Distance Measures between Digital Fuzzy Objects ... - Semantic Scholar

Distance Measures between Digital Fuzzy Objects and Their Applicability in Image Processing ´ c1 , Joakim Lindblad2 , and Nataˇsa Sladoje3 Vladimir Curi´ 1

Centre for Image Analysis, Uppsala University, Sweden [email protected] 2 Centre for Image Analysis, Swedish University of Agricultural Sciences, Uppsala, Sweden and Mathematical Institute, Serbian Academy of Sciences and Arts, Belgrade, Serbia [email protected] 3 Faculty of Technical Sciences, University of Novi Sad, Serbia [email protected]

Abstract. We present two different extensions of the Sum of minimal distances and the Complement weighted sum of minimal distances to distances between fuzzy sets. We evaluate to what extent the proposed distances show monotonic behavior with respect to increasing translation and rotation of digital objects, in noise free, as well as in noisy conditions. Tests show that one of the extension approaches leads to distances exhibiting very good performance. Furthermore, we evaluate distance based classification of crisp and fuzzy representations of objects at a range of resolutions. We conclude that the proposed distances are able to utilize the additional information available in a fuzzy representation, thereby leading to improved performance of related image processing tasks. Keywords: Fuzzy sets, set distance, registration, classification.

1

Introduction

Distances between sets are useful for many different applications in image processing, for instance, object matching [3, 8], image registration [12] and image retrieval [11]. However, it is a challenging task to differentiate between crisp discrete representations of similar objects if the spatial resolution is insufficient. An example of such a situation can be seen in Fig. 1, where a crisp discrete representation of a disk, Fig. 1B, a crisp discrete representation of an octagon, Fig. 1E, are difficult to visually associate with the correct type of continuous shape (disk and octagon). In recent years, fuzzy approaches have gained increased popularity in image processing. In [16, 17, 18] it is shown that a fuzzy representation provides higher precision and accuracy of different feature estimates than a crisp one. Keeping this fact in mind, it can be assumed that distances between fuzzy objects representations provide better discriminatory power than distances between crisp J.K. Aggarwal et al. (Eds.): IWCIA 2011, LNCS 6636, pp. 385–397, 2011. c Springer-Verlag Berlin Heidelberg 2011

386

´ c, J. Lindblad, and N. Sladoje V. Curi´

A

B

C

D

E

F

Fig. 1. A: Continuous crisp disk, B: Crisp discrete representation of a continuous disk (obtained by Gauss centre point digitization), C: Fuzzy discrete representation of a continuous disk (obtained by coverage digitization), D: Continuous crisp octagon, E: Crisp discrete representation of a continuous octagon, F: Fuzzy discrete representation of a continuous octagon

representations at the same resolution. This assumption is supported by the fact that a fuzzy discrete representation of an object (see Fig. 1C and Fig. 1F) often appears visually more similar to the corresponding continuous object, than a crisp representation at the same resolution. These observations motivate the study on distance measures between fuzzy sets presented in this paper. It is recently shown that the Sum of minimal distances and the Complement weighted sum of minimal distances have good performances for binary image registration [12,7]. We extend these distances to distances between fuzzy sets by following two different approaches. Having on mind matching and registration as possible applications, we evaluate distance measures with respect to several criteria of importance for such applications. We study whether the distances are monotonically increasing with respect to increasing translation and rotation of the object. Additionally, we investigate noise sensitivity of the observed distances. To further evaluate the proposed approach, we compare the correct classification rates, when utilizing the best performing fuzzy and crisp set distances for discriminating discrete representations of disks and octagons at a range of resolutions.

2 2.1

Background Basic Notions

A fuzzy set S on a reference set X, is a set of ordered pairs S = {(x, μS (x)) : x ∈ X}, where μS : X → [0, 1] is the membership function of the fuzzy set S, [19]. Many concepts of fuzzy sets are based on α−cuts. An α−cut of a fuzzy set S, is the set αS = {x ∈ X : μS (x) ≥ α}, α ∈ (0, 1]. Height of a fuzzy set S is h(S) = max μS (x), while the support of S is defined as Supp(S) = {x ∈ X : μS (x) > 0}. x∈X

The complement S of a fuzzy set S, is S = {(x, 1 − μS (x)) : x ∈ X}. The fuzzy representation of objects used in this paper is pixel coverage representation [16], where membership of a pixel is equal to the relative area of the pixel that is covered by the object. More formally, for a given continuous object O ⊂ R2 , inscribed into an integer grid with pixels p(i,j) , the n−level quantized pixel coverage digitization of the object O is

Distance Measures between Digital Fuzzy Objects

A(p(i,j) ∩ O) 1 1 2 + (i, j), n : (i, j) ∈ Z , n A(p(i,j) ) 2

387

(1)

where A(X) denotes the area of the set X, and x denotes the largest integer which is not greater than x. The set {0, n1 , ..., n−1 n , 1} represents the pixel coverage values for n−level quantized pixel coverage digitization. This set corresponds to the set of non-zero membership levels, e.g., n = 1 for a binary images, while n = 255 provides the set of membership levels for an 8-bit image. We use this representation since it has been shown to provide higher precision and accuracy of different feature estimates than the crisp object representation [16, 17]. It is also corresponds well with the outcome of many imaging situations. However, the applicability of the methods presented in this study is not restricted to this type of fuzzy representations. 2.2

Related Work on Distances between Crisp Sets

Distances between two crisp sets of points A, B ⊂ Zn , A, B = ∅, are mostly based on the point-to-set distance; the point-to-set distance between point a and set B is defined as (2) d(a, B) = inf d(a, b). b∈B

For the underlying point-to-point distance d(a, b), we use the Euclidean distance, i.e., d(a, b) = a − b2 . The first proposed distance between two sets A and B, and widely used in different applications, is the Hausdorff distance, dH , defined as dH (A, B) = max(sup d(a, B), sup d(b, A)). a∈A

b∈B

(3)

The Hausdorff distance is highly dependent on two points from the observed sets and, hence, sensitive to outliers. Several modifications of the Hausdorff distance are introduced to reduce the influence of outliers to the distance measure [8]. In [12] it is suggested that a distance measure applicable for image registration related problems should simultaneously: (i) utilize all points from the set; (ii) consider spatial position of the points and the sets. One distance that fulfills these conditions is the Sum of minimal distances, proposed in [9],

1 d(a, B) + d(b, A) . (4) dSMD (A, B) = 2 a∈A

b∈B

This distance has good performance for image registration [12]. The Sum of minimal distances is investigated further and the Complement weighted sum of minimal distances, dCW , is proposed in [7]. In dCW each point in the set is weighted by the distance to the complement of the set ⎛ ⎞ d(b, A) · d(b, B) d(a, B) · d(a, A) 1 ⎜ a∈A b∈B ⎟ dCW (A, B) = ⎝ + (5) ⎠. 2 d(a, A) d(b, B) a∈A

b∈B

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Based on the empirical evaluation in [7], it is concluded that the Complement weighted sum of minimal distances, among the considered distance measures, has the best performance for binary image registration. 2.3

Related Work on Distances between Fuzzy Sets

There exist a number of different distances defined for fuzzy sets. A good overview on distance measures between fuzzy sets can be found in [2]. Some of the measures provide a fuzzy number as output [15]. However, we consider only distances between fuzzy sets that provide crisp distance values; for the observed application (object matching) it is not obvious how to use distance values represented by fuzzy numbers. There exist several different possibilities how to extend a crisp distance to a fuzzy one [1]. The most common approach is to use integration over α−cuts [14]. Integration over α−cuts is a general principle for extension of properties and relations on crisp sets to corresponding ones on fuzzy sets. A distance measure between two fuzzy sets A and B can be defined by integration over all α−cuts, 1 α d(αA,αB) dα, (6) d (A, B) = 0

where d is a crisp set distance. For example, the Hausdorff distance between two fuzzy sets A and B is defined as, [14] 1 α dH (A, B) = dH (αA,αB) dα. (7) 0

A main drawback with this approach is that dα H (A, B) = ∞ if the heights of the two observed fuzzy sets are not the same. Several variations have been proposed to solve this problem [4, 6, 15, 10]. However, a perfect solution for this problem is not found yet [5]. Even if most of the distances between crisp sets of points rely on the pointto-set distance, the distances between two fuzzy sets are usually not defined using the point-to-set distance for fuzzy sets. Two definitions of the point-toset distance for fuzzy sets are proposed in [1]. The first definition is based on integration over α−cuts, where the distance between point a and fuzzy set B is defined as 1

d(a, B) =

d(a,αB) dα =

0

1

min d(a, b) dα.

α 0 b∈ B

(8)

The second definition is based on weighting of the points from the support, Supp(B), of the fuzzy set B with their membership values, d(a, B) =

min

b∈Supp(B)

(d(a, b) · F (μB (b)),

(9)

where F (t) is a decreasing function of t. The point-to-point distance d(a, b) is the spatial distance between two points and is independent on their membership values. A point-to-set distance can also be defined using fuzzy morphology, but a value of such distance is in general represented by a fuzzy number [1].


3

389

The Sum of Minimal Distances and Complement Weighted Sum of Minimal Distances for Fuzzy Sets

In this section we extend the Sum of minimal distances and the Complement weighted sum of minimal distances to distances between fuzzy sets. For that purpose we use (6) and (8). A requirement imposed by both (6) and (8) is that observed fuzzy sets have the same height, since, otherwise, integration over α−cuts leads to that the distance is equal to infinity (similar as for the Hausdorff distance on fuzzy sets). Sum of minimal distances and Complement weighted sum of minimal distances for fuzzy sets can be defined using (6) as 1 (A, B) = dSMD (αA,αB) dα, (10) dα SMD 0

dα CW (A, B) =

1

dCW (αA,αB) dα.

(11)

0

Instead, using (8), Sum of minimal distances and Complement weighted sum of minimal distances for fuzzy sets are defined as ⎛ ⎞ 1 ⎝ d(a, B) + d(b, A)⎠ , dps (12) SMD (A, B) = 2 ⎛ dps CW (A, B) =

a∈Supp(A)

b∈Supp(B)

d(a, B) · d(a, A)

1 ⎜ a∈Supp(A) ⎝ 2

a∈Supp(A)

d(a, A)

+

b∈Supp(B)

d(b, A) · d(b, B)

b∈Supp(B)

d(b, B)

⎞ ⎟ ⎠. (13)

Similarly, the Hausdorff distance using (8) has the form dps H (A, B)

= max

sup a∈Supp(A)

d(a, B),

sup b∈Supp(B)

d(b, A) .

(14)

We do not use point-to-set distance (9) since it is not clear which function F to use. We performed test with different decreasing functions, but we did not observe any good performance. Furthermore, if a ∈ Supp(B), then for b = a d(a, b) = 0 ⇒ d(a, b) · F (μ(b)) = 0 ⇒ d(a, B) = 0, which, we feel, further reduces the discriminative power, and thereby the usefulness of this distance definition.

4

Evaluation

This section presents the results of an empirical study of the observed distances. The distances are studied with respect to monotonicity, as well as with respect to noise sensitivity. In addition we compare classification performance of the best of the proposed distances for fuzzy sets, with the best performing one for crisp sets, on fuzzy and crisp discrete representations of objects, respectively.

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Fig. 2. Examples of tested fuzzy objects

4.1

Evaluation of Monotonicity in Noise Free and Noisy Conditions

For a good matching or registration performance it is desirable that the distance monotonically increases with increasing translation and rotation of the object. Therefore, we study how the distance changes when an object is translated and rotated with respect to the non-transformed object. We say that monotonicity of a distance measure, w.r.t. translation (rotation) is fulfilled for a particular object, if the distance between the object and a translated (rotated) version of the same object does not decrease with increasing translation (rotation). Translation is performed in steps of one pixel horizontally, up to the width of the considered object. Rotation is performed in positive direction around the center of mass of the object, in steps of one degree, up to 23, and up 45 degrees, in two separate tests. At each step the distance is computed between the transformed and the non-transformed image. This is similar to tests performed in [12] for crisp set distances. Rotation of the discrete objects requires interpolation. We tested linear and nearest neighbor interpolation and we selected nearest neighbor interpolation since it provided better performance. The tests are performed on fuzzy objects obtained from 100 binary images taken from [13]. To obtain fuzzy objects we perform pixel coverage digitization using sub-sampling by a factor 6, which provides 36 different membership levels. Some of the observed fuzzy objects can be seen in Fig. 2. Since some of the objects are rotationally symmetric (see Fig. 2C), monotonicity of the distance with respect to rotation of such objects can not be expected. Since the membership functions of all the observed objects are of the same height (equal to one), the requirement for set distances based on integration over α-cuts is satisfied. For any real application it is important that the distance measure is not too sensitive to noise present in the images. We perform tests with two types of noise perturbing the membership values of the observed objects: 1. Additive noise – pixel values are perturbed by Gaussian noise with zero mean and 0.01 variance. 2. Multiplicative noise – pixel values are perturbed by multiplicative uniformly distributed noise with zero mean and variance 0.04.


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Fig. 3. Different noise conditions. A: Noise free image, B: Image with Additive noise, C: Image with Multiplicative noise

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80 60 40 20 0

dH

dSMD

dCW

dα H

dα SMD

dα CW

dps H

dps SMD

dps CW

Distance measure

Fig. 4. Percentage of images where monotonicity is fulfilled in noise free case, for each distance and each of the observed rigid transformations

We assume that the pixels with membership value 1 are correctly classified as inner pixels of the object and we, therefore, only apply noise on the boundary of the fuzzy objects. Fig. 3 illustrates the different types of noise. The proposed distances for fuzzy sets are computed for the fuzzy objects, while corresponding crisp set distances are computed for crisp objects obtained from the considered fuzzy objects using α-cuts at α = 0.5. In Fig. 4 we present, for each of the observed distance measures, the percentage of images from the test set for which the distance measure shows monotonic behavior with increasing translation and rotation. In Fig. 5–6 we present, similarly, percentage of images where distances show monotonicity w.r.t. translation and rotation, for the two observed types of noise, and each of the observed distance measures. We notice that for the noise free case, all the observed distances perform well for the considered test with respect to translation. For the performed roα tation tests, the proposed distance measures dα CW and dSMD perform better, ps ps while distances dCW and dSMD perform worse than the corresponding crisp sets distances. The Hausdorff style distances, in general, perform significantly worse than the distances based on all the points of the sets. We conclude that dα CW exhibits the best overall performance.


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Monotonicity in %

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Fig. 5. Percentage of images where monotonicity is fulfilled when Additive noise is applied, for each distance and each of the observed rigid transformations

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Distance measure

Fig. 6. Percentage of images where monotonicity is fulfilled when Multiplicative noise is applied, for each distance and each of the observed rigid transformations

For both Additive and Multiplicative noise, we observe that the noise has a strong negative effect on monotonicity w.r.t. larger rotations, see Fig. 5–6. Also, we observe that the crisp set distances have significantly better performance than the fuzzy sets distances in the test on monotonicity with respect to rotation up to 45 degrees. Based on these observations, we form the hypothesis that using a reduced number of membership levels can improve performance of the observed distances in noisy conditions. Therefore, we perform tests for monotonicity of the observed distances where the membership levels in the noisy images are quantized to n non-zero levels, where n takes a number of values between 1 (binary case) and 36 (the original number of levels given by the subsampling). Monotonicity of the proposed distances with respect to different number of quantization levels, for both observed types of noise, is presented in Fig. 7–8. We consider only distances defined by (6), since they have the best performance for the noise


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Fig. 7. Percentage of images where distance is non-decreasing with increasing rotation up to 23 degrees, with respect to different number of membership levels for Additive (left) and Multiplicative (right) noise 70

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dH dαSMD dαCW 50

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1 2 3 4 5 6 8 9 11 12 1416 1821 2326 32 36


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Fig. 8. Percentage of images where distance is non-decreasing with increasing rotation up to 45 degrees, with respect to different number of membership levels for Additive (left) and Multiplicative (right) noise

free case. Using different numbers of quantizations levels has essentially no influence on the monotonicity with respect to translation and we do not present this α result graphically. We conclude that dα CW and dSMD , for the given conditions, perform best for membership values quantized to approximately 9 levels. For that case, the distances clearly outperform the corresponding crisp cases (only one non-zero membership level), as can be seen in Fig. 9. Interestingly, dα H does not follow the same pattern; monotonicity of dα decreases drastically if more H than one membership level is used, and the crisp version performs clearly best. Fig. 10 shows distance as a function of increasing rotation for one test object with Additive noise, when non-quantized membership levels and memberships quantized to 9 levels, respectively, are used. For this example monotonicity w.r.t. rotation up to 45 degrees is not fulfilled for the non-quantized case, whereas the quantized case does provide monotonicity.


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Fig. 9. Percentage of images where monotonicity is fulfilled when Additive (left) and Multiplicative (right) noise is applied and membership values are quantized to one (crisp case) and nine levels, respectively

Observing that the size of the quantization step for the 9-level case is roughly of the same size as the standard deviation of the tested noise, we form the hypothesis that the optimal quantization should essentially hide most of the noise but not more than that. In other words, keeping more membership levels gives the distance measure more information, but only reasonably reliable information leads to corresponding improved performance. Noticing that the membership quantization has a large impact on the performance (compare Fig. 5–6 with Fig. 9), we feel that this issue deserves further studies, and therefore is placed high on our list of future work. 4.2

Comparative Evaluation on Matching Crisp and Fuzzy Objects

In this section we evaluate the performance of the observed distances for object matching. We compare, for a given spatial resolution, classification performance based on fuzzy discrete object representations with classification performance based on crisp discrete object representations. Fuzzy representations of disks and octagons are generated using pixel coverage digitization of continuous crisp disks and octagons, respectively. The corresponding crisp object representations are obtained by taking the α-cut at α = 0.5, of the fuzzy object representations (see Fig. 11). The same procedure is performed for: (i) crisp object representations and (ii) fuzzy object representations, for each studied object size: Five discrete representations of each object type (disk and octagon) are generated as template objects (discretized at random position in Z2 ). 1000 discrete representations of each object type, observations, are generated in the same way. Each observation is then classified as either disk or octagon depending on to which template object an aligned version of the observation has the smallest set distance. In this test


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1 All membership levels

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0.8

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0.6

0.4

0.2

0 0

5

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Fig. 10. Rotation of Butterfly image (Fig. 2A) with Additive noise, for different quantization levels. Distance measure for this test is dα CW (distance values are scaled to the range [0, 1]). The quantized version (dashed line) exhibits monotonic behavior for this image, whereas the non-quantized (solid line) does not.

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Fig. 11. Classification example. A: Fuzzy discrete representation of a disk, B: Crisp discrete representation of a disk, C: Fuzzy discrete representation of an octagon, D: Crisp discrete representation of an octagon. Objects A, C, and D are correctly classified, while the crisp representation of a disk, B, is incorrectly classified as an octagon.

we use set distances dCW and dα CW , since they show the best performances in the evaluation of monotonicity (see also [7]). The alignment is performed using greedy search, where monotonicity of the distance measure is essential for success of the procedure. An object is correctly classified if the template object at minimal distance is of the same type (disk or octagon) as the observation. The number of correctly classified objects, for crisp and fuzzy discrete object representations of a number of sizes, is presented in Fig. 12. We see that the combination of a fuzzy object representation with the proposed distance between fuzzy sets, provides significantly better object discrimination; a higher correct classification ratio, reaching more than 10% of improvement for objects smaller than 10 pixels in diameter, is achieved when using a fuzzy approach than when using the corresponding crisp representation and set distance, at the same spatial resolution. In Fig. 11 an example is shown where, in the presented matching process, fuzzy representations of a disk and an octagon (Fig. 11A and Fig. 11C), as well as a crisp representation of an octagon (Fig. 11D), are correctly classified, whereas a crisp representation of a disk (Fig. 11B) is incorrectly classified as an octagon.

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Fig. 12. Correct classification ratios for distance based object classification utilizing fuzzy and crisp discrete object representations, for a range of spatial resolutions

5

Summary and Conclusions

We have presented two different extensions of Sum of minimal distances and Complement weighted sum of minimal distances to fuzzy sets and we have performed an empirical evaluation of the monotonicity of the distance measures with respect to translation and rotation of discrete objects, as well as evaluation of α noise sensitivity. The proposed distances, dα CW and dSMD perform better, while ps ps distances dCW and dSMD perform worse than corresponding crisp sets distances. We conclude that the proposed distance dα CW has the best overall performance. Based on the observed performance for noisy conditions, we hypothesize that additional quantization of the membership levels may actually lead to improved performance for noisy data. Therefore, we have performed test on noise sensitivity using a reduced number of membership levels. Tests showed that the performance of the observed distance measures depend on the used number of membership levels and that it seems that a quantization step roughly of the same size as the standard deviation of the noise gives best performance. As part of future work, we intend to explore the relationship between the noise level and the appropriate number of quantization levels used in a fuzzy object representation. We have performed distance based object classification for crisp and fuzzy discrete object representations. We conclude that the combined utilization of a fuzzy object representation and the proposed distance measure, dα CW , leads to significantly improved performance compared to a corresponding classification based on a crisp object representation. This demonstrates that the proposed distances are capable of utilizing the additional information that a fuzzy object representation provides and that this information can provide improved performance for different related applications in image processing.


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Acknowledgments J. Lindblad and N. Sladoje are financially supported by the Ministry of Science of the Republic of Serbia through the Projects ON174008 and III44006 of the Mathematical Institute of the Serbian Academy of Science and Arts. Scientific support from Prof. Gunilla Borgefors is gratefully acknowledged.

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