spported by the National High Technology Research and Deelopment Program of China (3 Program), with grant nmber 7AAZ53. The sorce code of PBEDT is aailable on http://cil.pk.ed.cn/algorithm/distancetransform/ Ying Tan is the corresponding athor. are excellent tools for a ariety of applications, sch as image processing, compter ision, pattern recognition, morphological filtering and robotics, etc. [9][9][]. In practice, seeral distance metrics, sch as the city-block (L ), the chessboard (L ), the octagonal and the Eclidean metric, are all sed for different sitations. One of the most natral and appropriate one of these is the Eclidean metric, which is radially symmetric and irtally inariant to rotation [][9][], and sed in many applications. Howeer, the DT in exact Eclidean metric, called EDT, is time consming. Intitionally, treated as a global operation, EDT can be compted by sing an exhastiely brte-force searching algorithm: for each pixel of the image, calclate the distance between that pixel and each featre pixel. This reqires O(N ) time (N is the nmber of image pixels) [][]. Nmeros algorithms hae been proposed to restrict the searching for the closest featre pixel in order to realize a fast EDT comptation. In terms of searching mode, all these algorithms can be roghly classified into three categories, ordered propagation[][], raster scan [5][][] and independent scan [][][][3][5][][]. Howeer, so far there is no algorithm which can compte the EDT with good efficiency and good precision [9]. Some extensie sreys are introdced in [] and [9]. They indicate that an efficient EDT shold be based on obtaining featre pixels information from limited region, and shold aoid global searching. Althogh many of these algorithms are of liner time complexity, some are not stable when the image content is changed or hae a large constant term [][9]. In this paper, we propose an efficient algorithm for compting the Eclidean distance transform of two-dimensional binary image, called PBEDT (Perpendiclar Bisector Eclidean Distance Transform). PBEDT is a two-stage independent scan algorithm. In the first stage, PBEDT comptes the distance from each point to its closest featre point in the same colmn sing one time colmn-wise scan. In the second stage, PBEDT comptes the distance transform for 5
each point by row with intermediate reslts of the preios stage. By sing the geometric properties of the perpendiclar bisector, PBEDT directly comptes the segmentation by featre points for each row and each segment corresponding to one featre point. Frthermore, by sing integer arithmetic to aoid time consming float operations, PBEDT still achiees exact reslts. The remainder of this paper is organized as follows. In Section, the detail of PBEDT is introdced. The refined implementation of the proposed algorithm is presented in Section 3. Experiments of comparison with state-of-the-art algorithms on ariant featre objects and in-depth discssion are reported in Section. Finally, Section 5 gies the conclding remarks of this paper.. PBEDT by perpendiclar bisector segmentation.. Preliminary As sal, this algorithm takes a n by m binary image as the inpt and otpts a distance transform, sally in sqared distance. Let I denote the point set of the image, I r denote the points in row r, and F I denote the set of featre pixels. (Generally, let ppercase letter denote point set and lower case letter denote point.) Let f() denote the closest featre point of, and is the Eclidean metric: f() = arg F I min, I. () Ths, the distance transform of image I can be compted by: DT () = f(), I () The featre points in the colmn c are denoted by : C c = {.x = c, F I }, c < n (3) PBEDT is a two-stage independent scan algorithm. In the first stage, it comptes the closest featre points for each row. Then in the second stage, it comptes the distance of each point to its closest featre point by row [][]... First stage For row r, gien two featre points, C c, if.y r <.y r, then any point in row r is closer to than to. If C c,.y r.y r, is called r s closest featre point in colmn c. All closest featre points in colmns of row r are denoted by S r g, and S r g n. Therefore, f() is rewriten as: f() = arg min, I r. () Sg r.3. Second stage In this stage, we gie an effectie method to compte (). Let Sf r denote the closest featre points of row r, and Sf r Sr g. S r f = I r f() (5) Let points in Sg r and Sf r are increasingly ordered by x coordinate. Let Pt r denote the set of points in row r whose closest featre point is t, called t s region of inflence in row r. P r t = { f() = t, I r }, t S r g () If t S r f, then P r t ; otherwise, P r t =. Hence, S r f = Sr g { P r =, S r g}. r p(c,r) B Figre. The perpendiclar bisector of and intersects row r. We take adantage of the geometrical property of perpendiclar bisector to erify the points whose region of inflence is empty. Let B denote the perpendiclar bisector of and (Fig.). B is the p set of points which hae e- qal distance to and. wconseqently, the points located at the side of cb are closer to than to. Let c r de- note the x coordinate of the intersection of B with row r. Hence, point p(c r, r) (Fig.) is the separation point of P r and P r (P r, P r,.x <.x), and we assign p P, r p P r. Gien.x <.x, the points in I r at the left of c r are closer to than to. The sitations in Fig.(a) clearly indicate a point s region of inflence in row r is empty. If c r > c r w(,, w S r g), then P r = (Fig.(d)). Next, and w will be compared with other points in S r g. If c r < c r w(,, w S r g), then P r (Fig.(c)). Howeer, point will not be compared with all the points in S r g, since the region of inflence has some special properties: Property. The points in set P r t are continos in x- coordinate, or, {.x.x w.x,,w P r t, P r t }. (If exists, then s = f(). Ths, B st intersects row r between and, and between and w, too. This iolates the fact that two lines only intersects at most once.)
(a) one by one. If an added point reslts in point p of S whose region of inflence in row r is empty, then p shold be remoed from S. When S is empty, we psh into S c as the first intersection point. In the next loop, we se the newly added point w and the stack top of S point to compte c r w, and compare c r w to the stack top of S c. The at the bottom of stack labels the edge of the processing row. Next, we proe S = S r f. (c) w (d) w Theorem. S = S r f Proof: This proof has two steps:. P r =, S r g S ; In the algorithm COMPUTES, the points of S are moed to S one by one, and only the point whose region of inflence on r is empty is remoed from S. Therefore, P r =, S r g S. Figre. For sitations when a point is added in S. (a) P r = ; P r = ; (c) P r ; (d) P r = ; Moreoer, Property. If, Sf r,.x <.x, then s.x t.x, s P r and t P r. Therefore, if the points in S r g at the left of point are all erified, then none of them is closer to the right of c r than (Property., Property.). CompteS S and S c are stacks. Initially, S and S c are both empty. while S is not empty withdraw the lowest point w from S ; if S is empty psh(s, w); psh(s c, ); else is top of S and c r is top of S c ; calclate c r w with and w; pop(s ), pop(s c ), add w back to the front of S, when c r w < c r ; retrn S ; pop(s ), pop(s c ), psh(s, w), psh(s c, c r w), when c r w = c r ; psh(s, w), psh(s c, c r w), when c r w > c r and c r w < cols; Conseqently, we gie an algorithm COMPUTES to obtain S r f from Sr g. Let S and S c be stack strctres. Initially, let S = S r g, and S =. Points are moed from S to S. P r, S ; Following algorithm COMPUTES, each point (except the endpoints) and its adjacent points, w (,, x S and.x <.x < w.x) hae two separation position c r and c r w. c r < c r w, ths c r < P r.x c r w. The left endpoint and its adjacent point w hae a separation position c r w. c r w >, ths < P r.x c r w. (The right end point is in a similar way.) Therefore, P r, S. Therefore, S = Sf r. Ths, each point in I r is compared with two adjacent points in Sf r at most to get its closest featre point (Property., Property.). By processing each row with COMPUTES, we get the closest featre point for each point of I. Ths, we calclate the EDT by (). 3. Implementation of PBEDT In this section, we introdce the implementation of PBEDT. 3.. First stage algorithm Different from former independent scan algorithms [][], which compte the sqared distance in -D by twice scan, in the first stage, we compte the relatie sqared distance between the closest featre points in colmn and the left end point of each row by a forward scan with back propagation. IN F T Y sed in SQUARE-DISTANCE-D(I) is an integer bigger than the sqared diagonal distance of the image. 7
SQUARE-DISTANCE-D(I) for c = n- mid 3 for r = m- if (I(r, c) = ) 5 if (mid > ) mid (r + mid)/; 7 for r = mid + r I(r, c) (r r) + c c; 9 mid x; else if (mid = ) I(r, c) INF T Y ; 3 else I(r, c) (r mid) + c c; 3.. Second stage algorithm We implement the second stage algorithm introdced by COMPUTES in an effectie way. 3.. Calclate the intersection points B intersects row r at (c r, r), ths (c r, r) = (c r, r) (Fig.), and we obtain c r : c r = {(.x) (.x) r (.y.y) +(.y r) (.y r) }/{ (.x.x)} We se the relatie distance to replace the y coordinate, ȳ = y r, then c r can be simplified as:, then c r = ((.x) + (.y) ) ((.x) + (.y) ) ; (.x.x) Moreoer, let d = (.x) + (.ȳ) (7) c r = d d (.x.x). () This calclation can be organized as a fnction (Intersection), which only has diision, mltiplication, and sbstraction. Intersection(x, x, d, d) retrn (d d)/( (x x)); From (7), d is the sqared distance of to the left most point of row r which can be compted in the first stage (SQUARE-DISTANCE-D). 3.. Integer calclation We se integer diision to replace the float diision in (). The integer diision abandons the decimal and keep the integer, which is faster bt not always accrate. In or algorithm, we se the efficiency of integer arithmetic and keep the accracy meanwhile. Fig.3 shows two different sitations, bt c r = c r w = in eery sitation by integer diision. In both of sitations, we assign point p(,5) is close to the leftmost point of this triple, since the point coordinates of I are all integers. Een if B and B w all intersect at point p(,5), the assignment is alid. Therefore, an eqialent integer diision will be more efficient than float diision. (a) c w c w Figre 3. Different sitations of assigning same closer featre point: (a) c r > c r w; c r < c r w Negatie ales between two integers will be ronded to the lager integer (-.9 will be ronded to ), while positie ales in between two integers will be ronded to the smaller integer (5.9 will be ronded to 5). Therefore, we se - to label the left edge of the processing row while we se in COMPUTES. Moreoer, we prejdge the sign of c r before calclating it, which promotes the exection speed. In (), x < x is known, ths the sign of c r is relatie to (d d). Therefore, Intersection is rewritten as: Intersection-INT(x, x, d, d) if (d > d) retrn (d d)/( (x x)); else retrn -; 3..3 Eclidean distance comptation We also se d which comes from (7) in the distance comptation. Gien is a point on row r (.y = r) and is the closest featre point of, then = (.x.x) + (.y.y) = (.x) (.x)(.x) + (.x) + (.y) = (.x)(.x.x) + d
This can be organized as a fnction, which only has one addition, one sbstraction and two mltiplications. Distance(x, x, d) retrn x (x x ) + d; 3.. Second stage algorithm Based on the technical details aboe, we gie the second stage algorithm (SQUARE-DISTANCE-D). SQUARE-DISTANCE-D(I) for r = to m stack c ; stack cx ; 3 stack g ; p for c = to n 5 if (I(r, c) < INF T Y ) while (TRUE) 7 if (p ) cx Intersection-INT(stack c[p], c,stack g[p], abs(i(r, c))) 9 if (cx = stack cx[p]) p p ; else if (cx < stack cx[p]) p p ; 3 contine; else if (cx (n )) 5 break; else 7 cx ; p p + ; 9 stack c[p] c; stack cx[p] cx; stack g[p] I(r, c); break; 3 if (p < ) retrn FALSE; 5 c ; for k = to p 7 if (k = p) cx n ; 9 else 3 cx stack cx[k + ]; 3 while (c cx) 3 I(r, c) Distance(c, stack c[k], stack g[k]); 33 c c + ; 3.3. Comptational complexity We discss the comptational complexity of PBEDT: The time complexity of PBEDT is O(N) times.. In the first stage, SQUARE-DISTANCE-D takes O(N) time, since it scans forward once and propagates backward at most.5n. Ths, each element is accessed no more than twice, leading to an aerage nmber of.5 access times per element (comptational complexity of O(N)).. In the second stage, SQUARE-DISTANCE-D processes m rows one by one. There are two processes for each row, one comptes stack c x and the other comptes sqared distance. In process one, Sg r n, and each point in Sg r can only be added to stack c once. Each point in stack c can only be remoed once, and stack c n. Hence, these adding and remoing operations are exected at most n times. Ths, process one takes O(n) time. Process two comptes the distance between each point w in row r and w s closest featre point compte once, and takes O(n) time. Hence, in the second stage, SQUARE- DISTANCE-D takes O(N) (O(m n)) times. The space reqirement of PBEDT is ery low. In each stage, PBEDT recycles the memory of inpt image. In the second stage, PBEDT need a temporary memory whose size is 3n to store stack c, stack cx, stack g.. Experiments and discssion To ealate its performance, PBEDT was compared with state-of-the-art EDT algorithms, sch as Marer et al. s [], Saito and Toriwaki s [], Cisenaire and Macq s [], Lotfo and Zampirolli s [], Meijster s [7] and Felzenszwalb et al. s []. In order to improe readability, these algorithms are abbreiated as MAU- RER3, SAITO99, CUISENAIRE, LOTUFOZAM- PIROLLI, MEIJSTER, FELZENSZWALB, respectiely. Felzenszwalb et al. gie the implementation of their algorithm [], and other algorithms are implemented by Fabbri et al. []. The tests were performed on a compter with an Intel Core Do.53GHz processor, GB RAM, Ubnt Linx OS with kernel..3. All algorithms are implemented in ANSI C/C++, and bilt by GCC... The performance of PBEDT was measred with images oer a wide range of sizes and contents, as Fabbri et al. recommends in [9]. Comparing the otpts of PBEDT to those of other algorithms, no difference has been obsered from all these tests. Additional tests with thosands of randomly generated images arying in width and nmber of featre points also indicate the correctness of or algorithm. 9
(a) Figre. Randomly generated sample images. (a) Random points, featre pixel proportion: %, size: 5 5; Random sqares, featre pixel proportion: 5%, sqare angle: 5, size: 5 5 (a) Figre 5. Test image Lenna. (a) Original image; Edge image of Lenna The following images are chosen for precisely analysis:. Random points. The image size aries from, 5 5,..., to with randomly generated featre points where the nmber of featre points comprises %, %,..., 9%, and 99% of the image. One sample is shown in Fig. (a). This test proides an idea of the performance of the algorithms relatie to the nmber of featre pixels [9][].. Random sqares. These images are generated by randomly choosing the centers and sizes of black sqares rotated by θ [, 9]. The sqares are filled and plotted into the image ntil the black pixels add p to a percentage ale p. One sample is shown in Fig.. This test is based on a synthetic image haing more similarity to real images with some orientation [7][9]. 3. Special featre contents: A featre sqare located at the corner of an image. In this case, EDT prodces the largest and s- mallest possible distances for a gien image size: diagonal and, respectiely [9][]. Binary images of real objects. The edge image from the Lenna image obtained by thresholding the response of an edge detector is sed, as shown in Fig. 5. Lenna is chosen since it has been niersally sed as an impartial benchmark for image analysis algorithms [9]. A white disk inscribed in the image. It is a perfect test for exactness, since the Voronoi diagram of the pixels along a circle is ery reglar in the continos plane, howeer, the discrete Voronoi regions in this case are irreglar, specially near the center of the disk [3][9][]. Half-filled image. This is the worst case of the brte-force algorithm, and Marer3 also sffers a setback in this test [9]. As shown in Fig. and Fig. 7, PBEDT is more stable than other algorithms with different proportions of featre pixels, and it is faster than MAURER3, MEIJSTER and FELZENSZWALB at all parameter settings sed, faster than SAITO99 and LOTUFOZAMPIROLLI at small proportion featre pixels, slower than CUSENAIRE only at ery large proportion of featre points. Besides, PBEDT is more stable than others with different slant angle of random sqares. SAITO99 and LOTUFOZAMPIROLLI are faster than PBEDT when slant angle is or 9, bt s- lower when other angles are sed. Only when the proportion of featre pixels is higher than 7%, SAITO99, CUSENAIRE and LOTUFOZAMPIROLLI are faster than PBEDT at eery slant angel. Fig. (a) shows that PBEDT is the fastest with the test of edge image of Lenna at all sizes. Fig. shows that PBEDT is the fastest in all featre contents. In all these tests, PBEDT shows excellent performance. It is the fastest in most cases, and the most stable in terms of image contents. Neither the nmber of featre pixels nor the orientation of content objects can affect its performance. It is faster than MAURER3, MEI- JSTER and FELZENSZWALB in all cases, and faster than SAITO99, LOTUFOZAMPIROLLI and CUISENAIRE in most cases. SAITO99 and LOTUFOZAMPIROLLI are seriosly affected by the orientation of featre objects, which show poor performance in most of angles. Moreoer, their performance aries with different proportion of featre pixels. Table. shows the mean exection time on randomly generated sqares with arios featre point proportions and slant angels at size 3 3. Obiosly, of all algorithms, PBEDT is the fastest one. 5. Conclsion We hae introdced a two-stage independent scan algorithm PBEDT, which is simpler than preios works: se partial Voronoi diagram [][] or lower-enelop of parabolas [][][7][]. The time complexity of PBEDT is linear in the nmber of image points with a small constant 3
. proportion5%. proportion3%. angle. angle5................ proportion5% proportion7% angle3 angle5................ proportion95% angle.... MAURER3 SAITO99 CUISENAIRE LOTUFOZAMPIROLLI MEIJSTER FELZENSZWALB PBEDT.... MAURER3 SAITO99 CUISENAIRE LOTUFOZAMPIROLLI MEIJSTER FELZENSZWALB PBEDT Figre. Exection time for random sqares images with arying featre point proportion, size 3 3, slant angle aried from to 9. Algorithms Aerage Comparison with time(s) PBEDT (%) MAURER3.9.% SAITO99. 5.% CUISENAIRE.57 9.% LOTUFOZAMPIROLLI. 7.3% MEIJSTER.39 39.9% FELZENSZWALB.3 73.% PBEDT.79 % Table. Comparison of aerage exection time for randomly generated sqares images. term and the memory reqirement is ery low. Compared with other state-of-the-art algorithms, PBEDT is the fastest in most cases, and also the most stable one with respect to image contents. The main innoations of this algorith- Figre 7. Exection time for random sqares images with arying slant angle, size 3 3, featre points proportion ranged from 5% to 95%. m are: the geometric property of perpendiclar bisector is sed to compte the segmentation of rows directly; the integer arithmetic is sed to aoid time consming float operations and still keeps exactness. All these innoations redce the comptational complexity significantly. A parallel ersion of PBEDT can be easily implemented [][]. This algorithm can also be extended to three or higher dimensional binary images. Crrently, we are ealating the n-d ersion of this algorithm. Reslts will be pblished in a more elaborate paper which is crrently in preparation. References [] H. Bre, J. Gil, D. Kirkpatrick, and M. Werman. Linear time eclidean distance transform algorithms. IEEE Transactions on Pattern Analysis and Machine Intelligence, 7(5):59 533, 995. [] O. Cisenaire. Distance transformations: fast algorithms 3
.5.5.5 Test with edge image of Lenna MAURER3 SAITO99 CUISENAIRE LOTUFOZAMPIROLLI MEIJSTER FELZENSZWALB PBEDT 5 5 3 35 Image width.5..3.. (a) Test with special content images MAURER3 SAITO99 CUISENAIRE LOTUFOZAMPIROLLI MEIJSTER FELZENSZWALB PBEDT Top left Bottom right Lenna White disk Half filled Image content Figre. Test reslts. (a) Reslts with Lenna edge images of ranged size; Reslts with images of special featres, size 5 5. and applications to medical image processing. PhD thesis, Loain-la-Nee, Belgim, 999. [3] O. Cisenaire and B. Macq. Fast and exact signed eclidean distance transformation with linear complexity. In ICASSP 99: Proceedings of the Acostics, Speech, and Signal Processing, 999, IEEE International Conference, pages 393 39, Washington, DC, USA, 999. IEEE Compter Society. [] O. Cisenaire and B. M. Macq. Fast eclidean distance transformation by propagation sing mltiple neighborhoods. Compter Vision and Image Understanding, 7():3 7, 999. [5] P.-E. Danielsson. Eclidean distance mapping. Compter Vision, Graphics, and Image Processing, :7, 9. [] R. de Alencar Lotfo and F. A. Zampirolli. Fast mltidimensional parallel eclidean distance transform based on mathematical morphology. In SIBGRAPI, pages 5. IEEE Compter Society,. [7] H. Eggers. Two fast eclidean distance transformations in z based on sfficient propagation. Compter Vision and Image Understanding, 9():, 99. [] R. Fabbri, L. da Fontora Costa, J. C. Torelli, and O. M. Brno. Complete reslts of the benchmark between exact edt algorithms. http://distance.sorceforge. net,. [9] R. Fabbri, L. da Fontora Costa, J. C. Torelli, and O. M. Brno. d eclidean distance transform algorithms: A comparatie srey. ACM Compt. Sr., (),. [] P. F. Felzenszwalb and D. P. Httenlocher. Distance transforms of sampled fnctions. Technical report, Cornell Compting and Information Science, September. [] P. F. Felzenszwalb and D. P. Httenlocher. Distance transforms of sampled fnctions (program). http://people. cs.chicago.ed/ pff/dt/,. [] M. L. Gariloa and M. H. Alswaiyel. Two algorithms for compting the eclidean distance transform. Int. J. Image Graphics, pages 35 5,. [3] W. H. Hesselink and J. B. T. M. Roerdink. Eclidean skeletons of digital image and olme data in linear time by the integer medial axis transform. IEEE Transactions on Pattern Analysis and Machine Intelligence, 3(): 7,. [] T. Hirata. A nified linear-time algorithm for compting distance maps. Inf. Process. Lett., 5(3):9 33, 99. [5] Y. Lcet. New seqential exact eclidean distance transform algorithms based on conex analysis. Image Vision Compt., 7(-):37, 9. [] J. Marer, R. Qi, and V. Raghaan. A linear time algorithm for compting exact eclidean distance transforms of binary images in arbitrary dimensions. IEEE Transactions on Pattern Analysis and Machine Intelligence, 5():5 7, 3. [7] A. Meijster, J. B. T. M. Roerdink, and W. H. Hesselink. A general algorithm for compting distance transforms in linear time. Comptational Imaging and Vision, ():33 3,. [] M. Miyazawa, P. Zeng, N. Iso, and T. Hirata. A systolic algorithm for eclidean distance transform. IEEE Transactions on Pattern Analysis and Machine Intelligence, (7):7 3,. [9] D. W. Paglieroni. Distance transforms: Properties and machine ision applications. CVGIP: Graphical Model and Image Processing, 5():5 7, 99. [] I. Ragnemalm. Neighborhoods for distance transformations sing ordered propagation. CVGIP: Image Underst., 5(3):399 9, 99. [] A. Rosenfeld and J. L. Pfaltz. Seqential operations in digital pictre processing. J. ACM, 3:7 9, 9. [] T. Saito and J. ichiro Toriwaki. New algorithms for eclidean distance transformation of an n-dimensional digitized pictre with applications. Pattern Recognition, 7():55 55, 99. 3