Qin-Zhong Ye. Dept. of Electrical Engineering Linkiiping University S LinkGping, Sweden. 2. The signed Euclidean distance transform

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1 The Signed Euclidean Distance Transform and Its Applications QinZhong Ye Dept. of Electrical Engineering Linkiiping University S LinkGping, Sweden Abstract The signed Euclidean distance transform is a modified version of Danielsson s Euclidean distance transform [I]. This distance transform produces a distance map in which each pixel is a vector of two integer components. If a distance map is created inside the objects, the two integer values of a pixel in the distance map represent the displacements of the pixel from the nearest background point in x and y directions, respectively. The unique feature of this distance transform that a vector in the distance map is always pointing to the nearest background point is exploited in several applications, such as the detection of dominant points in digital curves, curve smoothing, computing Dirichlet tessellations and finding convex hulls. 1. Introduction A distance transform produces a distance map from a binary image. For each pixel inside the objects in the binary image, the corresponding pixel in the distance map has a value equal to the minimum distance to the background. Similarly, we can also define a distance map for the background. Generally, we can define a zerodistance set in the distance map. The zerodistance set may correspond to the background or the objects in the binary image depending on the applications. From now on we ll say that a distance map is created in the complement of the zerodistance set. Different metrics have been defined for the discrete plane, for example, city block distance and chessboard distance. These distances deviate quite substantially from the Euclidean distance. Even though they can be used in certain applications, the usual ideal is Euclidean metric. Octagonal distance is a combination of the two distances mentioned above. Still, relative errors between the octagonal distance and the Euclidean distance are about 10% and large absolute errors occur for large distances. Better approximations to the Euclidean metric are proposed by Borgefors [2], [3], but errors are still proportional to distances. Euclidean distance transform on the Cartesian discrete plane was proposed by Danielsson [l] in This transform produces a distance map in which each pixel is a vector of two positive integer components. A fourpoint sequential Euclidean distance mapping algorithm and an eightpoint sequential algorithm are described and analysed in detail in [I]. Parallel Euclidean distance transform algorithms are also discussed [I]. For the eightpoint sequential algorithm, errors of less than 0.09 pixel units may occur at a few very rare locations due to the complex geometry of the objects. For most of practical applications, this algorithm produces an errorfree distance map. A modified version of the ordinary Euclidean distance transform, namely, the signed Euclidean distance transform is also briefly mentioned in 111. A discussion on the signed Euclidean distance transform and its parallel algorithm can also be found in [13]. In this paper, we discuss the signed Euclidean distance transform and its applications. We ll discuss the unique feature of this transform in section 2, where a sequential signed Euclidean distance transform algorithm is described. The implementation of the signed Euclidean distance transform is also discussed. Some of the implementation details are applicable to the original Euclidean distance transform as well. In sections 3 to 6, we describe several algorithms which employ the unique feature of the signed Euclidean distance transform. 2. The signed Euclidean distance transform In the Signed Euclidean Distance Transform, the distance map is defined to be a twocomponent image. The two components represent the displacement in x direction and y direction to the nearest zerodistance point, respectively. We can regard each pixel as a twocomponent vector, that is w, Y) = (L&,Y),L,(Z,Y)), where Lz(z, y) and L,(z, y) are integers. The real distance to the zerodistance set or the length of the vector is defined by We have found that this transform has an interesting and unique feature which has not been explored before, that is, a vector in the distance map serves as a pointer to its origin, i.e. the nearest zerodistance pixel. For a vector (L,(z,y),L,(z,y)) at (z,y), we can find its nearest zerodistance pixel (20, yo) through the following relation: (Z0,YO) = (ZlY) + (L,(Z Y),L,(Z,Y)). CH $ IEEE 49 5

2 This relation is illustrated in Fig. 1. Fig. 1. The eightpoint sequential signed Euclidean distance transform algorithm (8SSED) is described as follows. Initially, for each pixel in the distance map, set L,(z, y) = 0, L,(s, y) = 0, if the pixel (2, y) in the binary image is a zerodistance point; and set L,(s, y) = 2, Ly(z, y) = Z, if the pixel (2, y) is in the complement of the zerodistance set. 2 is a big value which is larger than the predetermined maximum distance. Then, four masks (see Fig. 2) are moved in the distance map in the following way. Starting from the bottom of the map, scan the whole map line by line by moving mask 1 from left to right along a line and then mask 2 from right to left along the same line. When the forward scan is finished, starting from the top of the map, scan the whole map line by line by moving mask 3 from right to left along a line and then mask 4 from left to right along the same line. At each pixel of the map, if the center pixel of the mask is in the complement of the zerodistance set, add each vector of the mask to the corresponding vector in the distance map and find among all summed vectors in the neighborhood defined by the mask the summed vector which has the minimum length. If the minimum length is less than fi(2 l), set the center pixel in the distance map to the summed vector. An example is given in Fig. 3, where a distance map is computed from the two zeredistance points labelled with (0,O). Mask4 " Mask 3 center pixel Mask 1, t Mask 2 Fig. 2. The masks and scan directions for 8SSED $ 2, , , , 2, 0 3, c Ll 1. a , 2 1. : 1, 4 5. ( , 1 2, 1 Fig. 3. An example of the distance map obtained using the algorithm 8SSED. The implementation of the above algorithm requires two squares (multiplications), one addition and one square root to evaluate the length of a vector. This sounds very expensive, but the square root computation is actually not necessary. One can me the sums of squares of the vectors for searching the minimum. The multiplications can be avoided by using differentials. For instance, if L: + Li is saved together with L,, L,, then (LZ + + L; = L: + L; + 2L, + 1. When the two scans (forward and backward scans) are finished, if one needs the length of the vector as an output of the transform, one can obtain the length by the use of a twoindex lookup table. The table is indexed by the two components of a vector and the content of each element in the table is the length of the vector. Only positive integers are used to represent the lengths, which means that rounding is included in the table lookup. 3. Detection of dominant points in digital curves The detection of dominant points in digital curves is to detect high curvature points. The common way to deal with this problem is to follow the curve (or trace the chain code of the curve) and try to measure the angle or curvature at each point of the curve [4,5]. Another way is to find a polygonal approximation of the digitized curve directly [SI. Here, a new algorithm is proposed which utilizes the unique feature of the signed Euclidean distance transform. Algorithm Curvature measurement Suppose the curve is a close digital curve in a binary image. First, regard the curve as the zerodistance set and Then, apply the following procedure. Procedure Compute curvature image Initialization Set every pixel in the curvature image to zero. The curvature image has the same size as the binary image containing the curve. 496

3 Accumulate For each point (z,y) at a certain distance, e.g. at di5 tance 5, to the curve, increment the pixel value by 1 in the curvature image at (z + L,(z,y),y + L,(z,y)), where L,(z,y) and L,(z,y) are the two integer components of a vector in the distance map. Now we have obtained a curvature image. At the points corresponding to the curve in the binary image, we find different values in the curvature image. High values correspond to the points of high curvature and low values to the points of low curvature. The reason for this can be explained as follows. First we discuss the continuous case. The two circles in Fig. 4 have radii rl and r2 respectively. Curvatures of the two circles are respectively c1 = $ and c2 = &. Since rl is less than r2, cl is larger than c2. At distance d to the 4." R,= rl t d I I Fig. 4. Two circles rl and r2. ' R,= r2 + d two circles we draw another two dotted circles RI and R2. If we fix a unit arc length 6 along circles rl and r2, we can find that, corresponding to the unit arc lengths on circles rl and r2, the arc lengths A1 and A2 along circles R1 and R2 are respectively ri+d x6 A1 = rl and A2 zz r2 + x 6. r2 Obviously, A1 is larger than A2. If we compare A1 to the curvature el, we find A1 = 6(1+ d x el). Thus, A1 is a linear function of the curvature c1, and it can be used as a measure of the curvature. This means that the higher the curvature is at a certain portion of the object contour, the more points in the distance map will refer to this very portion. For the practical use, the vectors at distance d to the curve or the vectors within a range of distances can be used to calculate the curvature measure. Obviously, the curvature measure algorithm is not applicable when curves are too close to each other so that there is no room for the distance map. In the procedure Compute curvature image, we have made no distinction between the points at distance d outside and inside the original curve. Actually, one can easily derive an algorithm to accumulate for the points outside the original curve and for those inside the original curve, separately. The dominant points, finally, can be computed from the curvature image by simple thresholding. More sophisticated methods can be invented, however, for instance examining the values of the curvature measure along the curve. 4. Curve smoothing Curve smoothing is desirable if a digitized curve is noisy or an object in a binary image has a noisy boundary. It is also needed if one wants to disregard the details (small features) of a curve. The fact that one can use distance transforms for smoothing the boundary of an object is commonly known (81, but three distance transforms and three thresholdings are needed. Furthermore, the result of the smoothing depends on to which side of the curve the expansion (dilation) is first taken. The following is a new curve smoothing method which uses only once the signed Euclidean distance transform and a traceback procedure. The basic idea of the method is to let two circles from both sides of the original curve touch the curve. Small features of the curve can be touched only from one side of the curve or not touchable from both sides. These small features will be smoothed out by our smoothing method. After the signed Euclidean distance transform, a labeling procedure will label the points of the curve as touchable from outside of the curve (label I), touchable from inside (label 2), touchable from both sides (label 3) and not touchable (label 0). See Fig. 5. Then some interpolated points are found between labeled segments. These points are connected with straight lines or smooth curves. The obtained curve cut away from the original curve all the features smaller than the predetermined diameters of the touching circles. If the interpolated points are connected with straight lines, these lines may form new sharp corners, but the new curve is still a smoothed representation of the original curve. Fig. 5. A labeled curve. Algorithm Curve Smoothing. c n Suppose the curve is a close digital,curve in a binary image. First, regard the curve as the zerodistance set and Then, apply the following procedure. 49 7

4 Procedure Trace back Initialization Set every pixel in a label image to zero. The label image has the same size as the binary image containing the curve. Labeling For each point (z,y) at a certain distance dout, e.g. at distance 10, from outside of the curve, label the pixel with 1 in the label image at (z + L,(z,y),y + L,(z,y)), where L,(z, y) and L,(z, y) are the two integer components in the distance map. For each point (2, y) at a certain distance d;,, e.g. at distance 10, from inside of the curve, label the pixel with 2 in the label image at (z + L,(z,y),y + L,(z,y)). If the pixel in the label image has already been labeled from the outside of the curve, i.e. the value of the pixel is 1, label the pixel with 3. Note that the degree of smoothing depends on dout and din. Segmenting and interpolating In the label image, along the trace of the original curve, find the starting points and end points of segments of label 1. Inside the segments, points of label 0 are ignored. Similarly, find the starting points and end points of segments of label 2. Inside the segments, points of label 0 are ignored. Interpolate between the end point of a segment of label 1 and the starting point of the next segment of label 2 along the curve. Also interpolate between the end point of a segment of label 2 and the starting point of the next segment of label 1 along the curve. Connecting these ordered interpolated points with straight lines or smooth curves, e.g. pieces of the circle of radius 10 or cubic spline curves. Segments of label 3 are touched by the circles of radii dout and din from both sides of the curve, so there is no need to smooth them out and they are plotted at their original positions. The ends of a segment of label 3 are connected to the interpolated points between the ends of the other segments (of label 1 or 2) and the ends of the segment of label 3. A segment of label 3 should be a sequence of connected points of label 3. Points or segments of label 0 between two segments of label 3 are replaced by a straight line connecting the two close ends of the two segments of label 3. Note that one can do the above things by tracing the curve only once. 5. Computing Dirichlet tessellations In image processing, Dirichlet tessellation is mostly used in texture analysis 171. The Dirichlet tessellation is defined as a subdivision of the plane determined by a finite set of distinct points (sites): each point has associated with it the region of the plane nearer to that point than to any other. It has been shown that on the discrete grid, Pseudo Dirichlet tessellations can be computed by using parallel distance transforms (31. A parallel distance transform prop agates the labels of the sites in a label image simultaneously to the computation of the distances to the sites. When the program of the parallel distance transform terminates after a number of iterations, the labeled areas form the tiles of the pseudodirichlet tessellation. This method has an obvious disadvantage: The parallel distance transform algorithms are painstakingly slow if they are implemented on a sequential machine. We have found that sequential noneuclidean distance transforms can also be used for computing the pseudo Dirichlet tessellation. The labels of the sites can be propagated in the two image scans. When propagating a label, if there are two or more points (possibly including the center point) in the neighborhood trying to give different labels to the center pixel, the center pixel will be given a special label indicating that the pixel is at the same distance to two or more sites. The common drawback of the above two methods is the inaccuracy associated with the noneuclidean distance transforms. The sequential signed Euclidean distance transform is well suited for computing Dirichlet tessellations on the discrete grid. The unique feature of the signed Euclidean distance transform fits perfectly in this application: The distances that are propagated across the plane are selfreferencing to their origins. No extra labels are needed. The points which are closest to site i are pointing to the site i with their twocomponent vectors in the distance map. Algorithm Computing Dirichlet tessellations Suppose there are a number of distinctively labeled points (sites) in the original image. First, regard the labeled points as the zerodistance set and Then, apply Procedure Tessellation For each point (z,y) in the tessellation image, set the Fig. 6. Dirichlet tessellation. The black points are the sites. 498

5 value of the pixel equal to the value (label of a site) of the pixel at (z + L2(z, y), y + Ly(z, y)) in the original image, where L,(z, y) and L,(z,y) are the two integer components of a vector in the distance map. The Dirichlet tessellation computed with the above algorithm is shown in Fig A convex hull algorithm Convex hulls have been utilized in various applications [9], [lo]. A convex hull of an arbitrary set S in the real plane is the smallest convex set conu(s) which contains S. A convex hull conu(s) is also defined as the intersection of all the halfplan containing S. In the digital plane, convex hulls are analogously defined, but in a digital plane neither a halfplane nor a segment of straight line is always uniquely defined [9]. One must be careful to allow for quantization effects. To compute a convex hull of a set of points is to find the set of extreme points situated at the border of the convex hull. By employing the unique feature of the signed Euclidean distance transform, we have derived an algorithm for finding the convex hull of an arbitrary object or a set of distinct points in a twodimensional digital plane. S could even be a mixture of these data types. The input data to the algorithm is an N x N binary image with the set S having value 1 and the complement of S having value 0. The new algorithm has three major stages. The first is the Signed Euclidean Distance Transform. The second has two procedures: one finds a starting point and the other collects a sequence of candidate extreme points by tracing the border of the distance map. The third stage is a procedure for finding a convex hull from a polygon [ll]. Algorithm Computing convex hulls First, regard the set S as the sercdistance set and The second stage starts by finding a point (zo,y~) in the bottom edge of the distance map whose pointer points to a zerodistance point with the minimum ycoordinate. Then, the following procedure traces the border of the image in the counterclockwise direction and collects a sequence of candidate points which are close to the border of the image. Procedure Image Border Tracing Starting from the point (zo,yo), the border points of the image are counterclockwise sequentially visited until (20, yo) is reached again. In this sequence of points if two neighboring points (z;, y;) and (z;+l,y;+l) have their vectors pointing to different zerodistance points, the point pointed by (zi,y;) is saved in an input list, i.e. if (zi + LZ(zi,yi),vi + LY(zi,yi)) # (zi+l+ Lz(zi+l, Yi+l),Yi+l+ Ly(Si+l, yi+l))i save (zi + Lz(z;, yi), y; + L,(zi, y;)) in the input list. Now, we have a sequence of n points selected from the original set of points. This reduced sequence of points are the ones which are closest to the border of the image. One may regard these points as the vertices of a polygon. In the third stage, we use the vector crossproduct rule Ill] to check concavities in the polygon. This rule is briefly described as follows. Let (z,, y,) = (SO, yo) so that we have n + 1 points in the sequence. Starting with j = 0, do the following steps (1 4) for every three consecutive points j, j + 1 and j + 2 in the sequence: Step 1. P = (zj+l zj)(~j+2yj+l) + (Yjyj+l)(zj+2 zj+ 1) Step 2. If P > 0 and point j + 2 is the last point in the sequence, stop. Step 3. If P > 0, move one point forward and go to Step 1. Step 4. Delete point j + 1. If j # 0, move one point backward; else move one point forward. Go to Step 1. The detailed convex hull algorithm and a generalized border tracing procedure can be found in ( Conclusions We have described in this paper the signed Euclidean distance transform and some of its applications. The unique feature of the transform has made many applications possible of which the ones discussed in this paper is just a subset. Others, like shape decompositions, skeletonizing will be discussed in forthcoming papers. In the near future, we ll also extend the transform to three dimensions. References [ 1 P.E. Danielsson, =Euclidean Distance Mapping, Computer Graphics and Image Processing 14, 1980, pp (21 G. Borgefors, Distance Transformations in Arbitrary Dimensions, Computer Vision, Graphics and Image Processing 27,1984, pp [3] G. Borgefors, Distance Transformations in Digital Images, CVGIP 34, N0.3, 1986, pp A. Rosenfeld and J.S. Weszka, An Improved Method of Angle Detection on Digital Curves, IEEE Trans. Comput., Vo1.C24, 1975, pp P.V. Sankar and C.V. Sharma, A Parallel Procedure for the Detection of Dominant Points on a Digital Curve, CGIP 7, 1978, pp K. Wall and P.E. Danielsson, A Fast Sequential Method for Polygonal Approximation of Digitised Curves, CGIP 28, 1984, pp N. Ahuja and B.J. Schachter, Pattern Models, John Wiley & Sons, New York, G. Borgefors, Distance Transformations in Digital Images, FOA report, C 30401E1, August A. Rosenfeld and A.C. Kak, Digital Picture Processing, Academic Press, Inc., New York, R.O. Duda and P.E. Hart, Pattern Classification and Scene Analysis, John Wiley, New York, J. Sklansky, Measuring Concavity on a rectangular mosaic, IEEE Trans. Computers, Vol. C21, December 1972, pp Q.Z. Ye, A Convex Hull Algorithm Using the Signed Euclidean Distance Transform,Internal Report,LiTHISY10919, LinkGping University, April H. Yamada, Complete Euclidean Distance Transformation by Parallel Operation, Proc. of 7th Int. Cod. on Pattern Recognition, Montreal, Canada, July 1984, pp