A Local Multiscale Characterization of Edges applying the ... - CiteSeerX

Appeared in: Proc. Nato A.S.I, Fractal Image Encoding and Analysis, Trondheim, July 1995

A Local Multiscale Characterization of Edges applying the Wavelet Transform Carl J.G. Evertsz, Kathrin Berkner and Wilhelm Berghorn Center for Complex Systems and Visualization, University of Bremen FB III, Box 330 440, D-28334 Bremen, Germany

Abstract – The multiscale detection and characterization of edges in images according to their strength, their scale and their H¨older exponent, is presented. The method is based on both older ideas involving scale-space filtering in the field of human vision, and on more recent developments in the crossroads of wavelet theory, singular functions and the analysis of fractal time series. Edges and their properties on all scales of observation are looked for, at the same time avoiding a proliferation of redundant information and computation. This is achieved by only looking a modulus maxima lines in the wavelet transform. In the end three descriptive parameters are left for each pixel in the image at which a modulus maxima line starts. These three edge-parameters make it possible to select edges according to the largest distance at which they are still visible, their strength, and their sharpness. In generic cases, this makes it possible to, e.g., separate edges due to noise from real ones. The numerical implementation, and the applications to test-images and medical-images are discussed. The latter involve the detection of boundaries of bone- and liver-tumors in X-rays images and CT-scans.

Introduction In image and time-series analysis there is often a need for scale dependent analysis and description. For the characterization of texture the small scales are typically more impor1

tant, while large objects are more easily segmented and analyzed by considering the larger scales. The importance of multiscale analysis of images, and in particular their edges, has long been recognized in the field of human vision (see e.g. References [19]). In the last decade, the multiscale approach to images got new impetus from both the introduction of fractal geometry[18] and the concept of wavelets[4]. The discovery of the fractal geometry of nature showed that many naturally occurring geometries, and therefore also there images are self-similar to some degree. The wavelet transform decomposes a signal or image in functions which are translations and dilations of one and the same localized mother function called a wavelet. In recent years some important connections have been made between wavelets, fractals, and image analysis. This paper discusses an edge detection algorithm that is based on both scaling ideas from fractal geometry and the theory of one-dimensional wavelets. It goes back to a paper by Witkin[23] which addressed the problem of how events found in a smoothed version of a signal, can be located in the original signal. The solution proposed there was to track the extrema in the convolution of the signal with a Gaussian smoothing kernel, as a function of decreasing kernel width[23, 2, 24]. One of the main aims there was to reduce the proliferation of redundant data inherent in scale-space considerations, by means of a “scale-space filtering.” In this paper, we combine Witkin’s tracking of extrema, with recent developments using modulus-maxima lines[11, 12, 13, 14, 16, 21] of wavelet transforms to estimate the local properties of singularities in functions. Such singularities give rise to a scaling behavior of the wavelet transform of the signal as a function of scale. The corresponding scaling exponent, a H¨older exponent, is a single number that provides a concise characterization of a singularity, and has been widely applied in the characterization of fractal signals and measures[1, 21, 8, 7]. Using the derivatives of the Gauss function as analyzing wavelets, one is able to find, and to follow edges across scales. Furthermore, the theory of modulus-maxima lines allows one to characterize the scale, strength, and through the local H¨older exponent also the scaling behavior of the edge-singularities. Applications to medical images and a grey-scale digitized picture of Lena are discussed. Also the effects of noise are discussed.

Gaussian Kernel Smoothing The convolution of a signal f(t) with a Gaussian function of width a can be interpreted as coarse graining at scale a. It acts like a low bandpass filter in the frequency domain. Therefore, the result of this transform is a new signal where all details on time- or spatial1 2 scales smaller than a are removed. Denoting the Gaussian kernel θ(x) = √12π e− 2 x and

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the convolution in the point b at scale a by Z 1 ∞ x−b θ( )f(x)dx, (1) Tθ [f](b, a) = a −∞ a we see that for fixed a, Tθ [f] is a smoothed version of the original signal, with a determining the degree of smoothing. If the one-dimensional signal f(t) is a cut through the intensity field of a grey-scales image, then it is well-known (e.g. [16]) that edges are located at, or near those points in the signal where the intensity field changes most rapidly. If the resulting signal is continuously differentiable, then these points of rapid variation are those in which the modulus of the first derivative has a local maximum. Such a point is called a modulus maximum point. The second derivative is always zero in such points. Therefore, the modulus maxima points of the first derivative are always zero-crossing points of the second derivative. However, as is illustrated in the next figure, zero-crossing points are not necessarily modulus maxima points. Modulus maxima points and the zeros of the second derivative can be determined in all smoothed versions Tθ [f](., a) of the signal. A zero-crossing point at scale a is then a 2 [f ](b,a) point b where ∂ Tθ∂b = 0. Figure 1 shows the smoothed version of a signal with two 2

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Figure 1: Transforms Tθ [f], (Tθ [f])′ and (Tθ [f])′′ at a given scale a of a signal f with two sharp variation points. step edges, together with its first and second derivative. The two outer zero-crossings of the second derivative at x1 and x2 , are modulus maxima points, while the center one is not. Only the modulus maxima points relate to edges in the “image.” 3

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For the Gaussian smoothing kernel, it has been shown by Witkin and Baboud et al.[23, 2] that zero-crossing points form continuous curves as a function of the smoothing scale a. The possible shapes of these curves are shown in Figure 2. It should be noted

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Figure 2: The only possible forms ( left: “type a)”, right: “type b)”) of zero-crossing lines of a convolution with a Gaussian derivative ( a triangle stands for a zero-crossing from + to −, a dot for a zero-crossing from − to +). It should be noted that the configuration shown in the right plot, practically never occurs in real applications.

that the configuration shown in the right plot, practically never occurs in real applications. However, they do occur in functions with special symmetry properties, like the devil-staircase function associated with the cantor-set[21]. This very special property of the Gaussian smoothing kernel guarantees that one can find all zero-crossing lines by starting at the zero-crossings at the lowest scale, and following them up to higher scales. Conversely, given a zero-crossing at large scale a, one is able to zoom in, i.e., take a → 0 along the corresponding line, and find its more precise location. As was mentioned before, not all zero-crossing points are modulus maxima points, but the converse is always the case, and as will be discussed shortly, also modulus maxima points form continuous curves[11, 12, 13, 14, 16, 21] as a function of scale a.

Edge detection and filtering with wavelets The Wavelet transform Is is not difficult to show that the partial derivatives of order n of the smoothed signal Tθ [f] can also be obtained by smoothing f with the nth derivative of the Gaussian. More precisely: ∂ nTθ [f] 1 (b, a) = (− )n Tθ(n) [f](b, a) n ∂b a

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In practice one therefore uses the right-hand side to numerically estimate the partial derivatives of smoothed signals. How this provides the link with the wavelet transform, and ultimately to the estimation of local regularity properties, is discussed next. We first review some basic properties of wavelets. A wavelet is a localized function of zero integral, out of which a whole family of functions can be obtained through translations and dilations, which under certain general conditions [4, 6, 5], form an orthonormal bases in L2 (R). More precisely, Definition:

ψ ∈ L2(R) is called a wavelet, if its Fourier-transform ψˆ satisfies: Z

∞

0

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Z 0 2 2 ˆ ˆ |ψ(ω)| |ψ(ω)| dω = dω = cψ < ∞ |ω| |ω| −∞

1 Z∞ x−b Wψ [f](b, a) = q )f(x)dx, ψ( a |a| −∞

(2)

is called the wavelet transform of f. The parameter a controls the dilation of the analyzing wavelet ψ; its inverse being a measure of frequency. The parameter b controls the location of the dilated wavelet. One can show (see next lemma) that the derivatives θ(n) =

∂ nθ(x) ∂xn

(3)

of the Gaussian are wavelets for n > 0. Up to a normalization factor, the second derivative of the Gaussian is the well-known Mexican hat wavelet. √ Up to a factor a, the wavelet transform, Equation 2, is the same as the convolution in Equation 1 with θ replaced by ψ. Therefore, the zero-crossings discussed before, are zero-crossings in the wavelet transform of the signal. For more details about wavelet transforms we suggest the reader to consult, e.g., References [4, 6, 5].

H¨ older exponents of singularities and edges The main reason for discussing this connection with wavelets in the present context, is that we would like to use some recent results due to Holschneider & Tchamitchian[10, 9], Jaffard[11, 12, 13, 14] and Mallat & Hwang [16], which link the scaling behavior of the wavelet transform, especially along modulus maxima lines ([16]), with H¨older exponents of singularities in signals. We first discuss the concept of regularity of functions that will be used for the characterization of edges.

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Definition: Let n > 0 an integer, α ∈ R>0 , n ≤ α ≤ n + 1. A function f : R → R is called Lipschitz α in x0, if there are constants A ∈ R and δ ∈ R>0 and a polynomial Pn(x) of order n , that for all h ∈ R with |h| < δ holds: | f(x0 + h) − Pn(h) |≤ A|h|α

(4)

α∗ (x0) = sup{α| f(x)is Lipschitz α in x0 } is called H¨older exponent of f in x0. A point where f has a H¨older exponent α < ∞ is a singularity of f. Points where f is discontinuous are related to H¨older exponent 0. For example the image-row in figure 1 with two “edges” can be interpreted as a function which has two singularities with the same H¨older exponent α = 0. The results due to Holschneider & Tchamitchian[10, 9], Jaffard[11, 12, 13, 14] and Mallat & Hwang [16] to be discussed now, provide a method to determine such H¨older exponents using wavelets, and can be used to make numerical estimates. It should be noted that in the following applications of the wavelet transform for the computation of H¨older exponents, it is more convenient[1, 3, 21] to use the alter1 native renormalization, |a|−1 , in Equation 1, instead of |a|− 2 in the wavelet transform, Equation 2. Therefore, throughout the rest of the paper, we use the convolution Tψ [f], in Equation 1 for the wavelet transform. First we state some useful definitions, and a lemma ( [15] ). Definition: Let n ∈ N. A wavelet ψ has n vanishing moments (is of order n,) if for all integers k < n Z

∞

ψ(x)xk dx = 0 and

−∞

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∞

−∞

ψ(x)xndx 6= 0.

Lemma: Let φ a n-times differentiable function and φ, φ(n) ∈ L2 (R), φ(n) 6= 0. Then it follows that ψ = φ(k) is a wavelet. Note that applying this lemma to the Gaussian function θ yields that all its derivatives are wavelets. It is easily to prove that these derivatives of the Gaussian function are wavelets with a number of vanishing moments equal to the order of the derivative. We can now state the main results from Reference [16] using the scaling properties of the wavelet transform of functions with local H¨older exponents from [9, 10, 11, 12], which are relevant for this paper. Let ψ be a wavelet with n vanishing moments,and compact support, which is the nth derivative of a n-times continuously differentiable smoothing function. 6

1. If all points of an interval (b − ǫ, b + ǫ) (ǫ > 0) are not origins of modulus-maxima lines of the transform Tψ [f], then f is Lipschitz n in all points of (b − ǫ, b + ǫ). 2. If f has singularities with h¨older exponents α < n, then these points are starting points of modulus-maxima lines of Tψ [f]. 3. Estimation of local H¨older exponents via modulus maxima lines in non-oscillating isolated singularities: Let Γ(a) be the parameterization of a modulus-maxima line as a function of scale a. If f has H¨older exponent α < n in x0 = Γ(0) and there exists a constant C and a scale a0, such that for all a < a0 , |Γ(a) − x0| < Ca, then it follows: f is Lipschitz α in x0, if and only if there exists a constant B, such that for all a < a0 |Tψ [f](Γ(a), a)| ≤ Baα. If f has non-isolated singularities, it is necessary to make some extra assumptions (see [16]) on the wavelet in order to get the Lipschitz exponents from the decay of the wavelet transform on the modulus-maxima lines. For example, problems arise when on tries to measure the Lipschitz-exponent e.g. in an oscillating singularity. However, this type of singularities almost never occurs in images. Result (2) shows that each singularity yields modulus-maxima lines, as long as the analyzing wavelet has enough vanishing moments. Therefore singularities and edges can be located by finding the modulus-maxima lines. Furthermore, result (3) shows that the H¨older exponent α of a singularity can be estimated from the scaling behavior of the wavelet transform on the modulus-maxima line as a function of a → 0.

Edge-parameters We now discuss the significance of three parameters related to modulus maxima lines, that can be used to describe and to selectively segment edges in images. For convenience we denote the parameterization of a modulus maxima line as a function of scale, by Γ(a). The edge-parameters are • the scale of the edge • the strength of the edge • the H¨older exponent associated with the regularity of the edge When the smoothing parameter a increases, the image becomes increasingly blurred, and edges of small spatial extend tend to become dominated by those of larger extend. 7

As a consequence, at large enough scales, the contribution of small edges diminishes to the extent, that they stop producing modulus-maxima lines. Therefore, a measure for the distance from which an edge in an image is still distinguishable, is provided by the magnitude of the scale at which the modulus-maxima line stops existing. The size of the wavelet transform, |Tψ [f](Γ(a), a)|, at the smallest numerically available scale a = 1 of a modulus maxima line Γ(a), is a measure for the variation in the intensity field. For example, in case the wavelet ψ is the derivative of the Gaussian, this value is the local derivative of the signal, and therefore characterizes the strength of the edge. The H¨older exponent at the pixel located at position Γ(1), describes the sort of singularity, if any, associated with the intensity field in its vicinity. Small H¨older exponents are related to sharp edges and large H¨older exponents to smooth edges. It is important to remark that one can only detect singularities with H¨older exponents smaller than the number of vanishing moments of the analyzing wavelet (see point 3 in previous section). A simple analytical and numerical example illustrating the use of these edge-parameters is discussed next.

Numerical Implementation Finding Modulus-Maxima lines Our applications involved digitized images I(x, y) of size nx × ny pixels, with nb < 12 bits quantization of the intensity I(x, y). In order to get 1-dimensional signals, we scanned the images in the four different ways shown in Figure 3. These are a horizontal scan, a vertical scan, and two diagonal scans. Each of these scans, yields a 1-dimensional signal of size nx ny . It is essential to scan the image in these various ways, because, e.g. horizontal

Figure 3: The four serpentine scans used in the edge detection. From left to right these are the horizontal, the vertical, and two diagonal scans. edges are missed in the horizontal scan. Also, experience showed that the horizontal and vertical scan are not sufficient, and that, in addition diagonal scans are needed. In each of these 1-d signals, modulus-maxima points are located at the smallest scale, and a special 8

procedure tracks each one of them to the largest scale at which it still exists. This has the advantage that the time consuming wavelet transform only needs to be computed in the direct vicinity of the modulus maxima lines, instead of for all values of a and b. In this way the amount of computations grows as Namax, where N is the length of the signal and amax is the maximum scale a. The reason for transforming 2-dimensional images into 1-dimensional signals in this paper, is that it is much easier to find and parameterize the modulus maxima lines. In the 2-dimensional case, complications arise due to the fact that the modulus maxima points can form surfaces. However, as is discuss in e.g. References [16] and [17], it is possible to apply the two-dimensional wavelet transform to edge detection, denoising, and enhancement of images. Figure 4 shows the result of the tracking of a modulus maxima line for a simple signal with isolated singularities with various H¨older exponents, α(x1 ) = 0.7, α(x2 ) = α(x3 ) = 100

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Figure 4: Top: Modulus maxima lines in isolated singularities with analyzing wavelets θ(1) (left) and θ(2) (right). Bottom: log-log plot of modulus maxima lines of Tθ(2) [f] starting in the singularities.

0.3, α(x4 ) = 1.5, α(x5 ) = 0. The upper-left plot shows the modulus maxima lines tracked, using the wavelet θ(1), and the upper-right plot those using θ(2) . According to the result (1) in the previous section, the absence of modulus maxima lines in between the singularities, 9

confirms that there the signal is at least Lipschitz 1 in the case θ(1), and at least Lipschitz 2 in the case θ(2) . The number of modulus maxima lines emanating from singularities grows linearly with the number of vanishing moment in the wavelet[16]. This proliferation of lines is not desirable, because it increases the number of lines to be tracked. Also, the errors in the numerical (discrete) approximation of the wavelets, increases at smaller scales because of the increasing number (n) of maxima in the wavelet. Nevertheless, depending on the application, one may need higher orders. In the example at hand, the singularity in x4 is missed by the wavelet of order n = 1, because the H¨older exponent is larger than 1 there. Increasing the order to 2 in the upper-right figure suffices to detect this singularity. For edge detection in images, experience shows that it suffices to detect H¨older exponents smaller than 1[16]. In practice we therefore use the wavelet-transform modulus maxima of θ(1).

Estimating the H¨ older exponents and other edge-parameters The results (1),(2) and (3) discussed in the previous section, concerning the estimation of the H¨older exponents of the singularities in a signal, have only been proved for analyzing wavelets with compact support [9-14, 16]. Strictly speaking, the derivatives of the Gaussian do not have compact support. However in our numerical applications, the support is cut off at distances exceeding ±5a around the center b of the wavelet. Because of the 2 exponentially decreasing tails (e−x ) of the Gaussian, there is essentially no mass outside 5 standard deviations from the center. The lower-left plot in Figure 4 are double-logarithmic plots of the modulus of the wavelet transform of order 2, along the modulus-maxima lines starting at the various singularities. According to the result (3), the slopes of these lines are estimates of the H¨older exponents. The quality of the scaling behavior varies substantially between singularities. The wavelet transform along the modulus-maxima lines related to x2 and x3 do not behave linearly in the double logarithmic plots. The upward swing at the smaller scales is due to the discretization of the signal f around x2 and x3. Using the scales 22 − 25 for the least square fit, one finds α ≈ 0.3 for the H¨older exponents of these singularities. In practical edge detection in images, one can not visually inspect all the double-logarithmic plots of the modulus-maxima lines. As an approximation, we decided to do a least-square fit from the smallest scale up to that scale, where the last change occurs in the sign of the local slope in the double-logarithmic plot. Negative slopes corresponding to negative H¨older-exponents (α < 0) are also possible[13, 14, 16, 21]. In those cases we first multiply the wavelet-transform with the scale, i.e., a|Tψ [f](Γ(a), a)| ≤ Baα+1, so as to get a positive slope α + 1 > 0. The least-square fit is then done as before. The edge-parameters discussed in the previous section can be used to distinguish 10

between the various edges in Figure 4. Even though both the edges x2 and x3 have the same H¨older exponent, they can be distinguished by considering their strength. In the lower plots of Figure 4, the natural logarithm of this strength |Tψ [f](xi, a = 1)| is given by the intersection of the curves with the y−axis are. In order of decreasing strength, this parameter yields x5, x3 , x2, x1, x4. The edges x1 and x5 differ both in H¨older exponent and strength. The role of the scale parameter is lost in these plots because the scales a > 100 are not shown. However, one observes that the modulus maxima lines of edge x2 are shorter than those of edge x3 . The variation in the scale parameter will become clearly visible in the applications shown in Figure 6. In applications to images, the four different scans yield different modulus maxima lines in each pixel, and therefore (at most) 4 different sets of edge-parameter values. For a rotationally invariant analysis, it is desirable to reduce this to one set of parameter values. Therefore, to compute the scale of the edge we took the size of the longest modulus-maxima line starting in the pixel under consideration. For the strength, we took q 2 2 2 S1 + S2 + S3 + S42 , where the Si ’s are the values of the strengths along the four different directions of scanning through the pixel. To estimate the H¨older exponent we took the smallest value, because it is associated with the scanning path through the pixel along which the variation is sharpest. Then, by thresholding each of the parameters one can filter the image in multiple ways.

Noise The three edge-parameters can also be used, to filter out noise in signals. In a signal with additive Gaussian white noise, typically the noise produces shorter modulus-maxima lines with relatively lower strength than those of modulus-maxima lines of events related to the original signal. In addition, the points related to the Gaussian white noise tend to have lower H¨older exponents. These characteristics of Gaussian white noise in terms of the edge-parameters can be used to separate the information coming from the underlying signal from the noise, by appropriately thresholding the three parameters. Figure 5 left, shows the grey-scale image of two discs, on to which has been added Gaussian white noise of standard deviation σ = 96. The inner disc has grey-scale value Ii = 64, the outer one Io = 192, and the background Ib = 0. The diameter of the inner disc is 80, and that of the outer is 160 pixels. For the inner edge one has a signal to noise ratio (Io − Ii)/σ = 1.33, and for the outer one (Io − Ib )/σ = 2. The right plot in the same figure, shows the output of the filter after thresholding the edge parameters. All pixels are colored black, except those with strength bigger than 78, modulus-maxima line length bigger than 23, and H¨older exponent 0.2; these are colored white. With this parameter-setting we can detect points which are only related to the boundaries of the discs and not to the noise. 11

Figure 5: left) Test image for studying effects of noise. The inner disc has grey-scale value Ii = 64, the outer one Io = 192, and the background Ib = 0. The diameter of the inner disc is 80, and that of the outer is 160 pixels. right) Edges found with appropriate thresholding of the edge parameters. This test also shows that the detection of edges in this case is independent of the direction of the edges in the images, and that the four scanning directions, one horizontal, one vertical and two diagonal, are sufficient.

Applications In this section we will discuss the application of the edge-parameters in the segmentation of edges of bone tumors, liver tumors and of edges in the picture of Lena.

Bone Tumors The uppermost image in Figure 6 shows the X-ray image of a human bone with a tumor. The task is to find the boundary of the tumor so that the physician can assess the degree of infiltration. The center image shows the signal obtained by scanning the original X-ray along the line crossing the tumor shown in the upper plot. This signal is superimposed on the graph of the modulus-maxima lines. The boundaries of the tumor are marked with xl and xr , both of which are starting points of modulus maxima lines. Trying a segmentation by thresholding the length of modulus-maxima lines keeping only those with sizes > 15, yields the result shown in the bottom left image. The segmentation shown in the bottom right figure was obtained by keeping only those pixels with strengths above 3.2. 12

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Figure 6: top) X-ray of a human bone with tumor. middle) Modulus maxima lines of Tθ′ [f] from the signal obtained by scanning along of the row market in the top figure. bottom left) Pixels (white) with modulus maxima lines with length greater than 15 bottom right) Modulus maxima with strengths larger than 3.2.

Liver Tumor Figure 7 shows the original slice of a computed tomography scan (CT-scan) of the abdom13

Figure 7: left) Original CT slice of human abdominal region, containing the liver on the left. The white spots in the liver are blood vessels. The larger dark region in the central upper part of the liver is the tumor. right) All pixels from which modulus-maxima lines emanate. That is: No thresholding of the parameters. inal region of the human body. The big organ to the left is the liver. The white regions in the liver are blood vessels which are enhanced because of the application of a contrast medium in the blood. The large dark region in the liver is a section through a large liver tumor. The right plot shows all pixels having a modulus maxima line ending in them. The left and right plots in Figure 8 show the result for different settings of the edgeparameters. In the left plot only the strength and the scale have been thresholded. The

Figure 8: left) Edge-parameter setting: Length > 30, strength > 12.8. right) Length > 30, strength > 12.8, H¨older > 0. right plot shows how many irrelevant boundaries can eliminated by not taking into account 14

the very singular boundaries demarcating e.g. the spine. This is realized by an additional thresholding of the H¨older exponent, in this case only keeping those with values larger than 0.

Lena The 512x512 grey-scale image of Lena is shown in the left part of Figure 9. The right

Figure 9: left) Original Lena ( 512 × 512 ). right) Only those pixels with a modulusmaxima line length > 30 are shown in white. The rest is made black. plot shows the significance of the length-parameter. All pixels with modulus-maxima line lengths larger than 30 are shown. These correspond to edges that are still visible after extreme blurring. The effects of a reduction of this parameter is shown in the left plot in Figure 11. Figure 10 shows the significance of the other two parameters, the strength and the H¨older exponent. In the left plot all pixels with a strength larger than 5 are shown. These correspond to edges which have a higher contrast as most of the noise-pixels, which therefore are removed. The right plot in Figure 10, shows all pixels with H¨older exponent smaller or equal to 0. These represent the very sharp edges, and since all lengths and strengths are allowed, one finds that the noise becomes very dominant. The right plot in Figure 11 shows an optimum setting of the edge parameters obtained by interactive tuning of the parameters. A segmentation of the feathers is obtained by keeping pixels with shorter modulus-maxima line lengths < 14, and larger strengths > 22. The result is shown in the right part of Figure 11.

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Figure 10: left) Strength > 5. right) H¨older < 0.

Figure 11: left) A subjectively optimal setting of the edge parameters: strength > 7, modulus maxima line length > 2. right) Obtaining the feathers with modulus-maxima line length < 14, strength > 22.

Conclusions and summary The edge detection and segmentation method discussed in this paper combines relevant information obtained from a multi-scale analysis. Through this analysis, three tuning 16

parameters involving the scale, the strength and the H¨older exponent can be associated to each pixel from which a modulus maxima line emanates. The number of edge parameters could be further expanded or reduced, depending on the application at hand. Data redundancy is avoided by only considering the modulus maxima lines in the wavelet transform. In addition, the amount of computations is reduced by tracking the modulus maxima lines, starting from the lowest scale. Nevertheless, the amount computations still grows as the size of the signal times the maximum scale of interest. For a 512x512 image up to scale 100, pre-computing and storing the three edge parameter values of the image allows real-time manipulation of sliders, which facilitates the interactive study of the effects of various parameter settings. Once the right range of setting has been found for a particular type of image, one can optimize the algorithm for that specific task.

Acknowledgment We would like to thank Thomas Netsch for various discussions and computer support. Also, we would very much like to thank Heinz-Otto Peitgen for discussions and his support of this research. We thank K.J. Klose for the liver CT and J. Freyschmidt for bone X-ray.

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