A new generic scheme for incremental implementations of distance transforms (DT) is presented: Incremental Distance Transforms (IDT). This scheme is applied on the cityblock, Chamfer, and three recent exact Euclidean DT (E2DT). A benchmark shows that for all five DT, the incremental implementation results in a significant speedup: 3.4×−10×. However, significant differences (i.e., up to 12.5×) among the DT remain present. The FEED transform, one of the recent E2DT, even showed to be faster than both city-block and Chamfer DT. So, through a very efficient incremental processing scheme for DT, a relief is found for E2DT’s computational burden

Boyer, K.

Erçil, A.

Lee, S.-W.

Schouten, Theo E.

van den Broek, Egon

Çetin, M.

English

University of Twente Research Information

Incremental Distance Transforms (IDT)Theo E. SchoutenInstitute for Computing and Information Sciences (ICIS),Radboud University NijmegenP.O. Box 91046500 HE Nijmegen, The Netherlandst.schouten@cs.ru.nlEgon L. van den BroekHuman Media Interaction (HMI),University of TwenteP.O. Box 2177500 AE Enschede, The NetherlandsHuman-Centered Computing Consultancyhttp://www.human-centeredcomputing.com/Vienna, Austriavandenbroek@acm.orgAbstract—A new generic scheme for incremental implemen-tations of distance transforms (DT) is presented: IncrementalDistance Transforms (IDT). This scheme is applied on the city-block, Chamfer, and three recent exact Euclidean DT (E2DT).A benchmark shows that for all five DT, the incrementalimplementation results in a significant speedup: 3.4 × −10×.However, significant differences (i.e., up to 12.5×) among theDT remain present. The FEED transform, one of the recentE2DT, even showed to be faster than both city-block and Cham-fer DT. So, through a very efficient incremental processingscheme for DT, a relief is found for E2DT’s computationalburden.Keywords-Euclidean distance, Fast Exact Euclidean Distance(FEED), distance maps/transforms, incremental implementa-tion, Incremental Distance Transforms (IDT), video processingI. INTRODUCTIONA distance transformation (DT) [1] converts a binaryimage into a distance image (DI). In such a DI, the valueof each pixel represents its distance to the set O of objectpixels o in the binary image. Such a DT is calculated asfollows:DI(b) = mino ∈ OD(b, o), (1)where D can be any metric and b is a background pixel.Although Equation 1 is straightforward, it is hard todevelop an algorithm that calculates the DT in a fastmanner [2], [3], [4], [5], [6]. Rosenfeld and Pfaltz [1]introduced the first fast algorithms for the city-block (D4)and chessboard distance (D8) metrics. Two decades later,Borgefors [7] extended them to Chamfer DT (DC), whichprovide better approximations to the L2 metric. The D4, D8,and DC have in common that they all use raster scans overthe image to propagate distance using local information only.As the tiles of the Voronoi diagram are not always connectedsets on a discrete lattice, the exact Euclidean distance (ED)cannot be obtained by raster scans.The last decade, various algorithms have been presentedto obtain exact ED transforms (EDT); e.g.,. [4], [5], [6]. Wewill now introduce three of the most promising EDT. Mau-rer, Qi, and Raghavan [8] introduced an EDT for arbitrarydimensions (DM ). More recently, Lucet [9] presented severalsequential exact EDT algorithms, based on fundamentaltransforms of convex analysis; e.g., the LLT algorithm. Bothapproaches use dimensional decomposition to process eachrow independently of the other rows. Next, they process eachcolumn of the resulting image independently of the othercolumns, to produce the final DI.An alternative approach for EDT was introduced bySchouten and Van den Broek [10]: the Fast Exact EuclideanDistance (FEED) transformation. With FEED, each objectpixel feeds its ED to all pixels in the image, which in turncalculate the minimum of the received EDs. They showedthat by reducing the number of pixels that must be processed,a very fast exact EDT can be obtained. Nowadays, DTare not only used for image processing (e.g., [11], [12])but are also used to process video sequences; e.g., inrobot navigation or video surveillance applications [13].These videos often contain large sequences of frames withstationary objects and one or more moving objects. In thispaper, we develop a new class of algorithms, the IncrementalDistance Transforms (IDT), to calculate the DI for suchframes in a fast way by using the DI of the stationary objects.In the next section, the principle of IDT will be describedthrough its application on: FEED, D4, DC , DM , and LLT.Next, their incremental implementations are benchmarked inSection III. This paper ends with a discussion in Section IV,reflecting on the work presented.II. INCREMENTAL DISTANCE TRANSFORMS (IDT)With IDT, a new class of algorithms is introduced. Thisclass is inspired by the algorithm of FEED [10], [13], asthis enables a direct incremental implementation. Therefore,we will now first briefly describe an adapted version ofthe FEED algorithm, as this aids the understanding ofthe incremental implementations for EDT. Second, in Sec-tion II-B, the IDT for D4 and DC will be defined. Third, inSection II-C, the IDT for DM and LLT will be introduced.2010 International Conference on Pattern Recognition1051-4651/10 $26.00 © 2010 IEEEDOI 10.1109/ICPR.2010.67237A. IDT for FEEDThe FEED algorithm calculates the EDT starting directlyfrom the definition (see Equation 1), or rather its inverse.FEED on an image is initialized with distance DI(·) for(background) pixels b:DI(b) = if (b ∈ O) then 0 else ∞. (2)Subsequently, the adapted naive FEED algorithm is:foreach o ∈ Odetermine: Aoupdate: foreach a ∈ Ao doDI(a) = min{DI(a), ED2(o, a)},(3)where Ao is the area where o should feeds distances to. ED2instead of ED is used as this avoids square roots. In fact,all calculations in our FEED implementation are done inintegers.With o ∈ O in Equation 3, only the border pixels of Ohave to be considered becausemin{ED(b, o)} == ED(b, ob), (4)where ob is a border pixel of O; i.e., having at least one ofits four 4-connected pixels in the background.Having this implementation of FEED, its incrementalimplementation can be developed as follows:DIs+m(b) = DIs(b) = DT with Os, (5)where DT is in this case FEED and Os is the set of stationaryobjects. This initializes the DIs+m, the combined DI for thestationary (s) and moving (m) objects. Subsequently, FEEDas defined in Equation 3, is applied again:foreach o ∈ Omdetermine: Aoupdate: foreach a ∈ Ao doDIs+m(a) = min{DIs+m(a), ED2(o, a)}(6)where Om is the set of moving objects.As with the original FEED algorithm, also its incrementalimplementation (FEED∗) can be speeded up. Most impor-tantly, the area Ao can be restricted immediately to a circlewith radius dmax (i.e., the outer bounding box), being themaximum ED in DIs. This is, because combining Om andOs can only decrease the maximum ED, compared to themaximum ED in Os only.B. IDT for city-block / Chamfer distancesThe IDT for the city-block (D4) and Chamfer (DC) DTare similar. Therefore, they are introduced together.D4 is initialized as follows:if I[0][0] ∈ O then DI[0][0] = 0, else DI[0][0] = ∞, (7)where I[y][x] is the input image and DI[y][x] its DI. Next,a forward raster scan is conducted:if I[y][x] ∈ O then DI[y][x] = 0 elseDI[y][x] = min{1 + DI[y][x − 1], 1 + DI[y − 1][x]}(8)Subsequently, a backward raster scan is applied:if DI[y][x] > 0 then DI[y][x] =min{DI[y][x], 1 + DI[y][x + 1], 1 + DI[y + 1][x]}(9)For the IDT based on D4, first the area is determinedover which Om can change the DI. For this the boundingbox (bb) of Om is determined and, subsequently, extendedin all directions with dmax, the maximum distance in DIs.Then, inside this bb, the maximum occurring distance isdetermined. Next, the bb of Om is enlarged with thatdistance to provide the local area over which the modifiedD∗4 algorithm must be applied.First, the incremental D∗4 algorithm is initialized, usingEquation 5. Second, Equation 8 and 9 are applied on Om,instead of O. The forward raster scan starts at the lowest[ys][xs] point of the bb of Om and for the next lines startwith that xs. It can be stopped as soon as a scanline [ye]with no changes in DI[y][x] has been found. The backwardscan is stopped as soon as a scanline below Om with nochanges in DI is detected.The IDT for DC is similar to the IDT for D4. However,because also the two diagonal neighboring pixels are takeninto account in the raster scans, a larger part of the localarea has to be covered.C. IDT for DM and LLTBoth DM and LLT produce intermediate images of whichthe values cannot be regarded as distances, in the sense thatthey cannot be compared with the finally produced Euclideandistances.According to the definition of DI (see Equation 1), DIsand DIm can be calculated separately. DIs is initializedthrough Equation 5, with DT being either DM or LLT. Next,their full original algorithm is applied on the rectangulararea over which the moving object can have an effect, asdescribed in the previous section, which results in DIm.Subsequently, using the min operator, both DI are combinedto provide the final ED2 DI.III. BENCHMARKIn a benchmark, the five original algorithms and theirfive IDT implementations, as described in the previous twosections, will be compared with each other: D4/D∗4 [1],DC /D∗C[7], the algorithm of Maurer et al. [8] (DM /D∗M),LLT/LLT∗ [9], and FEED/FEED∗ [10], [13].All ten algorithms were implemented using single pre-cision (32 bit) integers, producing their output in a formatfastest for each method: ED2 for FEED/FEED∗, DM /D∗M,and LLT/LLT∗; ED×3 for DC /D∗C; and ED for D4/D∗4. The238set standard implementation incremental implementationFEED DM LLT DC D4 FEED∗ D∗MLLT∗ D∗CD∗4A 7.6 34.3 34.7 10.3 7.5 0.5 3.5 3.4 1.2 0.7B 7.1 32.9 33.5 10.3 7.5 0.6 4.7 4.6 1.5 1.0C 6.6 24.2 23.1 11.4 7.9 1.3 17.5 17.6 3.8 2.7D 7.4 29.2 28.0 11.5 7.9 0.6 9.4 9.3 1.9 1.3average 7.2 30.1 29.8 10.9 7.7 0.7 8.8 8.7 2.1 1.4Table ITIMING (IN NS/PIXEL) RESULTS OF THE BENCHMARK ON 4 TEST SEQUENCES AS WELL AS THEIR AVERAGE; SEE ALSO SECTION III. THEBENCHMARK WAS CONDUCTED ON BOTH THE STANDARD IMPLEMENTATION, AS REFERENCE VALUES, AND THE IDT IMPLEMENTATION (DENOTEDWITH ∗) OF THE FOLLOWING ALGORITHMS: FEED, DM , LLT, DC , AND D4 ; SEE ALSO SECTIONS I AND II.times measured for the IDT methods include the time forreuse of DIs for subsequent frames. This is achieved byrestoring the part changed by the IDT back from a changedcopy of DIs.The benchmark was executed on a PC with an Intel Core2 Duo E6550P R© 2.33GHz processor (2x 32KB data and 2x 32 KB instruction L1 cache, 4096 KB L2 cache) and with1024 MB memory, using the gcc compiler. The benchmarkemployed four sequences of 120 images, of size 652× 492pixels. Each of the four sequences contained a triangularobject moving throughout the frame; see also Figure 1. Theinput image was split in two images. One for the programparts handling Os, with 0 and 255 indicating respectivelyfixed object pixels and the background. The other for theprogram parts handling Om, with 0, 127, and 255 indicatinga pixel from respectively Os, Om, and the background.Table I shows the results of the benchmark in terms ofspeed. It shows that all IDT implementations are signifi-cantly faster than their original implementation. For exam-ple, FEED∗ is at least 5× faster than FEED. Further, Table Ishows that FEED∗ is at least 7× faster than both D∗M[8]and LLT∗ [9]. FEED∗ is also >2.4× faster than D∗C[7] andeven >1.4× faster than D∗4 [1], with FEED being the onlyalgorithm to give exact results. Please note that the obtainedtimes are dependent on the content of the images.IV. DISCUSSIONOn the one hand, distance transforms (DT) are a basicoperation in computational geometry. On the other hand,they are applied within various applications (e.g., [11], [12],[13]), either by itself or as intermediate method; e.g., videosurveillance, robot navigation, skeletonization, fMRI dataanalysis, neuromorphometry, and volume rendering. Thisarticle will extent current research by providing a newclass of fast and exact algorithms for incremental distancetransforms (IDT).Par excellence, IDT can be used to generate distanceimages (DI) for video sequences, in a fast manner. It isassumed that each frame of a sequence consists of stationaryobjects and 1 moving object (see Figure 1), although anextension to multiple moving objects is straightforward.Then, the DI of the stationary objects (DIs) is calculatedonce for each sequence, using a certain DT. This DIs isused in the corresponding IDT to calculate the complete DIfor each frame. Such an IDT is an adapted version of thecorresponding DT.The principle of IDT is applied on five DT: city-blockDT (D4) [1], Chamfer DT [7], a recent exact EuclideanDT [8], the LLT algorithm [9], and the FEED algorithm [10].A benchmark has been employed to enable a controlledcomparison among these DT; see also Table I. In general,the observed speedup is large, compared to calculating theDI for each frame separately with the corresponding DT.However, it should be noted that the timing results dependon both the used DT and the content of the image/videosequence.The lowest speedup is obtained for the DM and LLTexact ED transformations. From the DIs and the positionof moving object, a rectangular area has been determined inwhich the DT is applied as the IDT in order to obtain theDI of the moving objects (DIm) in that area. Next, the twoDIs have been combined, using the minimum operator. Notethat this method is generic, as it can be used to convert anyDT method into a corresponding IDT method.For the raster scan methods DC and D4, the obtainedspeedup is larger. The IDT is applied to a rectangular area,determined in the same way as for DM and LLT, and consistsof a modified version of the corresponding DT. The startingDIm is initialized with DIs, which reduces the number ofupdates on it. Further, the raster scans can be restricted toonly part of the rectangular area. Also, by placing the datafor the rectangular area in the same memory locations asDIs, the final output is directly available.For FEED the obtained speedup is the largest. This islargely due to the fact that to obtain the final result, onlythe border pixels of the moving object have to feed theirEDs in a restricted area. With FEED on full images alreadybeing faster than the other two exact EDs, FEED’s IDTimplementation showed to be > 7× faster than the IDTsbased on both DM and LLT. Moreover, FEED∗ was alsofaster than both D∗Cand D∗4.It would of interest to apply IDT on various other DT,both those that provide exact ED and those that approximateED. Throughout the last decade, a range of interesting DT239Figure 1. Sample test images from set A (left) and B (right). The triangle is the object Om, which moves throughout the frame during the sequence.The other objects are all stationary (o ∈ Os). Sets C and D contain the same images as sets A respectively B with the border objects left out.have been presented either in separate papers (e.g., [3],[4]), in special issues of journals or dedicated workshops(e.g., [5], [6]), or as core of applications (e.g., [11], [12]).Consequently, a more exhaustive benchmark would be ofinterest. The current benchmark could be easily extendedthrough, for example: i) Incorporating a larger and morevarying set of image sequences than the current four sets;cf. Section III and Figure 1; ii) Including alternative DTs,such as just mentioned; and iii) Use image sequences withmultiple moving objects.As said earlier, one should note that the timings are alsodependent on the content of the images. This effect needsfurther investigation, especially for FEED, as its foundedon a stochastic process and, consequently, its computationalcomplexity cannot be proven. Consequently, FEED has onlyshown to be fast through experimental results. Althoughthis has been done repeatedly, as also in this article, incontrast with the other methods, it cannot be proved thatthe arithmetic complexity is O(n).Taken together, with the class of IDT algorithms, fastexact EDT can be done on video sequences. Hence, noapproximations of EDT are needed due to its computationalburden [2]. As such, an important new class of videoprocessing algorithms is launched, which will find its wayto many applications in both image and video analysis.ACKNOWLEDGMENTWe gratefully acknowledge the reviewers, for their valu-able comments on an earlier version of this article.REFERENCES[1] A. Rosenfeld and J. L. Pfaltz, “Sequential operations in digitalpicture processing,” Journal of the ACM, vol. 13, no. 4, pp.471–494, 1966.[2] A. Hajdu and T. Tóth, “Approximating non-metricalMinkowski distances in 2D,” Pattern Recognition Letters,vol. 29, no. 6, pp. 813–821, 2008.[3] O. Cuisenaire, “Locally adaptable mathematical morphologyusing distance transformations,” Pattern Recognition, vol. 39,no. 3, pp. 405–416, 2006.[4] F. Porikli and T. Kocak, “Fast distance transform computationusing dual scan line propagation,” Proceedings of SPIE (RealTime Imaging), vol. 6496, pp. 649 608–1–649 608–8, 2007.[5] G. Borgefors, I. Nyström, and G. Sanniti di Baja, “Discretegeometry for computer imagery,” Pattern Recognition Letters,vol. 23, no. 6, p. [special issue], 2002.[6] ——, “Discrete geometry for computer imagery,” DiscreteApplied Mathematics, vol. 125, no. 1, p. [special issue], 2003.[7] G. Borgefors, “Distance transformations in digital images,”Computer Vision, Graphics, and Image Processing: An Inter-national Journal, vol. 34, no. 3, pp. 344–371, 1986.[8] C. R. Maurer, Jr., R. Qi, and V. Raghavan, “A linear timealgorithm for computing exact Euclidean distance transformsof binary images in arbitrary dimensions,” IEEE Transactionson Pattern Analysis and Machine Intelligence, vol. 25, no. 2,pp. 265–270, 2003.[9] Y. Lucet, “New sequential exact Euclidean distance transformalgorithms based on convex analysis,” Image and VisionComputing, vol. 27, no. 1–2, pp. 37–44, 2009.[10] T. E. Schouten and E. L. van den Broek, “Fast Exact Eu-clidean Distance (FEED) Transformation,” in Proceedings ofthe 17th IEEE International Conference on Pattern Recogni-tion (ICPR 2004), J. Kittler, M. Petrou, and M. Nixon, Eds.,vol. 3, Cambridge, United Kingdom, 2004, pp. 594–597.[11] E. L. van den Broek, T. E. Schouten, and P. M. F. Kisters,“Modeling human color categorization,” Pattern RecognitionLetters, vol. 29, no. 8, pp. 1136–1144, 2008.[12] M. Chen, W. Lu, Q. Chen, K. Ruchala, and G. Olivera, “Effi-cient gamma index calculation using fast Euclidean distancetransform,” Physics in Medicine and Biology, vol. 54, no. 7,pp. 2037–2047, 2009.[13] T. E. Schouten, H. C. Kuppens, and E. L. van den Broek,“Video surveillance using distance maps,” Proceedings ofSPIE (Real-Time Image Processing), vol. 6063, pp. 54–65,2006.240

Incremental Distance Transforms (IDT)

Schouten, T.E.

Broek, E.L. van den

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                  84341.pdf (publisher's version ) (Open Access)ICPR 2010 : 20th International conference on Pattern recognition, 23 augustus 201

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A linear time algorithm for computing exact Euclidean distance transforms of binary images in arbitrary dimensions,”

Approximating non-metrical Minkowski distances in 2D,”

Discrete geometry for computer imagery,”

Distance transformations in digital images,”

Efﬁ-cient gamma index calculation using fast Euclidean distance transform,”

Fast distance transform computation using dual scan line propagation,”

Fast Exact Euclidean Distance (FEED) Transformation,”

Locally adaptable mathematical morphology using distance transformations,”

Modeling human color categorization,”

New sequential exact Euclidean distance transform algorithms based on convex analysis,”

Sequential operations in digital picture processing,”

Video surveillance using distance maps,”

Incremental Distance Transforms (IDT)

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