Edge Detection by Multi-Dimensional Wavelets

Transcription

1 Edge Detection by Multi-Dimensional Wavelets Marlana Anderson, Chris Brasfield, Pierre Gremaud, Demetrio Labate, Katherine Maschmeyer, Kevin McGoff, Julie Siloti August 1, 2007 Abstract It is well known that one-dimensional wavelet techniques are suboptimal in the representation of images. Recently a new generation of intrinsically two-dimensional wavelets, e.g. shearlets, has been introduced to alleviate these deficiencies. In this project, new edge detection methods were developed based on the shearlet transform. As a refinement of these methods, subdomain decomposition was introduced to preserve less dominant edges. Furthermore, several basic post-processing schemes were used to provide more distinct edges. All of the above methods were applied to both artificially generated and natural images. In order to measure the accuracy of the various methods, the Hausdorff distance between the actual and approximate edges of artificial images was computed. Through this analysis, it was concluded that edge detection methods based on shearlets are at least as accurate as popular methods, such as Canny and Sobel. On images with sharp corners, the results suggest that shearlet methods may actually improve upon traditional methods. 1 Introduction Wavelets are collections of functions that can be used to decompose signals into various frequency components at an appropriate resolution for a range of spatial scales. Edges can be defined as sharp changes of the intensity in a signal. Since wavelets provide excellent representations of discontinuous signals [8], they have been used in modern edge detection Albany State University, REU Group Member North Carolina State University, Graduate Assistant North Carolina State University, Project Mentor North Carolina State University, Consultant Washington University in St. Louis, REU Group Member University of Maryland, REG Group Member Pomona College, REU Group Member 1

2 methods [11, 12]. Applications of edge detection technology can be found in many fields, including medical imaging. The objective of this project was to explore the latest generation of wavelets in order to create improved edge detectors. Toward this end, several known signal-processing methods were studied and applied. These included methods based on the well-known Fourier transform and wavelet transforms in both one and two dimensions. Theoretical results concerning edge detection in one dimension were reviewed and the corresponding algorithms were implemented. Furthermore, tests were run on images by applying one-dimensional decompositions in both the horizontal and vertical directions independently. These results were compared to edge detection schemes based on gradient methods, which seek to capture sharp changes in intensity by approximating the gradient in the neighborhood of each pixel. It is well known that one-dimensional wavelet techniques are suboptimal in the representation of images [8]. Recently a new generation of intrinsically two-dimensional wavelets, e.g. shearlets [8], has been introduced to alleviate these deficiencies. In this project, new edge detection methods were developed based on the shearlet transform. As a refinement of these methods, subdomain decomposition was introduced to preserve less dominant edges. Furthermore, several basic post-processing schemes were used to provide more distinct edges. All of the above methods were applied to both artificially generated and natural images. In order to measure the accuracy of the various methods, the Hausdorff distance between the actual and approximate edges of artificial images was computed. Section 2 recalls the necessary background information, including a construction of wavelets and shearlets, as well as a description of various previous edge detection methods. For more information on wavelets, see [3, 4, 5, 10, 19, 20]. For a detailed discussion of shearlets, see [8]. Section 3 contains a description of the new edge detection algorithm using shearlets, including a discussion of the theory, implementation, and results of this new method. 2 Background Information 2.1 Wavelets The purpose of wavelets is to provide a decomposition of a function or signal into its various frequency components and at a range of appropriate spatial scales, or resolutions. For a given function f, wavelets give a collection of functions {ψ jk } such that f can be represented as the superposition of some linear combination of ψ jk. Such a collection is often called a wavelet basis. More precisely, for a given f, there exist coefficients c jk such that for all x, f(x) = j,k c jk ψ jk (x). (1) We say that a single term in the sum, c jk ψ jk, is the frequency component of f at a certain spatial scale and position. Thus, Equation (1) can be viewed as the decomposition of f into its various frequency components at a range of scales. 2

3 WAVELETS AND DILATION EQUATIONS 619 (1, I, 1, 1) is a left eigenvector with X = I. The right eigenvector yields the values $(I),..., $ ( N )at the integers. The recursion determines $ at all dyadic points. Values at other points are never used Wavelets and orthogonality. Finally we define a wavelet. It comes from the scaling function $ by taking "differences": To understand how, exactly, wavelets yield such a decomposition, more definitions are required. First, consider a function φ, which is normalized, i.e. We write Win place of the usual rl/, to distinguish more φ(x)dx clearly = 1, from $. Note the 2x on the right, and especially (- l)k. Examples show the effect of alternating signs: and which satisfies the Dilation Equation for some coefficients a k : φ(x) = k a k φ(2x k). (2) Such a φ will be called a scaling function. From this function we can define another function ψ as follows: ψ(x) = ( 1) k a 1 k φ(2x k). (3) k Such a ψ will be called a mother wavelet, because it is used to generate the entire collection {ψ jk }. A wide variety of functions meet these criteria. For example, the Daubechies 4 wavelet (see Fig. 1, created by [19]) can be obtained by choosing coefficients 1(1 + 3), 1(3 + 3), (3 3), 1(1 3), and this yields a nowhere differentiable function with a fractal-like 4 4 Haar wavelet structure from box [3, function 4, 5, 19]. On the other hand, Wavelet it can from be shown hat function that the derivative of a Gaussian function satisfies the above conditions (with an infinite number of coefficients), [17], and W,(x)=$(2x)-$(2x- I) ~=$(2x)-$$(2x- 1)-$$(2x+ 1) these wavelets are smooth. W4(x) from $ = D4 Orthogonal wavelet Figure 1: The Daubechies 4 wavelet The wavelet from the hat function does not belong here. It is not orthogonal to W(x + 1). The point is that the other two do belong. The Haar function is orthogonal to its own translations Now, given and dilations. a mother wavelet Historically ψ, weit define was the theoriginal wavelet basis wavelet by dilations (but with and translations: p = 1 and poor approximation). The orthogonal wavelet W4 has p = 2 and secondorder approximation. ψ jk (x) = 2 j/2 ψ(2 j x k). Without formulas for D4 and W4, how is the orthogonality of their translates known? We need a test that applies to the recursion coefficients 3 ck, or to the symbol P([)= $ 2 ckelk(.

4 Note that in this case the coefficients in Equation (1) are in fact given by c jk =< f, ψ jk >, (4) where <, > is the inner product on L 2 (R). The idea of decomposing a signal into frequency components has been heavily exploited with the use of Fourier decompositions, which use sines and cosines as their basis functions. The clear advantage of wavelets over traditional Fourier methods is that they are very welllocalized in both space (or time) and frequency. Recall that trigonometric functions capture frequencies perfectly, but they are truly global in the sense that they do not decay over time. Thus, these types of bases give poor representations of functions that have short blips, or any discontinuities. On the other hand, wavelet bases can be constructed so that they capture only local information in both space and frequency domains, and therefore wavelets provide a much better representation of discontinuous or non-periodic functions. Furthermore, this locality property in the space domain allows for so-called multi-resolution analysis [14], which simply means that frequency information is captured for a full range of scales (indexed by j in Equation 4). The same process can be carried out for functions of two variables, except that the theory requires use of three mother wavelets, ψ 1, ψ 2, and ψ 3, which are called horizontal, vertical, and diagonal, respectively. The wavelet basis then has three indices: where k is now an element of Z Shearlets ψ ijk (x) = 2 j/2 ψ i (2 j x k), (5) While one-dimensional wavelets provide optimal representation of discontinous signals in one dimension (a statement that will not be made precise in this paper), the two-dimensional wavelets described above provide provably suboptimal representation of two-dimensional signals [8]. For this reason, a wide variety of alternative approaches to two-dimensional signal decompositions have appeared in recent years, notably bandlets [13], contourlets [7], ridgelets [1], curvelets [18], and shearlets [8]. Each of these approaches attempts to take advantage of some two-dimensional geometric information to provide better representation. These approaches can provide better representation properties, but usually at some cost. For example, shearlets provide provably optimal representations of signals, but at the cost of orthogonality of the basis functions [8]. The shearlet decomposition functions form a Parseval frame, but not a true basis. Recall that a Parseval frame for L 2 (R 2 ) is a collection of functions {ψ j } such that for any f in L 2 (R 2 ), f 2 = j < f, ψ j > 2 Considering Parseval frames instead of bases is typically not a large disadvantage, since the only difference is that Parseval frames contain a certain amount of redundancy. In this paper we focus our attention on shearlets. 4

5 Constructing shearlet bases is a non-trivial task, but it will be presented here nonetheless. First, let ψ be a mother shearlet, which has certain technical properties (as defined in [8]). Then define the matrices A j, which is the dilation matrix, and B l, which is the shear matrix, as: and A j = B l = ( 2 j j/2 ( 1 l 0 1 Now we can define shearlets to be the functions ). ), ψ jkl = det A j/2 ψ(b l A j x k), (6) where j, l are in Z and k is in Z 2. With this definition, shearlets will be well-localized in both space and frequency domains. The shearlet ψ jkl is supported in frequency domain on trapezoids at scale 2 j, with orientation indexed by l, and associated with spatial location indexed by the vector k. Thus shearlets provide a decomposition of any L 2 (R 2 ) function into its frequency components according to the tiling of frequency domain by such trapezoids (see Fig. 2, created by the authors of [8]). The support of a shearlet in space domain is highly-localized; moreover, the support also maintains a specific orientation (loosely orthogonal to the line along which the corresponding trapezoids lie in frequency domain). This additional information constitutes the entire goal of shearlets: certain two-dimensional geometric information about the signal such as the orientation can be used to provide more efficient representation of the signal. 2.3 Edge Detection As mentioned in the introduction, edge detection has a wide variety of important applications. For this reason, it has been studied quite extensively, both theoretically and numerically. Theoretically, the edges can be defined (for piecewise smooth pictures, where the discontinuities occur only along smooth curves) as the collection of points at which the gradient is infinite. The idea of considering large gradients leads to various methods for edge detection in the discrete case. All of these methods define some discrete differential operator that approximates the gradient in some way. For example, the Sobel method [15] uses convolution with the following matrix to approximate the partial derivative in the horizontal direction: D x = Recall that the discrete convolution product of two matrices A and B is given by: (A B) ij = a k,l b i k,j l. (7) k l 5

6 Alternative tiling of the frequency plane induced by the shearlets: we tile the cone ξ 2 /ξ 1 1, next we tile the cone ξ 2 /ξ 1 > 1. Figure 2: Tiling of Frequency43Domain induced by Shearlets Once this operator has been applied to the signal at each point in the image, the points at which the gradient is large must be determined, and these points are then labeled as the edges. Gradient methods work quite well for truly smooth images, but they are highly susceptible to the effects of noise. In order to reduce the effects of noise, other methods have been explored, such as the spectral methods seen in [9] and the wavelet methods in [11, 12]. These methods work very well in one dimension, but in order to extend them to two dimensional signals, the one dimensional methods are applied twice (once in the horizontal direction and once in the vertical direction) and then combined. Due to this seemingly naive method of extension, these methods do not take into account the truly two dimensional information of edges. The fundamental idea for the spectral methods in one dimension is to note that the conjugate Fourier sum of a discontinuous function f (of one variable), S N f, converges to the function [f](x) = f(x + ) f(x ), which is only non-zero at the discontinuities, as N goes to. In the proposed versions of this method, the authors have generalized the conjugate Fourier sum so that the convergence to [f] will actually be exponential. These methods do decrease the effects of noise, but they are still susceptible to spurious oscillations near 6

7 the discontinuities, which leads the authors to use a combination of gradient and Fourier methods. The one dimensional wavelet methods introduced in [11, 12] rely on convoluting the signal with a smooth wavelet function. Recall that the convolution product of functions f and ψ is given by f ψ(x) = f(t)ψ(t x)dt Notice that if the wavelet ψ is the derivative of a smoothing function (like a Gaussian) φ, then the convolution of a signal f with ψ is equivalent to taking the derivative of the convolution of the f with φ: (f φ) = f φ = f ψ, (8) by the properties of the convolution product. Therefore the one-dimensional edge detection methods are theoretically equivalent to smoothing the signal, taking the derivative, and looking for large values of the derivative. They remain advantageous, though, because in practice they are fast and the smoothing reduces the effects of noise. Also, since the wavelet coefficients are localized in space and frequency domains, there are no spurious oscillations with which to contend. Given a function f, the first step is to compute the wavelet transform, W f(s, x) = f ψ s (x), (9) where ψ s (x) = 1ψ( x ). Note that this just gives the coefficients of the wavelet decomposition s s in the case that we choose s = 2 j. Next we define a modulus maxima of W f(s, x) at a fixed scale s to be a local maxima of W f(s, x), viewed as a function of x. The main theorem in [11, 12] then states that under certain conditions on ψ (such as having at least one vanishing moment and being the derivative of a smoothing function), the set of discontinuities of f is contained in the closure of the set of modulus maxima of W f(s, x). In practical terms, this theorem implies that finding the modulus maxima of W f(s, x) as s 0 should detect all of the edges of a function f (although it may detect more points, in general). In other words, we obtain an edge detector in the discrete case by simply looking at fine scales and finding the local maxima of W f(s, x) as a function of x. This method can be observed clearly by looking at the absolute value of these coefficients, as seen in Fig. 3, which was created using the Gaussian wavelet. The wavelet methods mentioned above have been applied to two dimensional images with some success, but since the method of extension from one to two dimensions seems not to take into account any truly two dimensional information, it appears that there is room for improvement. 3 Edge Detection with Shearlets 3.1 Theory The goal of the present work is to show that the newest generation of two-dimensional wavelets can yield viable edge detectors. To the best of the authors knowledge, the literature 7

8 Figure 3: Local maxima converge to the discontinuities currently contains no edge detection method based on any of these representations. Not only are there no theoretical results regarding the effectiveness of such methods, there are also no numerical results suggesting that such methods may work. According to preliminary studies, there may be a strong analogy between the methods found in [11, 12] and the methods attempted here. Such an analogy suggests that similar theoretical results may be achieved, although this remains a topic of continuing research. All known results do suggest that such methods may be fruitful. Indeed, shearlets provide provably optimal representation of two dimensional images by incorporating two dimensional geometrical information into a multiscale decomposition. Moreover, this additional geometric information pertains to the orientation of the signal at a particular point, and this type of information is precisely what the prior edge detection methods appear to be missing. 8

9 3.2 Implementation The idea for a shearlet edge detector has been inspired by the analogy to the wavelet method detailed in [11], and therefore the algorithm is also analogous to the wavelet method described in Section 2.3. Here we describe the algorithm used in this study. All numerical computations were done using MATLAB. The algorithm is organized as follows: 1. Gaussian smoothing: Reduce the effects of noise by convoluting the image with a Gaussian filter. 2. Calculate shearlet coefficients for 2 n distinct directions, where n is the level of detail. 3. Detect modulus maxima and thus edges using subdomain decomposition and thresholding. 4. Combine information from the 2 n directions: Flag points where at least one of the sets of directional coefficients detects an edge. 5. Neighborhood cleanup: Keep edges only at points where the magnitude of the neighboring difference is greater than the magnitude of the directional coefficients. Also, keep edges only at points where the magnitude of the the directional coefficients is greater than the thresholding parameter. 6. Post processing: further refine the image with techniques such as line thinning and deletion of isolated points. Each step of the algorithm is explained in further detail below. Gaussian Smoothing First, the image must be smoothed to reduce the effects of noise. This is done by convoluting the rows of the image with a 1D Gaussian filter as discussed previously, thereby creating a new image, and then convoluting the columns of the new image with the 1D Gaussian filter. This achieves the effect of convoluting the image with a 2D Gaussian filter because the Gaussian filter is separable. Calculate Shearlet Coefficients The next step of the algorithm is to compute the coefficients of the discrete shearlet transform at the jth level of detail, which can be done with a fast algorithm for images of size 2 n 2 n. The algorithm for computing this transform comes from work by the authors of [8]. The result of this computation is 2 j coefficient arrays (each of which has the same size as the original image), where j is the level of detail, i.e. the index of the scale in the notation of Section 2.2, and each array contains the shearlet coefficients at this level of detail with a 9

10 single orientation. In other words, at the scale 2 j, there are 2 j distinct orientations for which we compute the shearlet coefficients, and these are collected in arrays corresponding to the orientations. Thresholding At various points in the algorithm, it makes sense to ignore very small effects and consider only the dominant effects. This is a very general concept, which is present in any edge detector and is often called thresholding. If M = (m i,j ) is any two-dimensional array, then the transformation threshold M with parameter ɛ is given by { mi,j if m (T ɛ M) i,j = i,j ɛ 0 if m i,j < ɛ. For example, given an array of intensity values ranging from 0 to 256, it will not alter the picture in any significant way if those pixels with intensity values less than three are set to zero. Of course, this concept can be applied at multiple stages in an algorithm: any time when relatively small values should be set to zero. From a certain perspective, thresholding is necessary, because every pixel must be labeled as either lying on an edge or not lying on an edge, and somewhere a line must be drawn to distinguish the two. Furthermore, thresholding helps reduce the effects of noise, since often the features introduced to an image by noise are dominated by the actual features of the image. Detect Modulus Maxima The third step in the algorithm is to take the absolute values of the coefficients and find the local maxima. Note that in the one-dimensional case there is only one direction of approach for a given point in space, and thus the property P 1: x is a local maximum is the same as the property P 2: x is a local maximum along some line passing through x. This fact no longer holds in two dimensions. In two dimensions, P 1 holds when x is a point discontinuity, whereas P 2 holds when x is likely to be part of an edge. Thus, for the purposes of edge detection, the proper extension of x is a local maximum is actually P 2. Thus, the algorithm looks for local maxima in the shearlet coefficient arrays along eight directions: lines making angles kπ with the horizontal, for k = 0,..., 7. The purpose of looking 8 for maxima along eight directions is to capture the additional directional information given by shearlets. In fact, through direct comparison of methods, the edge detector that found 10

11 maxima in eight directions proved itself to be more robust with respect to the thresholding parameter than the edge detector that found maxima in only four directions (along lines making angles kπ with the horizontal, for k = 0,..., 3), as shown in Figure 5. 4 At this step there is also a certain relative thresholding that occurs, which is based on a method that decomposes the data into smaller subdomains, also known as subdomain decomposition. In this approach, each set of shearlet coefficients (for a given orientation and level of detail) is divided into four basic quadrants. The local maximum for each quadrant is determined and the largest of such maxima is flagged and recorded as the new maximum for its corresponding quadrant. The other local maxima are flagged as local maxima for their respective quadrants if and only if they are greater than a given fraction amount of this largest maximum. Otherwise, the largest local maximum is flagged as the local maximum for that entire quadrant. This process is applied recursively to each quadrant of the subdivision until either the local maximum is not sufficiently large (i.e. the largest maximum of the subdivision is used for the entire region of interest) or the size of the subdivision is smaller than a given minimum pixel size. This produces a matrix of values that indicate the local maximum to be used for each pixel of the image. The thresholding that follows eliminates extraneous local maxima by dictating that if the magnitude of the coefficient is less than that of its neighbor by a fractional amount of the corresponding local maximum, then it is not an edge. In other words, subdomain decomposition allows for local maxima to be determined based on truly local information. In particular, consider an image with high intensity edges in one region and low intensity edges in another. If the low intensity region is sufficiently close in magnitude to the high intensity region, then subdomain decomposition allows for the use of smaller threshold values for the low intensity region. As a result, more low intensity maxima will be detected, while still ruling out extraneous maxima which may occur due to noise or numerical error. Combine Information In the next step, the magnitude of the shearlet coefficients is reincorporated at each point. To be exact, each point that has been flagged as a local maxima in at least one of the coefficient arrays is then multiplied by the absolute value of the associated shearlet coefficient, and the arrays are added together (pointwise). (As an interesting note, points which have been flagged as edges in multiple directions may contain further geometric information, such as information about corners, but this has not been confirmed.) Then this array is once again thresholded to keep only the dominant edge candidates. Neighborhood Cleanup At this stage in the algorithm, the neighborhoods of actual edges (especially near corners) contain too many edge candidates. In order to eliminate these effects, a cleanup algorithm is implemented. Given the original image U = (u i,j ), first define the difference coefficients 11

12 D = (d i,j ) to be d i,j = max( u i,j u i+1,j, u i,j u i,j+1 ). Then, given the coefficient array M = (m i,j ) defined in the previous step of the algorithm, form the matrix N by the formula { mi,j if m (N) i,j = i,j < d i,j 0 if m i,j d i,j. After doing this type of cleanup, the second instance of thresholding occurs. In other words, the transformation T ɛ (defined above) is applied to the array N. As always, thresholding relies on the parameter ɛ. There is a default value for the parameter, which gives minimal error on artificial images (see Section 3.3), although the parameter can be manually varied if necessary. It is interesting to note that the error on artificial images does indeed attain a minimum, and this minimum error seems to be fairly robust with respect to changes in the parameter. Figure 7 illustrates this concept. In general, a very low threshold allows too much noise to be labeled as edges, and a very high threshold will cause the actual edges to be lost. Sometimes there is a satisfactory parameter value that balances these concerns, but if the noise is too large (relative to the actual edges), then it may be difficult to find an appropriate parameter value. This phenomenon occurs in some form in any known edge detector. At the end of this step in the algorithm, any non-zero element in the array N is set to the value 1. Post Processing Finally, the algorithm performs some elementary post-processing techniques. The first of these techniques involves deleting points which have no other points in their immediate neighborhood. The second technique is called line thinning in the literature, and it attempts to delete extraneous points hanging off the edge of a clear line. For example, suppose that the matrix M represents the edges of a signal, as computed above: M = Then line thinning would remove the central pixel. 3.3 Testing and Results Through rigorous testing, the shearlet edge detection method has been shown to be very competitive in detecting the edges of both artificial and natural images. The artificial images have been created using analytical formulas in such a way that the actual edges are known. 12.

13 For these images, the known edges are compared to the approximate edges (as detected by the shearlet method) using the familiar notion of the Hausdorff distance between two sets in R 2, i.e. if A and B are two sets of points in R 2, then the Hausdorff distance between A and B is computed as follows. For each a in A, let dist(a, B) = min a b. b B Then the distance from A to B (which may not be the distance from B to A) is given by dist(a, B) = max dist(a, B). a A The Hausdorff distance between A and B is then given by H(A, B) = max{dist(a, B), dist(b, A)}. Note that with these definitions, dist(a, B) dist(b, A) in general, but H(A, B) = H(B, A). This metric gives a precise method for measuring the effectiveness of an edge detector whenever the edges are known. For all artificial images tested so far, where the edges E are known, the shearlet methods introduced in this paper consistently yield edges F such that H(E, F ) = 1. An error of H = 1 indicates that given any pixel in the approximate edges, it is at most one pixel away from the true edges, and vice versa. The artificial images tested so far include regular n- gons with n = 3,..., 10 and n = 1000, as well as the irregular polygon seen in Figure 8. Traditional methods such as Canny perform equally as well on the regular polygons, although Canny gives at best an error of H = 2 on the irregular polygon tested. The plots of error versus threshold for both the Canny algorithm and the shearlet algorithm on the irregular polygon can be seen in figure 9. Furthermore, the shearlet methods seem to perform very competitively on natrual images. The results of these tests can be seen in Figure 8. Canny s algorithm has been implemented by MATLAB, and it should noted that the MATLAB postprocessing step is significantly more sophisticated than what has been implemented in the shearlet algorithm. Note that the default for the threshold is set to 0.7. This value is sufficient for regular n-gons with more than four sides and seems to work well for natural images. On artificial images with sharper angles, a threshold parameter value closer to zero works better. For example, to find the edges of the square and the irregular polygon images in Figure 8 the threshold parameter was set to The Gaussian smoothing uses standard deviation σ = 0.2. The minimum size of a subdomain in the subdomain decomposition is a 4 4 grid, and the subdomain threshold parameter is Conclusions The methods in this paper have been shown to be very competitive edge detectors on both natural and artificial images, with a numerical measure of error for the artificial images. 13

14 In fact, the irregular polygon tests suggest that shearlet edge detectors are more effective than known methods at capturing edges with sharp corners. In other words, it seems that the additional directional information given by shearlets may indeed yield improved edge detectors. Future work will include an expanded comparison of shearlet methods to known methods. Significantly more comprehensive testing on artificial images (including with sharp angles) will be performed. Also, a comparison will be done in which the post-processing techniques of the known methods are also applied to the results of the shearlet method. Furthermore, a full investigation of the theoretical potential of shearlet methods will be undertaken. References [1] E. Candes and D. Donoho, Standford University Ridgelets: a Key to Higher- Dimensional Intermittency, emmanual/papers/roysoc. pdf. [2] J. Canny, A Computational Approach to Edge Detection, IEEE Trans. Pattern Analysis and Machine Intell., 8(6): pp , [3] I. Daubechies, Orthonormal Bases of Compactly Supported Wavelets, Comm. Pure Appl. Math, vol. 41, pp , 1988 [4] I. Daubechies, Orthonormal Bases of Wavelets with Finite Support-Connection with Discrete Filter, Wavelets: Time-Frequency Methods in Phase Space, Marseille Colloquium, Berlin, [5] I. Daubechies and J. Lagarias, Two-Scale Difference Equations, I-II. Siam J. Math. Anal., vol. 22, no. 5, pp , [6] R. DeVore, Nonlinear Approximation, Acta Numercia, pp , [7] M. Do and M. Vetterli, Contourlets, Beyond Wavelets, Academic Press, [8] G. Easley, D. Labate, and W.Q. Lim, Sparse Directional Image Representations using the Discrete Shearlet Transform, [9] A. Gelb and E. Tadmor, Adaptive Edge Detectors for Piecewise Smooth Data Based on the minmod Limiter, Journal of Scientific Computing, vol. 28, no. 213, September [10] A. Graps, An Introduction to Wavelets, IEEE Comput. Sc. Eng., vol. 2, pp , [11] S. Mallat and W.L. Hwang, Singularity Detection and Processing with Wavelets, IEEE Trans. Info. Theory, vol. 38, no. 2, pp , March

15 [12] S. Mallat and S. Zhong, Characterization of Signals from Multiscale Edges, IEEE Transaction on Pattern Analysis and Machine Intelligence, vol. 14, no. 7, pp , July [13] S. Mallat and G. Peyre, A Review of Bandlet Methods for Geometrical Image Representation, Numer Algor, no. 18, April [14] S. Mallat, A theory for multiresolution signal decomposition : the wavelet representation, IEEE Transaction on Pattern Analysis and Machine Intelligence, vol. 11, p , July [15] K. K. Pingle, Visual Perception by Computer, In A. Grasselli, editor, Automatic Interpretation and Classification of Images, pp Academic Press, New York, [16] K. Sandberg, University of Colorado at Boulder, The Haar Wavelet Transform, [17] S. Song and P. Que, Wavelet based noise suppression technique and its application to ultrasonic flaw detection, Ultrasonic, vol. 44, Issue 2, February 2006, pg [18] J. Starck, M. Nquyen, and F. Murtagh, Wavelets and Curvelets for Image Deconvolution: a Combined Approach, Signal Processing, vol. 83, no. 10, pp , [19] G. Strang, Wavelets and Dilation Equations: A Brief Introduction, SIAM Review, vol. 28, pp , [20] D. F. Walnut, An Introduction to Wavelet Analysis, pp Springer, [21] Cameraman is copyright Massachusetts Institute of Technology. 15

16 Figure 4: Directional coefficient arrays at level three for a heptagon 16

17 Figure 5: Top: plot of error v. threshold parameter for algorithm that detects modulus maxima along 4 primary direction for a hexagon; Bottom: plot of error v. threshold parameter for algorithm that detects modulus maxima along 8 directions for a hexagon. 17

18 Figure 6: Implementation of Subdomain Decomposition: m 1, m 2, and m 3 are maxima for the first subdivision of the shearlet coefficients, and m 41, m 42, m 43, and m 44 are maxima for the next subdivision of the shearlet coefficients in the fourth quadrant 18

19 Figure 7: Top: plot of error vs. threshold parameter for a heptagon with noise; Bottom: edges vs. threshold parameter for a heptagon with noise 19

20 Figure 8: Column 1: a square, a heptagon, an irregular polygon, and cameraman; Column 2: the edges, as determined by the Canny method; Column 3: the edges, as determined by the shearlet method. 20

21 Figure 9: Top: plot of error vs. threshold parameter for Canny method applied to an irregular polygon; Bottom: plot of error vs. threshold parameter for shearlet algorithm applied to an irregular polygon; Note that the error continues at a constant value for threshold values up to 1 21