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Finding contour-based abstractions of planar patterns
0031 3203/93 $6.00+.00 Pergamon Press Ltd © 1993 Pattern Recognition Society
Pattern Recognition, Vol. 26, No. 10, pp. 1563 1577, 1993 Printed in Great Britain
FINDING CONTOUR-BASED ABSTRACTIONS OF PLANAR PATTERNS CARLO ARCELLIand GIULIANARAMELLA Istituto di Cibernetica, C.N.R., 1-80072 Arco Felice, Naples, Italy (Received 19 August 1992; in revised form 10 March 1993; receivedfor publication 7 April 1993) Abstract--An algorithm is described to detect a number of points, on the contour of a planar shape, which constitute the vertices of a schematic polygonal representation of the shape itself. A set of points, initially extracted from the chain-coded representation of the contour, is iteratively examined, while removing some points and inserting new ones. The number of selected points decreases in size from iteration to iteration, and the selection process converges towards an expected perceptually significant set of points. The polygon obtained by linking successive points approximates the contour in an intuitive way. It is not constrained within a given tolerance, and is likely to locally change from a coarse to a more faithful approximating shape, in correspondence with contour regions increasing in details.
Dominant points Perceptual significance Polygonal representation
Accumulated evidence
Iterated selection
with a sequence of line segments fitting the contour, so as to reduce a chosen error criterion, (ii) by detecting The attention of an observer is often attracted by the and linking the feature points. regions of an object perceived as discontinuities with The first approach, also known as polygonization, respect to an otherwise fiat environment. In the case has greatly developed in the past, and most of its facets of single-valued planar objects, where relevant shape can be found in textbooks, t~s'19~ On the contrary, the information can be found on the contour, curved enough second approach is still stimulating an increasing contour arcs are instances of such regions. These arcs number of papers t2°-2g) so that its characterization is include points which give an appealing abstraction of not yet completely neat. In this paper, we too follow the shape, as soon as they are linked by straight line the feature point detection approach. Roughly speaking, segments, in the order they are successively encountered one approach is the dual of the other (the first performs when going around the object. "'2~ We refer to these side detection and the second performs vertex detection), points as feature points. Feature points cause contour but it is possible to catch a glimpse of a basic difference partitioning and can be regarded as the vertices in a between the polygons they can originate. In the first polygonal representation of the object. case, all the sides of the polygon are within the same Since the early times of pattern recognition research, a priori given tolerance with respect to the contour arcs criteria for contour partitioning and representation they approximate; in the second case, the significance techniques have been topics of interest/3'4~ In particular, of the representation is given, rather than by the accura polygonal representation is mostly free of the effects acy of the approximation, by the stability of the vertices, of digitization noise and is more manageable, as it is "anchored" to contour positions understood as percepexpressed concisely by a small number of points. More- tually meaningful. Accordingly, the polygon is likely over, its structure is easier to describe in terms of to become an abstraction of the shape, where the sides its geometry or of the way suitable sets of sides are are allowed to approximate the contour arcs at different degrees of accuracy: more sides being needed only organized. Polygonal representation has been popular in ap- when required to account for the presence of more plication areas such as chromosome analysis, character details. recognition and industrial parts classification, and is In the literature, curvature extrema have often been still appealing for its simplicity and robustness. For regarded as candidate feature points, and several algorinstance, it has recently been taken into account to deal ithms have been designed which need one or more with tasks such as the recognition of key-sections,15~ input parameters to delimit the size of a region of image segmentation in the leather industry, 16~and face support, tuned to give evidence to the details of interest. identification on crystal cross-section images/7~ More- In fact, the methods adopted to compute curvature on over, besides its use in former times for shape decom- the digital plane are mostly angle detection procedures, position and description,ta-l°~ it is profitably employed and require that the computation be referred to a for shape comparison and identification."1-~6~ prespecified supporting arc. ~29~A shortcoming is that, Generally, polygonal representation is obtained in if the contour includes details of different size, the two ways: 117~(i) by approximating the contour curve selected region of support will be too large for the 1. I N T R O D U C T I O N
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detection of fine details (and the contour will locally be represented as a smoothed one), and will be too small to find only one feature point in correspondence with details detectable within a greater support (so that a number of redundant feature points may be found). When the problem domain requires a shape abstraction mirroring both coarse and fine details into the polygonal representation, one can resort to procedures needing no input parameters. These allow the detection of feature points having different perceptual relevance by using only the information concerning the geometry of the contour. 13°'31) Alternatively, a solution can be found by analyzing the contour at different resolution scales and finding the polygon either directly(3z) or after processing the derived scale-space diagram, where the feature points detected at different resolutions are organized in compact form. (33) We look for a representation of the overall shape, accounting for both the perceptual relevance of its constituting contour arcs and for the spatial extent of the regions one can perceive as enclosed by the selected contour arcs. The polygon should approximate the shape in a way which should not be constant and not constrained by an a priori given tolerance, and we regard the goal of capturing the essential nature of the shape as attained if the structure of the representing polygon does not meaningfully change under pattern rotation. This paper can be related to previous research (34-36) focusing on the use of criteria of perceptual significance, based on certain properties of points with respect to a chord, to detect the feature points. We describe an iterative procedure which allows one to find a number of feature points, relative to details of different size, without explicitly exploiting curvature information, and without using input parameters to delimit the regions over which the feature points are computed. At every iteration, the significance of a contour point is evaluated with respect to geometrical parameters which are derived from the global structure of the shape. The process terminates when evidence of the significance of each point has been accumulated during the previous iterations. Section 2 introduces the perceptual criteria taken into account to select the feature points, as well as the notion of dominance adopted in this paper. It also describes how the set of feature points is updated. Section 3 lists the steps accomplished during each iteration of the procedure, and Section 4 shows some results of the experimental work carried out. Section 5 includes some remarks on the contribution.
succeeding points with reference to this order in a circular way. For the sake of simplicity and without loss of generality, we refer to objects with no holes. In the following, we use interchangeably the terms point and pixel. Selected points are called vertices. A preliminary step, aimed at extracting the initial set of vertices from the chain-coded version of the contour, is premised to the iterative procedure. Let V(°) indicate this set, and no its cardinality. Starting from V(°), the procedure originates a sequence of new sets of vertices V (~), V (2).... , V(y), each one with decreasing cardinality (no > nl > n2 > '.. > nl), though not necessarily properly included by the preceding set. The (k + l)th iteration step (k = 0 . . . . . f ) has the set V ~k~as input, and the set V (k + 1)as output. The ( f + 1)th iteration step has the output V(I + 1)coinciding with the input V(I). The number of vertices decreases in size from iteration to iteration, and the vertex selection process converges towards an expected perceptually significant set of feature points. 2.1. Selection criteria Let V (k) = {vl kl, i = 1. . . . . nk} be the set of nk vertices input to the (k + 1)th iteration. We assume that every vertex vlk) is representative of the contour arc having as extremes its two adjacent vertices v~l,~i+t,',~k)and relate its significance to the geometry of the triangle TI k) including these vertices and vl k). The a r e a a~ k) of the triangle, the cosine cosl k) of the internal angle relative to vl k), and the distance dl k) of vJk) from the chord (vi_ (k) t, v~(k) + 1) are taken as parameters of perceptual significance (see Fig. 1). This choice is done to take into account factors influencing human perception such as the size of the region viewed by v~k~ (roughly related to the area alk) of a protrusion or of an intrusion of the object) and the "cornerity" (roughly related to cos~k)) of the contour arc surrounding v~k). Moreover, dl k) is related to an often used measure of perceptual significance, defined as the pointwise error between a contour arc and the corresponding side in the approximating polygon. The significance of a vertex is evaluated in the context of the overall shape, by referring to the mean value of the area and of the cosine (denoted below by Atk) and COS (k), respectively) relative to the triangles associated to all the vertices of V (kl. A (k) and COS (g) are terms of comparison for al k) and coslk), and guide the process of vertex selection by giving evidence to the vertices
I
(k) Vi+l
2. CONTOUR ANALYSIS
The picture including the object of interest is scanned in forward raster fashion. The object contour is assumed to be an 8-connected digital curve, which is traced anti-clockwise and chain-coded (37~ starting from the first pixel hit by the scan. As an order on all the contour points is introduced, we use the terms preceding and
~)
vi-I
II
I
~ill i i i i i I I I f (at) v 1
Fig. l. Triangle with reference to which the significance ofl)~k} is computed.
Finding contour-based abstractions of planar patterns vlk>, such that ali~ or cosl ~) overcome A ~ and COS ~kl, respectively. Among the possible types of comparison, the experiments carried out have suggested to take into account only the relations "greater than" and "equal to" as explained in Condition S1 below. As for dl~, it is used to accomplish a local check on the significance of vl~) by comparing its value with that of dl~ ~ and dl~ 1We call any vertex vl~) significant for which the following conditions are simultaneously satisfied.
If If If If
existed in the past. The strength of the dominance is expressed by an accumulator z! k~, which is associated to each vertex as soon as this is ascribed to Vtkl, k > 0. When k = 0, the initial value of zl kJ is set to zero. When k > 0, the initial value is set either to zero or to one, according to the status of the accumulators relative to the pixels in V ~k+ 11 (see below). During the (k + 1)th iteration, the value ofzl k) is updated as follows, depending on how Conditions S1 and $2 are satisfied by the triangle currently associated to vl kl.
both Conditions S1 and $2 are false Condition S1 is false and Condition $2 is true: Condition S1 is true and Condition $2 is false: both Conditions S 1 and $2 are true:
Condition SI: {a~kl > A 'kl} or
{cosl kl > C O S ~k~} or
{cosl k) = COS ~k~ and k > 0 and COS ~kl4: COS ~k- 1~}. Condition $2: dl~1 is not a local minimum. When Conditions S1 and $2 are not satisfied, vlk) identifies a triangle where alkl and cos~k) are both nonsignificant with respect to the global pictorial context, expressed by the mean values A ~k~and COS~k~;and the local m i n i m u m dl kl implies that the contour arc including vlk~ could be substituted by the chord ,-,-~,~i+,)~'!kl . (k~ without introducing a dramatic decrease in the approximation. To express computationally how the factors influencing our perception of shape operate is a problem which we propose to face by taking into account, besides the simultaneous fulfilment of both Conditions S1 and $2, also the individual fulfilment of either of Conditions S1 and $2. This is done by ascribing a score to the perceptual significance of Conditions S 1 and S2, and by considering an iterated process converging towards a stable vertex organization. Vertex significance depends on distribution and number of vertices under examination. Since vertex selection introduces a new distribution, the significance of a vertex may vanish at the next iteration. We define the dominance of a vertex as the quality of being significant at one or more iterations. A vertex present in successive sets of feature points is ascribed a dominance which is stronger the greater the n u m b e r of iterations during which it is significant. When a vertex is found to be not significant, we do not cancel immediately its dominance if the history of the vertex shows that a strong enough dominance
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zl~l = z I ~ 1 - 2 z1.1 = z l ~1 - 1 zl kl = zl kl + 1.
To decide whether a v e r t e x 11k} can be ascribed to the set V tk + 11, one has to check the updated value of the accumulator associated to that point. The vl kl with non-negative accumulator values are directly ascribed to the set V ~k+ 11, while the ones with negative accumulator values are no longer regarded as candidates. To avoid a too drastic reduction in the n u m b e r of vertices, what follows is applied to any set of non-isolated vertices with negative accumulator value (a vertex with negative accumulator value is termed isolated if the accumulator values of both its neighboring vertices are not negative). tk~ Let qo(k~ and q~+ 1, be two vertices with non-negative accumulator value which enclose a n u m b e r (m > I) of consecutive vertices q]kl,...,q~l having negative accumulator value. See Fig. 2(a), where m = 5. We check the significance of the medial points r~kt ( j = 1.... , m - 1) of the contour arcs delimited by the p a i r s (q(tkl, q~2kl),(q(2k), q~) . . . . . (q~)- t, q~). A medial point r~k) is regarded as significant and taken as a new vertex if it satisfies Condition $3 below, with reference to the triangle T~ k~ defined by r~k~ and by the two nearest points (to its right and its left) in ~,~/0 S~k> , ',~k) 1 , ' ' ' , r~kt m l,qm+l}, already confirmed as vertices. Condition $3: {~kl> A,kt} or {Cos~kI>COS 'kl} where a~k~and cos~k~denote the area of T~k~and the cosine of the angle relative to r~~, respectively. The initial value of the accumulator associated to each new vertex is set to zero if it is zero the value of at least one accumulator relative to the pixels already directly ascribed to V~ + 11; is set to one, otherwise. To reduce asymmetry in the distribution of the found vertices, we examine the medial points in alternate order from the left and from the right, i.e. the first, the last, the second, the penultimate and so on. For instance, let us consider the arc in Fig. 2(a), where the consecutive vertices q~, q z.... , q5 with negative accumulator value are enclosed by qo and q6 with non-negative accumulator value. The set of medial points {rl,rz . . . . . r4}
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a)
abstraction is not too drastic, an additional termination condition has been introduced to force the convergence of the algorithm towards a polygon having almost the same spatial extension of the input. Let -~(o) and x , ~~(o) y e(k + 1), I~y ~(k+ 1) be respectively the maximum horizontal x and vertical length of the polygon having as vertices the elements of V (°) and of the set V (k +1), k = 0 , . . . , f. Then, independently of the accumulator values, the process stops at the (k + 1)th iteration, and returns as output the set V(k),if the following condition is satisfied:
q$ -qel
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lel
i i lo
q2
~lllllJJ
I I
Condition T: l e ~ ) - e ~ + ' ) I > t . e ~ ) or le(r°, --e~k+~)I > t*e(r °) where t is a positive n u m b e r smaller than 1.
b) q0
I I I I I Iol
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At iteration (k + 1), the steps performed on the set V Ik) are as follows:
r3
q
l c)
qo
ill,~.,,I
IIIl
r1 Fig. 2. (a) Points ql, qz..... qs having negative accumulator value. They are substituted by the points rl and r 4 in (c), selected among the r~ shown in (b).
(shown separately in Fig. 2(b), for ease of illustration) is processed in the order r~, r4, r2, r 3. Significance evaluation (i.e. checking of Condition $3) of the point r~ is accomplished on the triangle defined by qo, rl, q6. Then, if r I is significant, the significance of r 4 is evaluated on the triangle r l , r 4 , q 6 , otherwise on qo, ra, q6. For the sake of simplicity, suppose that both r~ and r 4 are significant, this implies that the significance of r 2 is checked on r l, r 2, r 4. Then the significance of r3 is checked on r 1, r3, r 4 if r 2 is not significant, on r2, r 3, r 4 otherwise. If both r 2 and r 3 are not significant, the vertices found are those shown in Fig. 2(c). 3. ITERATION STEPS The initial set V (°) is obtained after visiting twice the chain-coded version of the contour. During the first inspection, for every run of equal chain-codes, the first extreme of the run is ascribed to V(°). During the second inspection, any pixel previously ascribed to V(°) whose differential chain-code is smaller than that of an adjacent pixel, is substituted in V (°) by the adjacent pixel. The process should terminate at the iteration ( f + 1), in correspondence of which none of the accumulators associated to the vertices belonging to V (:) assumes negative value, i.e. when V (f+ 1 ) coincides with V (f). This termination condition is generally adequate to originate a polygonal representation which accords well with h u m a n intuition. However, to ensure that the
(1) For all the elements of V(k), compute the area al k), the cosine cosl k), and the distance dl k) relative to the triangle associated to each vlk). (2) Compute the mean area A (k)and the mean cosine COS (k)" (3) For all the elements of V (k), check Conditions S1 and $2, and update their corresponding accumulator. Let V (k+ 1) be the set of pixels with non-negative accumulator value. (4) If V (k) = V (k + l), exit. Otherwise go to Step 5. (5) For any subset of consecutive points q~k). . . . . q~) with negative accumulator value, detect the new points satisfying Condition $3, and ascribe them to V (k÷~). Initialize the accumulator value of the new points. (6) Check Condition T on V (k+l). If it is satisfied, exit with the set V (k). Otherwise, set k = (k + l) and return to Step 1.
4. EXPERIMENTALRESULTS The algorithm has been tested on several patterns, some of which are already used in the literature for illustrative purposes (see Fig. 3). The pattern shown in Fig. 4 was used in reference (30) to compare different well-known algorithms with a new one, which does not require input parameters to delimit the region of support. Since that algorithm (3°) showed a good performance, we used those results as a reference for the ones we obtained. Figure 5(a) shows the vertices found on a pattern characterized by details of different size. A somewhat similar pattern is meaningfully partitioned in reference (32), by examining the contour at seven resolution scales. In our output, the vertices are placed in correspondence with perceptually meaningful regions, but it seems that two or three more vertices could originate a better polygonal representation. As one could expect from a selection mechanism placing shape abstraction before accuracy in the approximation, also this example shows that we get a smaller set of vertices with respect to the quoted algorithm, t3°) Figure 6 illustrates the results on two patterns which have been digitized after being rotated. O u r experiments showed that stability is a property difficult to
Finding contour-based abstractions of planar patterns
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Fig. 3. Feature points found on test patterns used in the literature.
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a)
b)
Fig. 4. Results on a symmetric pattern: (a) the proposed algorithm; (b) the algorithm given in reference (30).
a) Fig. 5. Results on a pattern having details of different size; (a) the proposed algorithm; (b) the algorithm given in reference (30).
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b) Fig. 5. (Continued.)
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Fig. 6. Feature points in differently oriented patterns.
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Finding contour-based abstractions of planar patterns
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Fig. 7(a). Feature points selected by the proposed algorithm.
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Finding contour-based abstractions of planar patterns
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Fig. 7(d). Feature points selected by the proposed algorithm (continued).
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Fig. 7(f). Feature points selected by the proposed algorithm (continued).
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ensure, since the number/or the location of the feature points may partially change under pattern rotation. However, even with a different n u m b e r of sides, the shapes of the resulting polygons generally preserved their similarity. This result appears non-negligible if we take into account the relatively small cardinality of the set of vertices we obtain. Finally, Fig. 7 illustrates the performance of the algorithm on several examples having different size and typology.
change its size and the point itself may be not significant. Notwithstanding this, the dominance of the point may persist until the last iteration. The points in the final set are d o m i n a n t points, in the sense that evidence of their significance has been accumulated through a sufficient number of iterations. Acknowledgements--The help of Mr Salvatore Piantedosi in
preparing the illustrations is gratefully acknowledged.
REFERENCES 5. CONCLUDING REMARKS
We have presented an algorithm to detect on a planar shape a set of suitable feature points, which constitutes a perceptually significant abstraction of the shape itself. The set of feature points is initially extracted from the chain-coded contour, by considering one of the extremes of the equally coded runs. Then, the set is iteratively examined, while removing some feature points and inserting new ones. The obtained polygonal representation appears sufficient to capture the essential features of the perceived shape, and to preserve symmetry, if this property characterizes the pattern. The approximation of the polygon to the contour is not constrained within a given tolerance, but is expected to locally change from a coarse to a more faithful one, as soon as one moves towards regions increasing in detail. This peculiarity makes a quantitative comparison with the existing algorithms difficult, and the comparison with the algorithm 13°1 included in this paper should be regarded as relatively meaningful. Generally, algorithms exhibit different performances depending on the values ascribed to the various parameters used to control some crucial steps in the procedure. Thus, in accordance with what has been argued in a recent paper, 13s) it should be necessary to face the whole problem of performance characterization, so as to establish a precise protocol to make possible well-founded comparisons. As comparisons are not fully reliable, we should regard the features of the method, rather than the obtained results, as the major contribution of this paper. The feature points have been extracted without explicitly exploiting curvature information and different paradigms for curve partitioning, as discussed in reference (32), have been taken into account when designing the phases of the point selection process. Points are found at locations along the contour from where they can dominate a significant triangular region, the significance of a region being derived both from the distribution of the currently existing feature points and from the geometry of the shape. The set of points is iteratively modified, by substituting a smaller set of new points to any set of consecutive feature points, whose dominance is regarded as weak in the global context. New feature points are inserted at locations along the contour that allow a more symmetric as possible subdivision of the contour arc to be partitioned. At a given iteration, the arc dominated by a feature point may
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About the Author--CARLO ARCELLIreceived the doctoral degree in physics from the University of Bologna, Italy, in 1969. Since 1970, he has been working at the Istituto di Cibernetica of the National Research Council, Naples, where he has done research in the field of picture processing. His current interests are mainly in shape analysis and description.
About the Author--GIULIANA RAMELLAreceived the doctoral degree in physics in 1990, from the University of Naples, Italy. In 1991, she joined the Istituto di Cibernetica of the National Research Council, Naples, to carry out research in the area of intermediate-level picture processing. Her current activity is concerned with the representation of bidimensional patterns in terms of boundary and region information.
Pattern Recognition, Vol. 26, No. 10, pp. 1563 1577, 1993 Printed in Great Britain
FINDING CONTOUR-BASED ABSTRACTIONS OF PLANAR PATTERNS CARLO ARCELLIand GIULIANARAMELLA Istituto di Cibernetica, C.N.R., 1-80072 Arco Felice, Naples, Italy (Received 19 August 1992; in revised form 10 March 1993; receivedfor publication 7 April 1993) Abstract--An algorithm is described to detect a number of points, on the contour of a planar shape, which constitute the vertices of a schematic polygonal representation of the shape itself. A set of points, initially extracted from the chain-coded representation of the contour, is iteratively examined, while removing some points and inserting new ones. The number of selected points decreases in size from iteration to iteration, and the selection process converges towards an expected perceptually significant set of points. The polygon obtained by linking successive points approximates the contour in an intuitive way. It is not constrained within a given tolerance, and is likely to locally change from a coarse to a more faithful approximating shape, in correspondence with contour regions increasing in details.
Dominant points Perceptual significance Polygonal representation
Accumulated evidence
Iterated selection
with a sequence of line segments fitting the contour, so as to reduce a chosen error criterion, (ii) by detecting The attention of an observer is often attracted by the and linking the feature points. regions of an object perceived as discontinuities with The first approach, also known as polygonization, respect to an otherwise fiat environment. In the case has greatly developed in the past, and most of its facets of single-valued planar objects, where relevant shape can be found in textbooks, t~s'19~ On the contrary, the information can be found on the contour, curved enough second approach is still stimulating an increasing contour arcs are instances of such regions. These arcs number of papers t2°-2g) so that its characterization is include points which give an appealing abstraction of not yet completely neat. In this paper, we too follow the shape, as soon as they are linked by straight line the feature point detection approach. Roughly speaking, segments, in the order they are successively encountered one approach is the dual of the other (the first performs when going around the object. "'2~ We refer to these side detection and the second performs vertex detection), points as feature points. Feature points cause contour but it is possible to catch a glimpse of a basic difference partitioning and can be regarded as the vertices in a between the polygons they can originate. In the first polygonal representation of the object. case, all the sides of the polygon are within the same Since the early times of pattern recognition research, a priori given tolerance with respect to the contour arcs criteria for contour partitioning and representation they approximate; in the second case, the significance techniques have been topics of interest/3'4~ In particular, of the representation is given, rather than by the accura polygonal representation is mostly free of the effects acy of the approximation, by the stability of the vertices, of digitization noise and is more manageable, as it is "anchored" to contour positions understood as percepexpressed concisely by a small number of points. More- tually meaningful. Accordingly, the polygon is likely over, its structure is easier to describe in terms of to become an abstraction of the shape, where the sides its geometry or of the way suitable sets of sides are are allowed to approximate the contour arcs at different degrees of accuracy: more sides being needed only organized. Polygonal representation has been popular in ap- when required to account for the presence of more plication areas such as chromosome analysis, character details. recognition and industrial parts classification, and is In the literature, curvature extrema have often been still appealing for its simplicity and robustness. For regarded as candidate feature points, and several algorinstance, it has recently been taken into account to deal ithms have been designed which need one or more with tasks such as the recognition of key-sections,15~ input parameters to delimit the size of a region of image segmentation in the leather industry, 16~and face support, tuned to give evidence to the details of interest. identification on crystal cross-section images/7~ More- In fact, the methods adopted to compute curvature on over, besides its use in former times for shape decom- the digital plane are mostly angle detection procedures, position and description,ta-l°~ it is profitably employed and require that the computation be referred to a for shape comparison and identification."1-~6~ prespecified supporting arc. ~29~A shortcoming is that, Generally, polygonal representation is obtained in if the contour includes details of different size, the two ways: 117~(i) by approximating the contour curve selected region of support will be too large for the 1. I N T R O D U C T I O N
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detection of fine details (and the contour will locally be represented as a smoothed one), and will be too small to find only one feature point in correspondence with details detectable within a greater support (so that a number of redundant feature points may be found). When the problem domain requires a shape abstraction mirroring both coarse and fine details into the polygonal representation, one can resort to procedures needing no input parameters. These allow the detection of feature points having different perceptual relevance by using only the information concerning the geometry of the contour. 13°'31) Alternatively, a solution can be found by analyzing the contour at different resolution scales and finding the polygon either directly(3z) or after processing the derived scale-space diagram, where the feature points detected at different resolutions are organized in compact form. (33) We look for a representation of the overall shape, accounting for both the perceptual relevance of its constituting contour arcs and for the spatial extent of the regions one can perceive as enclosed by the selected contour arcs. The polygon should approximate the shape in a way which should not be constant and not constrained by an a priori given tolerance, and we regard the goal of capturing the essential nature of the shape as attained if the structure of the representing polygon does not meaningfully change under pattern rotation. This paper can be related to previous research (34-36) focusing on the use of criteria of perceptual significance, based on certain properties of points with respect to a chord, to detect the feature points. We describe an iterative procedure which allows one to find a number of feature points, relative to details of different size, without explicitly exploiting curvature information, and without using input parameters to delimit the regions over which the feature points are computed. At every iteration, the significance of a contour point is evaluated with respect to geometrical parameters which are derived from the global structure of the shape. The process terminates when evidence of the significance of each point has been accumulated during the previous iterations. Section 2 introduces the perceptual criteria taken into account to select the feature points, as well as the notion of dominance adopted in this paper. It also describes how the set of feature points is updated. Section 3 lists the steps accomplished during each iteration of the procedure, and Section 4 shows some results of the experimental work carried out. Section 5 includes some remarks on the contribution.
succeeding points with reference to this order in a circular way. For the sake of simplicity and without loss of generality, we refer to objects with no holes. In the following, we use interchangeably the terms point and pixel. Selected points are called vertices. A preliminary step, aimed at extracting the initial set of vertices from the chain-coded version of the contour, is premised to the iterative procedure. Let V(°) indicate this set, and no its cardinality. Starting from V(°), the procedure originates a sequence of new sets of vertices V (~), V (2).... , V(y), each one with decreasing cardinality (no > nl > n2 > '.. > nl), though not necessarily properly included by the preceding set. The (k + l)th iteration step (k = 0 . . . . . f ) has the set V ~k~as input, and the set V (k + 1)as output. The ( f + 1)th iteration step has the output V(I + 1)coinciding with the input V(I). The number of vertices decreases in size from iteration to iteration, and the vertex selection process converges towards an expected perceptually significant set of feature points. 2.1. Selection criteria Let V (k) = {vl kl, i = 1. . . . . nk} be the set of nk vertices input to the (k + 1)th iteration. We assume that every vertex vlk) is representative of the contour arc having as extremes its two adjacent vertices v~l,~i+t,',~k)and relate its significance to the geometry of the triangle TI k) including these vertices and vl k). The a r e a a~ k) of the triangle, the cosine cosl k) of the internal angle relative to vl k), and the distance dl k) of vJk) from the chord (vi_ (k) t, v~(k) + 1) are taken as parameters of perceptual significance (see Fig. 1). This choice is done to take into account factors influencing human perception such as the size of the region viewed by v~k~ (roughly related to the area alk) of a protrusion or of an intrusion of the object) and the "cornerity" (roughly related to cos~k)) of the contour arc surrounding v~k). Moreover, dl k) is related to an often used measure of perceptual significance, defined as the pointwise error between a contour arc and the corresponding side in the approximating polygon. The significance of a vertex is evaluated in the context of the overall shape, by referring to the mean value of the area and of the cosine (denoted below by Atk) and COS (k), respectively) relative to the triangles associated to all the vertices of V (kl. A (k) and COS (g) are terms of comparison for al k) and coslk), and guide the process of vertex selection by giving evidence to the vertices
I
(k) Vi+l
2. CONTOUR ANALYSIS
The picture including the object of interest is scanned in forward raster fashion. The object contour is assumed to be an 8-connected digital curve, which is traced anti-clockwise and chain-coded (37~ starting from the first pixel hit by the scan. As an order on all the contour points is introduced, we use the terms preceding and
~)
vi-I
II
I
~ill i i i i i I I I f (at) v 1
Fig. l. Triangle with reference to which the significance ofl)~k} is computed.
Finding contour-based abstractions of planar patterns vlk>, such that ali~ or cosl ~) overcome A ~ and COS ~kl, respectively. Among the possible types of comparison, the experiments carried out have suggested to take into account only the relations "greater than" and "equal to" as explained in Condition S1 below. As for dl~, it is used to accomplish a local check on the significance of vl~) by comparing its value with that of dl~ ~ and dl~ 1We call any vertex vl~) significant for which the following conditions are simultaneously satisfied.
If If If If
existed in the past. The strength of the dominance is expressed by an accumulator z! k~, which is associated to each vertex as soon as this is ascribed to Vtkl, k > 0. When k = 0, the initial value of zl kJ is set to zero. When k > 0, the initial value is set either to zero or to one, according to the status of the accumulators relative to the pixels in V ~k+ 11 (see below). During the (k + 1)th iteration, the value ofzl k) is updated as follows, depending on how Conditions S1 and $2 are satisfied by the triangle currently associated to vl kl.
both Conditions S1 and $2 are false Condition S1 is false and Condition $2 is true: Condition S1 is true and Condition $2 is false: both Conditions S 1 and $2 are true:
Condition SI: {a~kl > A 'kl} or
{cosl kl > C O S ~k~} or
{cosl k) = COS ~k~ and k > 0 and COS ~kl4: COS ~k- 1~}. Condition $2: dl~1 is not a local minimum. When Conditions S1 and $2 are not satisfied, vlk) identifies a triangle where alkl and cos~k) are both nonsignificant with respect to the global pictorial context, expressed by the mean values A ~k~and COS~k~;and the local m i n i m u m dl kl implies that the contour arc including vlk~ could be substituted by the chord ,-,-~,~i+,)~'!kl . (k~ without introducing a dramatic decrease in the approximation. To express computationally how the factors influencing our perception of shape operate is a problem which we propose to face by taking into account, besides the simultaneous fulfilment of both Conditions S1 and $2, also the individual fulfilment of either of Conditions S1 and $2. This is done by ascribing a score to the perceptual significance of Conditions S 1 and S2, and by considering an iterated process converging towards a stable vertex organization. Vertex significance depends on distribution and number of vertices under examination. Since vertex selection introduces a new distribution, the significance of a vertex may vanish at the next iteration. We define the dominance of a vertex as the quality of being significant at one or more iterations. A vertex present in successive sets of feature points is ascribed a dominance which is stronger the greater the n u m b e r of iterations during which it is significant. When a vertex is found to be not significant, we do not cancel immediately its dominance if the history of the vertex shows that a strong enough dominance
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zl~l = z I ~ 1 - 2 z1.1 = z l ~1 - 1 zl kl = zl kl + 1.
To decide whether a v e r t e x 11k} can be ascribed to the set V tk + 11, one has to check the updated value of the accumulator associated to that point. The vl kl with non-negative accumulator values are directly ascribed to the set V ~k+ 11, while the ones with negative accumulator values are no longer regarded as candidates. To avoid a too drastic reduction in the n u m b e r of vertices, what follows is applied to any set of non-isolated vertices with negative accumulator value (a vertex with negative accumulator value is termed isolated if the accumulator values of both its neighboring vertices are not negative). tk~ Let qo(k~ and q~+ 1, be two vertices with non-negative accumulator value which enclose a n u m b e r (m > I) of consecutive vertices q]kl,...,q~l having negative accumulator value. See Fig. 2(a), where m = 5. We check the significance of the medial points r~kt ( j = 1.... , m - 1) of the contour arcs delimited by the p a i r s (q(tkl, q~2kl),(q(2k), q~) . . . . . (q~)- t, q~). A medial point r~k) is regarded as significant and taken as a new vertex if it satisfies Condition $3 below, with reference to the triangle T~ k~ defined by r~k~ and by the two nearest points (to its right and its left) in ~,~/0 S~k> , ',~k) 1 , ' ' ' , r~kt m l,qm+l}, already confirmed as vertices. Condition $3: {~kl> A,kt} or {Cos~kI>COS 'kl} where a~k~and cos~k~denote the area of T~k~and the cosine of the angle relative to r~~, respectively. The initial value of the accumulator associated to each new vertex is set to zero if it is zero the value of at least one accumulator relative to the pixels already directly ascribed to V~ + 11; is set to one, otherwise. To reduce asymmetry in the distribution of the found vertices, we examine the medial points in alternate order from the left and from the right, i.e. the first, the last, the second, the penultimate and so on. For instance, let us consider the arc in Fig. 2(a), where the consecutive vertices q~, q z.... , q5 with negative accumulator value are enclosed by qo and q6 with non-negative accumulator value. The set of medial points {rl,rz . . . . . r4}
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a)
abstraction is not too drastic, an additional termination condition has been introduced to force the convergence of the algorithm towards a polygon having almost the same spatial extension of the input. Let -~(o) and x , ~~(o) y e(k + 1), I~y ~(k+ 1) be respectively the maximum horizontal x and vertical length of the polygon having as vertices the elements of V (°) and of the set V (k +1), k = 0 , . . . , f. Then, independently of the accumulator values, the process stops at the (k + 1)th iteration, and returns as output the set V(k),if the following condition is satisfied:
q$ -qel
I I I { iol
%
lel
i i lo
q2
~lllllJJ
I I
Condition T: l e ~ ) - e ~ + ' ) I > t . e ~ ) or le(r°, --e~k+~)I > t*e(r °) where t is a positive n u m b e r smaller than 1.
b) q0
I I I I I Iol
I I
| IOl I I I
r2
At iteration (k + 1), the steps performed on the set V Ik) are as follows:
r3
q
l c)
qo
ill,~.,,I
IIIl
r1 Fig. 2. (a) Points ql, qz..... qs having negative accumulator value. They are substituted by the points rl and r 4 in (c), selected among the r~ shown in (b).
(shown separately in Fig. 2(b), for ease of illustration) is processed in the order r~, r4, r2, r 3. Significance evaluation (i.e. checking of Condition $3) of the point r~ is accomplished on the triangle defined by qo, rl, q6. Then, if r I is significant, the significance of r 4 is evaluated on the triangle r l , r 4 , q 6 , otherwise on qo, ra, q6. For the sake of simplicity, suppose that both r~ and r 4 are significant, this implies that the significance of r 2 is checked on r l, r 2, r 4. Then the significance of r3 is checked on r 1, r3, r 4 if r 2 is not significant, on r2, r 3, r 4 otherwise. If both r 2 and r 3 are not significant, the vertices found are those shown in Fig. 2(c). 3. ITERATION STEPS The initial set V (°) is obtained after visiting twice the chain-coded version of the contour. During the first inspection, for every run of equal chain-codes, the first extreme of the run is ascribed to V(°). During the second inspection, any pixel previously ascribed to V(°) whose differential chain-code is smaller than that of an adjacent pixel, is substituted in V (°) by the adjacent pixel. The process should terminate at the iteration ( f + 1), in correspondence of which none of the accumulators associated to the vertices belonging to V (:) assumes negative value, i.e. when V (f+ 1 ) coincides with V (f). This termination condition is generally adequate to originate a polygonal representation which accords well with h u m a n intuition. However, to ensure that the
(1) For all the elements of V(k), compute the area al k), the cosine cosl k), and the distance dl k) relative to the triangle associated to each vlk). (2) Compute the mean area A (k)and the mean cosine COS (k)" (3) For all the elements of V (k), check Conditions S1 and $2, and update their corresponding accumulator. Let V (k+ 1) be the set of pixels with non-negative accumulator value. (4) If V (k) = V (k + l), exit. Otherwise go to Step 5. (5) For any subset of consecutive points q~k). . . . . q~) with negative accumulator value, detect the new points satisfying Condition $3, and ascribe them to V (k÷~). Initialize the accumulator value of the new points. (6) Check Condition T on V (k+l). If it is satisfied, exit with the set V (k). Otherwise, set k = (k + l) and return to Step 1.
4. EXPERIMENTALRESULTS The algorithm has been tested on several patterns, some of which are already used in the literature for illustrative purposes (see Fig. 3). The pattern shown in Fig. 4 was used in reference (30) to compare different well-known algorithms with a new one, which does not require input parameters to delimit the region of support. Since that algorithm (3°) showed a good performance, we used those results as a reference for the ones we obtained. Figure 5(a) shows the vertices found on a pattern characterized by details of different size. A somewhat similar pattern is meaningfully partitioned in reference (32), by examining the contour at seven resolution scales. In our output, the vertices are placed in correspondence with perceptually meaningful regions, but it seems that two or three more vertices could originate a better polygonal representation. As one could expect from a selection mechanism placing shape abstraction before accuracy in the approximation, also this example shows that we get a smaller set of vertices with respect to the quoted algorithm, t3°) Figure 6 illustrates the results on two patterns which have been digitized after being rotated. O u r experiments showed that stability is a property difficult to
Finding contour-based abstractions of planar patterns
i
l~llllllllllllll~
Fig. 3. Feature points found on test patterns used in the literature.
ii|
a)
b)
Fig. 4. Results on a symmetric pattern: (a) the proposed algorithm; (b) the algorithm given in reference (30).
a) Fig. 5. Results on a pattern having details of different size; (a) the proposed algorithm; (b) the algorithm given in reference (30).
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b) Fig. 5. (Continued.)
m
l l E I l l
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i 111
~ , l l l | l l l |
~ 1 1 1 J l l l
Fig. 6. Feature points in differently oriented patterns.
| t i i 11
i 1 iTrn'T'rTr'n
Finding contour-based abstractions of planar patterns
~1~
ItI. iTrls ,Xjllllll~
Fig. 7(a). Feature points selected by the proposed algorithm.
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c~ m~
~J
o
o
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Finding contour-based abstractions of planar patterns
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iiiiiii rlllJ,,ixll9
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...... J va~-u-uo~nn~tal~ Fig. 7(c). Feature points selected by the proposed algorithm (continued).
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I l I l l t
Fig. 7(d). Feature points selected by the proposed algorithm (continued).
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g.
o.
~
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j~L~...... U~rr~ ..... ~.........
I nlT~rllT 1
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Fig. 7(f). Feature points selected by the proposed algorithm (continued).
•(panu!luoo) tuqluo~l~ pasodoad aql ;{q palaolos slu!od aanl~azl '(~)L "~!~l L1i i i ~
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suaall~d aeu~id jo suo!lo~alsq~ pasuq-anoluoo Su!pu!~I
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ensure, since the number/or the location of the feature points may partially change under pattern rotation. However, even with a different n u m b e r of sides, the shapes of the resulting polygons generally preserved their similarity. This result appears non-negligible if we take into account the relatively small cardinality of the set of vertices we obtain. Finally, Fig. 7 illustrates the performance of the algorithm on several examples having different size and typology.
change its size and the point itself may be not significant. Notwithstanding this, the dominance of the point may persist until the last iteration. The points in the final set are d o m i n a n t points, in the sense that evidence of their significance has been accumulated through a sufficient number of iterations. Acknowledgements--The help of Mr Salvatore Piantedosi in
preparing the illustrations is gratefully acknowledged.
REFERENCES 5. CONCLUDING REMARKS
We have presented an algorithm to detect on a planar shape a set of suitable feature points, which constitutes a perceptually significant abstraction of the shape itself. The set of feature points is initially extracted from the chain-coded contour, by considering one of the extremes of the equally coded runs. Then, the set is iteratively examined, while removing some feature points and inserting new ones. The obtained polygonal representation appears sufficient to capture the essential features of the perceived shape, and to preserve symmetry, if this property characterizes the pattern. The approximation of the polygon to the contour is not constrained within a given tolerance, but is expected to locally change from a coarse to a more faithful one, as soon as one moves towards regions increasing in detail. This peculiarity makes a quantitative comparison with the existing algorithms difficult, and the comparison with the algorithm 13°1 included in this paper should be regarded as relatively meaningful. Generally, algorithms exhibit different performances depending on the values ascribed to the various parameters used to control some crucial steps in the procedure. Thus, in accordance with what has been argued in a recent paper, 13s) it should be necessary to face the whole problem of performance characterization, so as to establish a precise protocol to make possible well-founded comparisons. As comparisons are not fully reliable, we should regard the features of the method, rather than the obtained results, as the major contribution of this paper. The feature points have been extracted without explicitly exploiting curvature information and different paradigms for curve partitioning, as discussed in reference (32), have been taken into account when designing the phases of the point selection process. Points are found at locations along the contour from where they can dominate a significant triangular region, the significance of a region being derived both from the distribution of the currently existing feature points and from the geometry of the shape. The set of points is iteratively modified, by substituting a smaller set of new points to any set of consecutive feature points, whose dominance is regarded as weak in the global context. New feature points are inserted at locations along the contour that allow a more symmetric as possible subdivision of the contour arc to be partitioned. At a given iteration, the arc dominated by a feature point may
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About the Author--CARLO ARCELLIreceived the doctoral degree in physics from the University of Bologna, Italy, in 1969. Since 1970, he has been working at the Istituto di Cibernetica of the National Research Council, Naples, where he has done research in the field of picture processing. His current interests are mainly in shape analysis and description.
About the Author--GIULIANA RAMELLAreceived the doctoral degree in physics in 1990, from the University of Naples, Italy. In 1991, she joined the Istituto di Cibernetica of the National Research Council, Naples, to carry out research in the area of intermediate-level picture processing. Her current activity is concerned with the representation of bidimensional patterns in terms of boundary and region information.