Choosing appropriate distance measurement in digital image segmentation

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1 Choosing appropriate distance measurement in digital image segmentation András Hajdu 1, János Kormos2, Benedek Nagy3, and Zoltán Zörgő 4 1 Institute of Informatics, University of Debrecen, H-4010 Debrecen PO Box 12. hajdua@inf.unideb.hu 2 Institute of Informatics, University of Debrecen, H-4010 Debrecen PO Box 12. kormos@inf.unideb.hu 3 Institute of Informatics, University of Debrecen, H-4010 Debrecen PO Box 12. nbenedek@inf.unideb.hu 4 Institute of Informatics, University of Debrecen, H-4010 Debrecen PO Box 12. zorgoz@inf.unideb.hu Abstract. In this paper we show how we can take advantage of using different distance functions in image processing applications. The proposed methods are based on well-known algorithms that use distance measurement. We focus on multidimensional image indexing and segmentation procedures, and also show examples for the extraction of such feature vectors that can be used in image retrieval. As a special family, we perform a detailed analysis using distance functions generated by neighbourhood sequences. The application of such distance functions is quite natural and descriptive for images, since e.g. the colour coordinates of the pixels are non-negative integers. An additional interesting property of neighbourhood sequences is that they do not generate metrics in general, so we can obtain many distance functions in this way. Our final purpose is to find distance functions that provide the best results for a given problem, so we present some tools that help with finding them. 1 Introduction Extracting relevant data from multidimensional (e.g. colour) images are important operations with growing interest [17]. If the problem is to find objects on the images then these operations are known as segmentation methods in the literature of digital image processing. A rich family of these procedures make use of the fact that objects are usually defined by a group of neighbouring image pixels with similar colour values. As a natural consequence, these algorithms need some kind of distance measurement to compare the colours of the pixels. Traditionally the most of these image segmentation techniques are based on classical (e.g. Euclidean) metrics. In some cases other metrics may provide better results [6], but very few suggestions can be found in the literature how to choose them. Especially, if the domain of the problem is a discrete set like many colour represantions in digital image processing integer valued distance functions are also might be worth considering. Beside the simpler handling of digital distance

2 2 András Hajdu, János Kormos, Benedek Nagy and Zoltán Zörgő functions they often have more illustrative behaviour than classic metrics. Another important point is that these segmentation techniques are based on only distance measurement and the satisfaction of triangle inequality is not a natural requirement to hold. In other words such distance functions also may behave well that are actually not metrics. The above observations has led us to the investigation of the applicability of a special family of digital distance functions in multidimensional image segmentation problems. Namely, our approach is based on distance function generated by neighbourhood sequences, as to every neighbourhood sequence a distance function can be assigned in a natural way. The usefulness of special neighbourhood sequences in image processing applications was already noted in the very first fundamental papers of digital image processing, e.g. in [18]. The precise mathematical background of neighbourhood sequences will be recalled in the following section. The main advantages of these types of functions that they can be introduced in arbitrary finite dimension and also on different types of grid. Moreover, we can also take advantage the fact that these distance functions are not metrics in general. We observe three classical image processing applications that are based on distance measurement, namely the definition of colour ranges, segmenting objects by region growing and classyfing image pixels by the tools of cluster analysis. Some of these procedures use parameters and to help with their selection we propose some tools, namely some histograms composed from the difference between the colours of the pixels. Using these tools we illustrate the behaviour of classical metrics and distance functions based on neighbourhood sequences, and also the advantages of the latter family. Moreover, we show how our proposed tools help with choosing parameters to obtain more optimal results. As a very general domain in our investigations we focus on the 3D Red-Gree-Blue (RGB) colour domain. The structure of the paper is as follows. In Section 2 we recall some concepts and results from the theory of neighbourhood sequences that we need in our further analysis. Section 3 explains the application of neighbourhood sequences in the RGB-domain for image segmentation purposes. In Section 4 we present concrete techniques for color image segmentation using different distance functions. Finally, in Section 5 we summarize our results and indicate some corresponding open problems. 2 Neighbourhood sequences In this section we recall some basic concepts and results of the theory of neighbourhood sequences based on [4,7, 18]. Though we will consider the RGB image representation in details, we give the notions for arbitrary dimension, since our procedures can be applied to arbitrary dimensional integer image representations. In the two dimensional grid the two possible neighbourhood relations were defined in [18] as cityblock and chessboard motions. Based on these motions distances were defined as the length of a shortest path between points building from

3 Choosing appropriate distance measurement in digital image segmentation 3 the possible motions. Later the higher dimensional spaces were also involved in digital geometry. Let n be an arbitrary positive integer. Let q and r be two points in ZZ n. The i-th coordinate of the point q is indicated by Pr i (q). Let m be an integer with 1 m n. The points q and r are m-neighbours, if the following two conditions hold: Pr i (q) Pr i (r) 1 (1 i n), n Pr i (p) Pr i (q) m. i=1 With the concept of the neighbourhood sequences one can vary the used neighbourhood criteria in a path in the following way. In the n dimensional digital space the sequence A = (A(i)) i=1, where A(i) {1,...,n} for all i IN, is called an n-dimensional (shortly nd) neighbourhood sequence. If for some l IN, A(i + l) = A(i) (i IN), then A is periodic with period l. In this case we briefly write A = {A(1)A(2)...A(l)}. For example, we write {12} for the neighbourhood sequence 1,2,1,2,1,2,.... Note that the periodic neighbourhood sequences were introduced and investigated in [4, 5, 3, 13], while the general, not neccesary periodic neighbourhood sequences are used in [7, 11, 14, 16]. A point sequence q = q 0, q 1,..., q m = r, where q i 1 and q i are A(i)- neighbours in ZZ n (1 i m), is called an A-path from q to r of length m. The A-distance d(q, r; A) of q and r is defined as the length of the shortest A-path(s) between them. Considering the three dimensional digital space we recall the formula to calculate the distance from [16]. For this purpose we need the concept of the limited neighbourhood sequences ([14]). The sequence A (m) with elements A (m) (i) = min(a(i), m) is the m-dimensional limited sequence of A. The A-distance of the{ points p and q is given by} d(p, q; A) = i 1 max {v(1), d 2, d 3 }, where d 2 = max i v(1) + v(2) > A (2) (j), and d 3 = j=1 { } i 1 max i v(1) + v(2) + v(3) > A(j). Where the values v(j) are a permutation of Pr i (p) Pr i (q) in non-decreasing way, i.e. v(1) = max( Pr i (p) Pr i (q) ) j=1 i and v(3) = min( Pr i (p) Pr i (q) ). i The distance functions generated by neighbourhood sequences are not metrics in general and the existence of this property can be checked by the following criterion [14]. The distance function based on an nd-neighbourhood sequence A is a metric on Z n l+j if and only if A (k) (i) j A (k) (i) for all j, k N with i=l+1 1 k n. Nevertheless, in some cases non-metric distance functions also provide nice results, thus it is not recommended to exclude them from analysis. Moreover, with these non-metrical distances we have a lot more distance functions to choose from to refine our results. i=1

4 4 András Hajdu, János Kormos, Benedek Nagy and Zoltán Zörgő Since in many applications the so-called Minkowski distances L p (q, r) = ( n ) 1/p q(i) r(i) p, p > 0, L (q, r) = max n q(i) r(i) are considered, we i=1 i=1 show the relationship between the L p metrics and the distance functions generated by neighbourhood sequences. The distances L p and the A-distances are in the following relations: L 1 (q, r) = d(q, r; (1)) d(q, r; A) d(q, r; (n)) = L (q, r) holds for any nd-neighbourhood sequence A. While the L p metrics form a well ordered set, i.e. L p (q, r) L p (q, r) for all points q, r if and only if p p, the distances based on neighbourhood sequences have only a partial ordering. In the three dimensional space we have the following statement. d(p, q; A 1 ) d(p, q; A 2 ) for all points p, q, if and only if i 1 j=1 A (2) 1 i 1 (j) j=1 A (2) 2 j=1 i 1 (j) and A 1 (j) i 1 A 2 (j). In this case the neighbourhood sequence A 1 is faster than A 2. (Note, that this relation was introduced by Das et al. in [4] in case of periodic neighbourhood sequences, while in [7] by arbitrary neigbourhood sequences.) Moreover based on this partial order in [10] the velocity of neighbourhood sequences was investigated. j=1 3 Using neighbourhood sequences in the RGB colour representation In this section, we will consider the 3-dimensional RGB image representation in detail. Some of colour image processing methods are based on the comparison of the colour of the pixels. Usually the 24-bit RGB cube (that is, the domain is between black=(0, 0, 0) and white=(255, 255, 255) with red=(255, 0, 0), green=(0, 255, 0), blue=(0, 0, 255), yellow=(255, 255, 0), magneta=(255, 0, 255) and turquoise=(0, 255, 255)) is used. Now, we illustrate the descriptive behaviour of neighbourhood sequences in measuring the distances of colours. In our first example, we show that not only the values of distances, but the respective distances, i.e. the concept closer and farther are highly depend on the used neighbourhood sequences. Let us consider the following three colours given by their RGB coordinates: C 1 (60, 60, 60) (dark grey), C 2 (180, 180, 180) (light grey), and C 3 (240, 30, 90) (raspberry red). Calculating the distances of these colours with respect to the neighbourhood sequences {1}, {2}, and {3}, we get the distances shown in Table 1. A 1 = {1} A 2 = {2} A 3 = {3} d(c 1, C 2) d(c 1, C 3) d(c 2, C 3) Table 1. The distances of colours depending on the neighbourhood sequence used

5 Choosing appropriate distance measurement in digital image segmentation 5 These values indicate that the distances of these colours highly depend on the neighbourhood sequence. More precisely, for the neighbourhood sequences {1}, {2}, and {3}, the colours (C 1, C 3 ), (C 2, C 3 ), and (C 1, C 2 ) are the closest, respectively. The above used three neighbourhood sequences generate metrics. In the following example we show a distance which does not meet the triangular inequality. Let A 4 = {3, 2, 1}, C 4 (30, 30, 30), C 5 (88, 69, 50) and C 6 (108, 109, 109). Then it is easy to calculate the distances: d(c 4, C 5 ; A 4 ) = 58, d(c 5, C 6 ; A 4 ) = 59 and d(c 4, C 6 ; A 4 ) = 118. As we can see the triangular inequality (d(c 4, C 5 ; A 4 ) + d(c 5, C 6 ; A 4 ) d(c 4, C 6 ; A 4 )) does not true for these values. In the next example we will use the non-periodic neighbourhood sequence A 5 = {1, 1, 1, 2, 2, 2, 2, 2, 3, 3,...}, with A(i) = 3 for all i > 8. Lets calculate some A 5 -distances in the RGB-cube. d((0, 0, 0), (255, 255, 255); A 5 ) = 259. d((255, 0, 0), (0, 0, 255); A 5 ) = 257. d((0, 255, 0), (0, 0, 0); A 5 ) = 255. d((100, 50, 50), (200, 250, 100); A 5 ) = 200. d((75, 175, 124), (149, 101, 199); A 5 ) = 75. Moreover, A 5 generates a metric on Z n and hence on the RGB-cube. 4 Applications in image segmentation based on distance functions Now we have the tools to perform more general distance measurements as usual in concrete colour image processing tasks. As we have a corresponding research on stone parts segmentation the applicability of methods will be shown for images that contain stone particles captured by a microscope. The main challenge here is to detect special type of stones and to separate different stone parts. The whole segmentation procedure considers many image processing techniques among which the colour based segmentation is a basic approach. Now we present some well-known tools of colour image segmentation that are based on distance measurement and we found appropriate to gain better results in some cases. Namely we show the advances that can be achieved by considering new families of distance functions in defining colour ranges, performing segmentation by region growing and separating image segments by clustering. Beside showing examples for the advantages of our approach, we also introduce some tools which nicely describe the behaviour of the chosen distance measurement in the given image segmentation procedure. As the whole approach is based on the differences between colour values with respect to the chosen distance function, we found histograms composed from these values to be natural descriptors. Refining this basic idea we define three histogram types one for defining colour regions, another one for region growing and a third one which does not need any additional parameters after fixing a distance function. 4.1 Colour ranges Using the basic version of this procedure [8] we can found those pixels in the image whose colours are within a given threshold distance to an initially fixed seed

6 6 András Hajdu, János Kormos, Benedek Nagy and Zoltán Zörgő colour. In the software realizations of this method usually one special distance function is fixed and the user should choose the seed colour and the threshold value without any guidelines. Our purpose is to make it possible to change not only the seed colour and threshold value, but also the distance function to gain more optimal results. Using a faster distance function with lower threshold levels and a slower distance function with a higher one, similar results can be obtained. Identical results rarely can be achieved in this way, since the occupied regions (spheres) of distance functions geometrically differ, but the differences can be very small. However, using alternative distance measurement (e.g. distance functions based on neighbourhood sequences) we can gain additional possibilities beside adjusting the speed of the distance function. Namely, distance functions do not generate metrics in general, thus we can realize such ideas that would have no sense in case of classical metrics. One such idea in image segmentation might be that starting from one seed colour (object point) we want to grab many points in a fast way as they are assumed to be object points, but then to avoid the inclusion of non-object point we want to decrease the inclusion of the number of pixels. To realize this idea a distance function is needed which becomes slower. However, this property contradicts triangle inequality thus cannot be got by a metric. To help with choosing the suitable distance function d we use colour range histograms. After fixing a seed colour p, the k-th column of this histogram represents the number of those image pixels with colour q for which k d(p, q) < k+1. Note that for integer valued distance functions we can use the simpler condition d(p, q) = k. Then a local minimum of this histogram is a natural choice for the threshold value, since local minima can be assumed to indicate those parts of the colour cube which colours are taken by small number of image pixels. Especially, the first mode of the histogram (the closest colours to the seed) is possible the most important in applications. As the modality of the colour range histogram depends on the chosen distance function, a distance function is expected to perform better if the desired local minimum can be detected better in its corresponding histogram. For example, if the first local minimum has greater importance, then the first valley of the histogram should be as deep and wide as possible. To illustrate the above ideas in choosing the suitable distance functions and corresponding threshold values we present an illustrative example. Let us consider the original image in Figure 1(a), where the aim is to separate the stone part indicated by a dashed (artificial) boundary. First we show the differences regarding to different distance functions, when one seed colour is fixed and the threshold value is adjusted by accordingly to obtain approximately the similar quality of result. It can be nicely observed that in this case the idea of choosing a distance function which becomes slower (namely the neighbourhood sequence {3 10 1} nicely worked, as it did not include that many non-desired points as the metric L 1 or L. To determine the suitable threshold values we used the corresponding colour range histograms shown in Figure 2.

7 Choosing appropriate distance measurement in digital image segmentation 7 (a) (b) (c) (d) Fig.1. Colour ranges from the seed colour =(221,120,162) based on different distance functions; (a) original image with desired stone part indicated, (b) L 1, threshold=48, (c) L, threshold=25, (d) {3 10 1}, threshold=35. (a) (b) (c) Fig.2. Colour range histograms according to different distance functions; (a) L 1, (b) L, (c) {3 10 1}. Note that the histogram on Figure 2(c) has the best behaviour when the focus is on the modality of the first mode. This behaviour has led to the better quality of result shown in Figure 1(d). The colours of objects are usually not homogenous. A usual reason is that illumination causes many colour shades which can be nicely observed in our images captured by a microscope. In this case it is a natural improvement to consider more seed colours to describe the objects. Using this technique, the threshold values obviously should be decreased and less non-object points will be involved. The choice of the suitable distance functions can be performed with respect to each of the seed colours, separately. Our following example illustrates the improvement that can be achieved if we use more seed colours with smaller threshold values. In Figure 3 we improve the results shown in Figure 1 by considering more seed colours. 4.2 Region based segmentation It is a very usual requirement in image segmentation methods that the resulted segments should be connected, where connectedness is defined based on the wellknown 4- or 8-neighboring relations in 2D which are equivalent to the 1- and 2-neighbouring relations, respectively, in the present terminology. A digital set is 4-connected (resp. 8-connected) if any two of its points can be reached by such a sequence of points of the set that are 4-neighbours (resp. 8-neighbours).

8 8 András Hajdu, János Kormos, Benedek Nagy and Zoltán Zörgő (a) (b) (c) (d) Fig.3. Colour range from using three seed colours; (a) seed colour: =(223,121,158), distance: L, threshold: 20, (b) seed colour: =(222,129,124), distance: L 1, threshold: 40, (c) seed colour: =(228,134,135), distance: L, threshold: 10, (d) union of (a), (b), (c). To preserve connectivity we insert our approach for obtaining colour ranges into a region growing algorithm. Region growing techniques are classical (see [8, 19]) procedures for finding connected segments that consist of pixels with similar colours. Starting from a seed point (inner point of the region) and fixing a threshold value we merge those neighbours of the seed pixel whose colours are within a distance to the colour of the seed. Then we continue with merging those pixels that are neighbours of previously merged ones and also have colours within the given threshold to the seed. The procedure stops, when no more pixels can be merged in this way. It is clear that a natural extension of our colour range approach leads to a more general region growing method, as well. Namely, we achieve region growing if we require spatial connectivity to the seed beside considering colour information. In this way we can also take advantage of the use of our alternative colour distance measurement in case of region growing. Colour range histograms also can be extended naturally for region growing purposes. If x is the seed pixel then the k-th column of the histogram represents the number of those pixels y for which k d(f(x), f(y)) < k + 1 and there exists a 4-path between x and y such that for any element z of the path d(f(x), f(z)) T, where d and T are an appropriate distance function and threshold value, respectively. In Figure 4 we show an example for a region growing histogram. Pixels reached (a) Distance (b) Fig. 4. Region growing of stone parts; (a) selecting seed image pixel with seed colour =(85,145,59), (b) region growing histogram of the metric L.

9 Choosing appropriate distance measurement in digital image segmentation 9 Note that the shape of the histogram is like a cumulative typed one, and its jumps indicates those distance values, where remarkable number of new pixels are merged. That is the suggested choice of thresholds are the distance values correspond to the very beginning of the constant sections of the histogram, as it is shown in Figure 4(b). The region occupied by the algorithm till the indicated values can be observed in Figure 5. (a) (b) (c) (d) Fig.5. Occupied regions of region growing after the distance; (a) 22, (b) 95, (c) 122, (d) 144. The proposed region growing procedure naturally can be improved by using the ideas for optimisation (more seed colours, modality analysis to choose appropriate distance functions, etc.) discussed at our colour range approach. 4.3 Classifying colours with cluster analysis In such segmentation tasks if no a priori information or user interaction are available, more automatic methods are needed than e.g. the above ones. In such situations the widely applied methods of cluster analysis can be used. A huge number of clustering approaches are known, basically all of them are based on such distance measurement. Thus the question also arises here, whether the application of other distance functions then the classical ones may lead to better results. As alternative possibilities, we used digital distance functions generated by neighbourhood sequences again, and achieved nice results. To compare the performance of different distance functions first we considered hierarchical clustering [1] with moving centroids that are rounded for integer valued distance functions. To initialize this clustering procedure we put every every colour in the image into separate clusters. Then at every further steps we merge those two clusters which are closest to each other with respect to the chosen distance function. The algorithm stops when an initially fixed number of clusters is reached or when the distance of the closest two clusters is larger than an initially fixed threshold value. Since hierarchical clustering does not need any more pre-processing or additional arguments we considered it as the most objective approach to compare the efficiency of the distance functions. In Figure 6 we show the results of hierarchical clustering under some distance functions. It is always a very sophisticated problem in image processing to decide which result is the better. The reason is that the decision highly depends on the sub-

10 10 András Hajdu, János Kormos, Benedek Nagy and Zoltán Zörgő (a) (b) (c) (d) Fig. 6. Results of hierarchical clustering into six cluster by different distance functions; (a) original image, (b) L 1, (c) L, (d) {123}. jective impressions of the human receiver. That is why it is a usual practice in image processing to ask a group of people for taking sides in such questions. Nevertheless, these surveys are rather expensive and time consuming ones, and thus many formulas were composed to make these decisions based on quantitative measures. Such a measure is the one proposed by Levine and Nazif [12] to evaluate the results of the clustering method. In our investigations we used this uniformity measure with a slight modification and considered the measure E = 1 L σi 2 i=1 Lσ 2, where L is the number of final clusters, σi 2 is the i-th within-class variance, while σ is the between-class variance. Note that 0 E 1, and the smaller E is, the better the corresponding result can be assumed. In Table 2 we present our results on clustering Figure 6(a) into six clusters with different distance functions. E L 1 L {12} {123} { } {31} Hierarchical 0, , , , , ,07480 K-means 0, , , , , ,02612 Table 2. Levine and Nazif error after classifying into six clusters using different distance functions and clustering methods. The quantitative analysis in Table 2 nicely show that neighbourhood sequences are worth taking into considerations in clustering procedures, as well. In our corresponding project, that is the segmentation of stone parts, hierarchical clustering does not seem to be a good choice in general, since no hierarchy can be observed these types of images. That is why we observed another type of namely the K-means clustering method (see [1]). The classic version of this clustering determines K initial cluster centroids and then range the elements into the clusters with the closest centroids. Table 2 also contains our experimental results according to Levine and Nazif uniformity measure for the same task as in the case of hierarchical clustering. From the table we can see that

11 Choosing appropriate distance measurement in digital image segmentation 11 the K-means clustering procedure yields reliable results (at least quantitatively) than the hierarchical one. Moreover, these results show that distance functions based on neighbourhood sequences can be also nicely applied here, as well. 5 Conclusions In this paper we showed how the special family of distance distance functions generated by neighbourhood sequences can be applied successfully in colour image segmentation beside classical metrics. Especially, L p metrics are also can be approximated by distance functions based on neighbourhood sequences, [3, 5,9, 13]. It is also explained how the choice of different distance functions may improve the results. These methods are based on the suitable choice of a distance function and a threshold for those colours which are within this range from fixed seed colours. The models are given for the RGB colour representation, but they can be extended to arbitrary dimensions [4,7] and also to other types of grids [15]. By the help of the relation introduced in [4, 7] or the velocity value in [10], we can choose faster or slower distance functions or we can even ignore metrical properties [7, 14]. As application areas, we mentioned stone parts segmentation, where the main task is to extract the different stone types from an image captured by a microscope. In this topic there are still some unsolved problems like finding the suitable seed colours, threshold values and neighbourhood sequences to achieve better results. The precise mathematical solution of these problems seems to be difficult even for the reason that the goodness of the result may be a subjective decision. We defined some theoretical tools (histograms) that help with finding the suitable parameters, as the modes of the histograms will represent segment classes in the image. References [1] M.R. Anderberg, Cluster Analysis for Application, Academic Press, New York, [2] S.H. Cha and S.N. Srihari, On measuring the distance between histograms, Pattern Recognition 35 (2002), [3] P.E. Danielsson, 3D octagonal metrics, Eighth Scandinavian Conf. Image Process. (1993), [4] P.P. Das, P.P. Chakrabarti and B.N. Chatterji, Distance functions in digital geometry, Inform. Sci. 42 (1987), [5] P.P. Das and B.N. Chatterji, Octagonal distances for digital pictures, Inform. Sci. 50 (1990), [6] B. Everitt, Cluster Analysis, Heinemann Educational Books Ltd, London, [7] A. Fazekas, A. Hajdu and L. Hajdu, Lattice of generalized neighbourhood sequences in nd and D, Publ. Math. Debrecen 60 (2002), [8] R.C. Gonzalez and R.E. Woods, Digital image processing, Addison-Wesley, Reading, MA, 1992.

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