Edge Dipole and Edge Field for Boundary Detection Toshiro Kubota Department of Computer Science University of South Carolina Columbia, SC 29208
Terry Huntsberger Department of Computer Science University of South Carolina Columbia, SC 29208
Abstract
Conventionally, edges are treated as either scalar or vector quantities. This paper presents a novel framework which treats edges as directional dipoles that induce the eld around themselves. An analogy can be made between this concept and the interaction of magnetic dipoles with the magnetic eld. The dipoles interact with the eld and align themselves into a smooth contour con guration. This paper shows the e ectiveness of the concept in edge linking and proposes ecient computational schemes for real-time implementation of the edge dipole interactions. It also proposes an image representation using the dipoles on a hexagonal lattice and a contour extraction algorithm implemented on the representation. The algorithm consists of three processes: noise removal, edge alignment and edge thinning/extension. The results of some experimental studies are also presented.
1. INTRODUCTION
An edge is an important cue for the human vision processing and has been an important feature for various computer vision tasks, especially for boundary detection.3,17 The purpose of the boundary detection process is to divide an image into a set of disjoint regions where each region corresponds to a physically meaningful area. The process has been known to be one of most dicult problems in computer vision.1 It has to remove spurious edges, group them to form a continuous contour in a physically meaningful way, and is often required to extend the contour to ll gaps. The processes are often called perceptual organization especially when its focus is to group edges based on the symmetry, curvature and proximity information suggested by the Gestalt psychology.4,7,21 The eld has been studied extensively in both psychology and computer vision.5,8,9,15,20 Conventionally, edges are treated as scalar quantities. With this treatment, an edge shows only the existence of a boundary at the location and only the proximity information can be used for the purpose of edge grouping and contour segmentation. Edges can be treated as vector quantities so that the curvature information can be obtained easily to improve the perceptual organization process.18,23,16 Each vector valued edge is evaluated based on some local measurement to compute the likeliness of how it should be extended to complete the boundary. The local measurement is often called saliency.2,7,21,22 This paper treats edges not only as vectors but also as dipoles which induce a vector eld around them. Thus, edges are called edge dipoles and the eld is an edge eld. An analogy can be made with magnetic dipoles and the magnetic eld. The eld induced by an edge dipole exhibits smooth continuous ows from the positive pole to the negative pole. The eld in turn can be used to rotate other dipoles to align with the eld. A similar treatment has been proposed by Guy and Medioni7 where each edge element produces a eld called extension eld and the edges are grouped based on the accumulation of the eld vector. The major di erence is that our scheme provides an interaction mechanism with the dipole and the eld, while the extension eld is passive and has no in uence on the edges. Another di erence is that the edge dipole eld has circular ows while the extension eld does not. The circular ows diverging at the positive pole and converging at the negative pole provides a convenient mechanism for lateral inhibition and discourages thick boundaries from forming. As a matter of fact, the extension eld can be easily computed using the implementation scheme presented in this paper, and the edge dipole eld can be considered as its super-set. Due to the active interaction of the edge dipole elds, it is more robust to noise in the initial edge con guration. This type of medium to long range neural interactions have been reported recently in various forms.6,10,24 The authors can be reached at
[email protected] and
[email protected]
The total edge eld is a linear sum of the edge eld associated with each edge dipole. From the magnetic analogy, the interaction can be described as the minimization of the energy, X E = ?F~ m ~i (1) i
where F~ and m ~ i are the total eld and the ith edge dipole, respectively. Starting with a set of sparsely placed dipoles, which can be the result of some edge detection process, the minimization of the energy (1) produces the con guration of the dipoles where the total eld ows strongly between pairs of dipoles. The eld strength can be used to group the dipoles into disjoint sets and ll gaps to complete contours in smooth perceptually agreeable manners. This idea was implemented in a simple iterative edge linking procedure.11 A drawback of this edge linking procedure is that a large part of the information contained in the original image has been lost at the edge detection stage. Therefore, the linking process does not utilize all the information available. This paper presents an image representation using the edge dipoles. It is basically a gradient map on a hexagonal lattice and is complete in a sense that it contains all the information of the original image except the intensity o set. We also presents a contour extraction method implemented on the dipole representation. It consists of three subprocesses: non-linear di usion for spurious edge removal, edge alignment, and edge thinning/extension. The paper is organized as follows: Section 2 describes the interaction of edge dipoles and the eld, and shows how a smooth contour con guration can be obtained through the interaction. Section 3 gives the underlying requirements on the edge dipole eld in order for the eld to express non-vanishing convergent ows. Section 4 provides ecient computational schemes of the edge dipole eld interaction. Section 5 discusses the dipole representation of an image and the contour extraction process. The experimental results of the process will be given. Finally, Section 6 gives the summary and the conclusions.
2. MAGNETIC FIELD
Denote the positional vector from the origin to the positive and negative poles as ~+ and ~? , respectively. Then the magnetic vector eld at r is de ned by # " ~f(r) = m ~r ? ~+ ? ~r ? ~? (2) j~r ? ~+ j3 j~r ? ~? j3 where m is the strength of the dipole. The eld decays quadratically with the distance from the dipole. Figure 1 shows a magnetic dipole and the eld induced by it. The thick short arrow in the center of the image represents the dipole, where the direction of the arrow is the direction from the negative pole to the positive pole of the dipole. The thiner and longer arrows represent the magnetic eld at the location. The eld wraps around smoothly from the positive dipole to the negative dipole. The total magnetic eld is the sum of the eld induced by each dipole. Thus, ~ = X f~i (r) (3) F(r) i
where f~i is the magnetic eld induced by the ith magnetic dipole. Imagine that magnets are placed on a smooth frictionless surface. Each magnet is pinned down to the surface through a hole at the center of the magnet so that it cannot shift its position but can rotate without any friction around its center. Then the magnets rotate themselves and quickly settle into a stable state. The state is the minimum of the energy (1). We say that the magnets are aligned at this stable state. This magnet and eld interaction can be applied nicely to the edge linking problem. Assume each magnet represents an edge element, the direction of the edge is normal to the gradient vector, and the cross product of the gradient and the edge direction, G m ~ , points upward towards the reader (as shown in Figure 2) where G is the gradient vector. The direction vector for the edge will always have the lower intensity region to its left. An edge
Figure 1. Single Magnetic Dipole and the Magnetic Field. This gure shows single magnetic dipole
shown with a thick short arrow at the center of the gure and the magnetic elds shown by thin long arrows.
Gradient Vector
p1 p2
Edge Dipole Vector
p1 p2
p1 > p2
p1 < p2
Figure 2. The Direction of the Edge Dipole. This gure describes how the dipole direction and polarity are determined through the intensity gradient of an image. (a) and (b) show two di erent examples.
con guration with a small energy measured by (1) corresponds to a smoother more likely edge con guration, while a con guration with a large energy corresponds to a non-smooth or an impossible con guration of edges. For example, the edge con guration shown in Figure 3(a) has a large energy. This con guration is impossible since the direction of the edge implies that p1 > p2 and p3 < p4 . However, since there is no edge between p1 and p3, it is assumed that p1 = p3 , and similarly, p2 = p4. Therefore, there is a contradiction and this con guration is not feasible. With the same argument, the con guration shown in Figure 3(b) is not feasible because p1 < p2 , p2 < p3, but p3 < p1 , which results in contradiction. Due to the repelling force from the 3 magnets, the con guration has very large energy. The con guration shown in Figure 3(c) is feasible and corresponds to a T-junction. The aligned state of the 3 magnets is shown in 3(d) and represents a Y-junction. If there is enough support from the surrounding area, the con guration stays as the T-junction. A nal set of examples are shown in Figure 4. The con guration shown in Figure 4(a) is possible and indicate that the region surrounded by the four dipoles has a smaller intensity than the outer region. On the other hand, the con guration shown in Figure 4(b) is impossible. As a matter of fact, the aligned con guration of 4(b) is the same as that of in Figure 4(a) with the exception of the leftmost vector which has the opposite orientation. Figure 4(c) shows a feasible con guration where two edges pointing the same direction lie closely together. This type of the con guration corresponds to a thick edge. The energy associated with this con guration is rather high, since the eld induced by the upper dipole at the location of the lower dipole runs in the opposite direction to the direction of the lower dipole, and vice versa. If one has a much higher pole strength than the other, at the aligned state, the
p1
p3
p2
p4
p1
(a)
p2
p1
p2
p1
p2
p3
p3
p3
(b)
(c)
(d)
Figure 3. Simple Edge Con gurations I. This gure shows simple edge con gurations. (a) an impossible con guration, (b) an impossible con guration, (c) a T-junction, (d) a Y-junction is the aligned state of (c).
(a)
(b)
(c)
Figure 4. Simple Edge Con gurations II. This gure shows more edge con gurations. (a) aligned edges, (b) not aligned edges, and (c) a con guration of a thick edge.
weaker dipole changes its direction to comply with the eld induced by the stronger dipole. Thus, a strong edge suppresses weak edges of the same direction nearby, thus inhibiting thick edges from forming. The simple examples shown above demonstrate how the edge con guration can be evaluated using the energy associated with the con guration if the edges are treated as magnetic dipoles. Therefore, by minimizing the energy of the con guration with respect to the orientation of each edge dipole, the edges align to form a smoother contour. Moreover, two aligned edge dipoles create a strong edge dipole eld between them and the eld strength can be used to select two edge segments to be connected. In general, the eld strength between two dipoles decreases as the curvature of the dipole vectors increases. Thus, the eld strength can be treated as a saliency measure. The amount of contribution by a dipole to the eld decreases as the distance from the dipole increases, which agrees with the saliency measure suggested by Shashua and Ullman.21
3. EDGE DIPOLE FIELD
This section describes constraints imposed on the edge dipole eld. They are imposed so that the resulting eld exhibits smooth convergent non-vanishing ows. Without a loss of generality, assume that the midpoint of the dipole is situated on the origin of the space coordinate so that the positive and negative poles are located at and ?, respectively. De ne a domain D as r 2 R2; r 6= and the eld is de ned in the domain D. Denote ~r as the positional vector from the origin to r. Thus, ~ represents the positional vector to the positive pole. The constraints based on the edge dipole eld are as follows: 1. Conservative eld or equivalently,
f~ = ?r (r)
(4)
r f~ = 0:
(5)
The scalar function, , is called the potential. 2. Balanced poles The positive and negative poles create the potentials + and ?, with + (~r) = ? ? (~r)
(6)
N2
N1 P1 N3
P2
P3 P4 P5 N5
N4
P6 N6
Hexagonal Lattice and its Neighborhood System. This gure shows the hexagonal lattice
Figure 5.
used for the experiments throughout this paper. For each grid, there are 6 neighbors Ni and the distance to every neighbor grid is the same.
3. Isotropy of the potential
+ (~r) = (j~rj)
where is a scalar function on r > 0, r 2 R. 4. Monotonic potential d (x) < 0; d2 (x) > 0 (x > 0) dx dx2 5. Local support limx!1 (x) = 0
(7)
(8) (9)
3.1. Properties
Based on the constraints listed above, the following properties on the edge eld are obtained.11 1. The eld exist everywhere in D. Moreover, the eld diverges from the positive pole and converges into either the negative pole or 1. 2. krf~k2 is bounded in D. Note that rf~ is a second order tensor. 3. The eld is anti-symmetric around the midpoint of the dipole. Since the midpoint is assumed to be at the ~ ?r). origin, f~(r) = f( 4. The eld equivalent to (2) can be generated with (x) = x1 .
3.2. Edge Linking
The concept of the edge eld/dipole interaction is applied to an edge linking problem. The algorithm consists of two stages: the edge alignment stage and the linking stage. At the alignment stage, each edge dipole rotates gradually to the edge eld to minimize the energy (1). At the linking stage, new edge dipoles are created at the location where the eld strength is more than a threshold value. This stage also deletes an edge if the eld at the location is opposite to the direction of the dipole.11 We employ a hexagonal lattice where each grid point has 6 immediate neighbors with the equal distance throughout the lattice (Figure 5). The advantage of the hexagonal lattice over more conventional rectangular lattices is that the dipole interaction with its neighbors does not have directional preferences. The rectangular lattice in general has directional preferences toward the horizontal and vertical directions over the diagonal directions since the diagonal distance between a diagonal neighbor pair is larger than a horizontal or vertical pair.
(a)
(b)
Figure 6. The Edge Pattern. This gure shows two sparse edge con gurations used for edge linking test.
(a)
(b)
The Edge Linking Results. This gure shows the results of the edge linking process applied to the con guration shown in Figure 6.
Figure 7.
Some test patterns are shown in Figure 6 and the results of the linking are shown in Figure 7. The threshold is 1:3m which is large enough not to create new edges around a single dipole but small enough to create edges along the direction with more than 4 aligned edges. The algorithm successfully connected sparsely located edge elements in a way that agrees with human perception.
4. IMPLEMENTATION
The direct computation of the edge dipole interactions is O(N 2M 2 ) where N 2 is the size of the image and M is the dipole interaction range which can be as big as N for long range interactions. The worst case, therefore, the complexity grows O(N 4 ). This section proposes two implementation schemes. Both implement the interaction using the idea of the steerable decomposition.13,14 One implements steerable base lters in the frequency domain and reduces the complexity to O(QN 2logN) where Q is the number of the base lters. The other implements the lters using a set of separable lters and constructs a multi-resolution representation of the eld in O(PQmN 2) where P is the separable approximation order and m is the length of each 1D wavelet lter for the multi-resolution decomposition. The latter is more suitable for a real-time implementation and applications can take advantage of the multi-resolution structure.
4.1. Fourier Domain Implementation
Each dipole contributes to the eld by (2). The horizontal component of the eld at r induced by a dipole at s can be written as ! ? +x r ? s r ? s x x x ? = m(r ? s; ) (10) fx (r; s) = m j~r ? s~+ j j~r ? s~? j
Figure 8. The Steerable Base Filters. These lters are used to steer the edge- eld generation lter, in 11.
where s+ and s? are the locations of the positive and negative poles, respectively, > 1 is a constant, and rx ? cos() rx + cos (r; ) = ? (11) = 2 2 2 ((rx ? cos) + (ry ? sin) ) ((rx + cos)2 + (ry + sin)2 ) =2 with rx and ry being the x and y components of r, respectively. With = 3, (10) is the same as the x component of (2). The x component of the total eld at r is Fx (r) =
X
i
mi fx (r; si) =
X
i
mi (r ? si ; i )
(12)
Since is 2 periodical with respect to , it can be decomposed into12 (r; ) =
Q X k
(r)qk ():
(13)
Now using (13), (12) becomes Fx(r) =
X
i
mi (r ? si ; i ) =
X
i
mi
X
k
k (r ? si )qk (i ) =
XX
k
i
mi qk (i )k (r ? si ):
(14)
The above equation, (14), is merely a sum of the images obtained by convolving mi qk (i ) with the space invariant lter, k (r). Therefore, the above operation can be implemented on the Frequency domain using FFT, and the complexity of the operation becomes O(QN 2logN). Similarly, the y component of the eld at r induced by a dipole at s is !
r ? s+ r ? s? fy (r; s) = m y ~+y ? y ~?y = m(r ? s; ) j~r ? s j j~r ? s j where
(15)
ry + sin ry ? sin ? : (16) = 2 2 2 ((rx ? cos) + (ry ? sin) ) ((rx + cos)2 + (ry + sin)2 ) =2 Figure 8 shows the three base lters, k with = 2. These three real lters cover 99.95 percent energy of the eld generation lter, , and 6 and 8 real lters cover 99.999 percent and 100.0 percent, respectively. (r; ) =
4.2. Multi-resolution Implementation
The base lters can be decomposed into a sum of separable lters by the Singular Value Decomposition, and each lter can be approximated by the correlation of a basic spline lter and a discrete lter. The discrete lter becomes the discrete wavelet basis for the multi-resolution decomposition of an image.12 With this technique, a long-range interaction of the edge eld can be implemented using a series of convolution operations with the discrete wavelet basis followed by decimation operations. The complexity of the eld interaction using this scheme becomes O(PQmN 2) where P is the order of the separable approximation, and m is the length of the wavelet lters. The separable approximation of order 6 is enough to cover 99.5 percent of the energy of the set of steerable lters, k .
5. DIPOLE REPRESENTATION AND CONTOUR EXTRACTION
This section describes an image representation using the edge dipoles and a contour extraction process implemented on the representation.
5.1. Dipole Representation
Denote I[m; n] as the original image intensities de ned on the lattice L. Assume that the original function I(x; y) can be approximated by an interpolation of some kernel (x; y). Then
and
I(x; y)
X
rI(x; y)
X
m;n m;n
I[m; n] (x ? m; y ? n)
(17)
I[m; n]r (x ? m; y ? n):
(18)
It is well known that a band-limited function I can be recovered with a sinc function if the sampling rate is above the Nyquist limit. Here, due to computational eciency, a linear interpolation is used on the hexagonal lattice. Then each triangle in the neighborhood system forms a planar surface. Moreover the gradient at each grid point is approximated by the average of the gradients of the 6 neighboring surfaces and can be computed by X (19) m ~ = 61 (Ik ? I)~pk k where Ik is the image intensity of the kth immediate neighbor and ~pk is the unit vector pointing to the kth neighbor (Figure 5). Using the approximated gradient, the edge dipole at every grid is obtained as described in Section 2 using Figure 5. It can be observed that the representation is complete in a sense that the original image can be recovered if the o set of the image or the intensity value at one location is known. In order to apply the computation schemes described in the previous section on the hexagonal lattice, the steering base lters, k has to be mapped onto the rectangular lattice, which produces two sets of lters, kodd and fkeveng where the former is used on the odd rows and the latter used on the even rows of the image.
5.2. Algorithm
Our task is to extract contours from the dipole representation described above. Initially, the representation contains noise or spurious edges and con gurations of thick edges. The task is divided into three processes: noise removal to remove spurious edge dipoles which are not part of real contours, edge alignment to adjust the eld into smoother
ows, and edge thinning and extension through a local interaction of the dipoles. The edge alignment has been described in Section 2. The other two processes are described below.
For the purpose of removing noise or spurious edges, non-linear di usion of the type r c(krf k)rf is applied where c(x) is a monotonically decreasing function of x with the range [1; 0).19 In the dipole representation, the magnitude of a dipole represent the magnitude of the gradient. Thus, the di usion corresponds approximately to dmi / ?c(m ) m > 0: (20) i i dt where m is the magnitude of the dipole. Here, the di usion coecient c(x) is rather a function of mF where F is the total eld strength at the location. The quantity, F=m can be considered as a saliency measure of the edge normalized by its edge strength. This way, weak edges will be kept if the saliency of the edge is high. In our experiments, c(x) is chosen to be 1=(1 + ex ). The di usion (20) constantly reduces m. In order to prevent all the edge dipoles to disappear as t ! 1, the di usion process is modi ed so that the process conserves the amount of the dipole strength P mi at any time point. The modi ed di usion process is dmi / ?c(m ) + (t) 0 < m < m ; (21) i i max dt where (t) is the average value of c(m). The idea of the non-linear di usion is also adapted for edge thinning and extension. In order to extend the contour, the di usion has to be performed along the tangent directions to the dipole. In order to apply thinning, negative di usion or lateral inhibition is applied along the perpendicular directions to the dipole. Here, the di usion coecient c is set to cos(2AB ) so as to provide a smooth transition from the positive di usion to the negative di usion, where AB is the angle between m ~ A , the dipole in question, and the vector from m ~ A to the other dipole, m ~ B . Similar di usion is also asserted from m ~ B to m ~ A with the di usion coecient cos(2BA ) where BA is the angle between m ~ B and the vector pointing to m ~ A from m ~ B . Note that the amount of the di usion between m ~ A and m ~ B are not balanced since AB 6= BA in general. Thus, the process does not conserve the total strength of the dipoles in the representation. Moreover, the amount of di usion should decrease as the distance between two dipoles increases. The following modi ed edge thinning/extension process conserves the total strength and the amount of the interaction decreases quadratically as the distance between the pair increases. dmA / X mB ? mA ?cos2 ? sin2 (22) AB BA dt rAB 2 B where rAB is the distance between m ~ A and m ~ B.
5.3. Results
The processes of the noise removal and the edge thinning/extension coupled with the dipole alignment discussed in Section 2 are used to extract contours from the dipole representation of an image. Three processes are applied iteratively on the representation starting with the noise removal followed by the alignment and the thinning/extension. Figure 9 shows an input image and the result of the contour extraction. The input consists of many spurious edges and low contrast fuzzy edges. The process extracted boundaries under shadows where the intensity di erences are very subtle as well as the roof boundaries at the sky. Figure 10 shows another result of the application of the multi-step process. The middle image is the initial dipole representation. Only the magnitudes of the dipoles are shown. The bottom image is the result of the noise removal, alignment and thinning/extension. The result shows clearer and more descriptive representation of the contours than the initial representation.
6. SUMMARY AND CONCLUSIONS
This paper presented the concept of the edge dipoles, the edge eld, and the interaction of the eld and the dipoles. It also proposed ecient computation schemes of the interactions, the image representation using the edge dipoles, and a contour extraction process applied directly on the representation. Experiments with both arti cial and real-world images demonstrated that the representation captures the underlying edge geometry and was robust in the presence of noise and gaps. The whole concept can be described in short as anisotropic and dynamic neighborhood interactions and can be applied to many other vision problems. More studies are underway to apply the concept for image restoration,
Figure 9. The Contour Extraction Result I. This gure shows the input image (left) and the dipole magnitude after the extraction process (right).
texture analysis and motion analysis. It is also interesting to unify other types of medium to long range anisotropic interactions based on the edge dipole eld framework. Our future research direction for the improvement of the contour extraction algorithm includes 1) improvement of the edge thinning/extension process where the current implementation exhibit slow convergence, and 2) development of multi-resolution based algorithms.
Acknowledgments
This research is supported in part under ONR Grant No. N00014-94-1-1163 and ARO Grant No. DAAH04-96-10326.
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Figure 10.
The Contour Extraction Result II.
This gure shows the input image (top), the initial dipole magnitude (middle) and the dipole magnitude after the extraction process (bottom).
