NCSU REU - Edge Detection with 2D Wavelets

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Georgiana Weaver
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1 NCSU REU - Edge Detection with 2D Wavelets Kevin McGoff July 12, 7 Contents 1 Introduction 1 2 Getting the coefficients 2 3 Organizing the coefficients 2 4 Detecting edges 2 5 Results 3 6 Conclusions 3 1 Introduction In this document we describe some numerical explorations undertaken by Katherine Maschmeyer and Kevin McGoff. We sought a two dimensional analogue of the method described by Mallat in the one dimensional case in [1]. The idea is as follows. Given an image, we first decompose the image into its discrete two dimensional wavelet decomposition. We then arrange these coefficients in such a way that each coefficient is associated with a spatial point on the original image. Finally, we seek to infer the location of the edges in the original image by finding certain local maxima in the modulus of the coefficients. 1

2 2 Getting the coefficients This step in the algorithm is quite standard by now, and we use the Matlab command wavedec to implement it. The only parameter involved in this step is the choice of wavelet. As described in [1], the theory seems to allow for any wavelet which is the derivative of a smothing function and which has at least one vanishing moment. Thus we ran our experiments with the wavelet labelled coif1 by Matlab. 3 Organizing the coefficients At this stage in the algorithm, we deal with a problem of size. If the original image is size n n, then the coefficient matrix at the level of detail j is size n 2 j n 2 j. In order to get precise spatial information out of these coefficients, we must upsample the coefficient matrix so that the coefficients are spread out appropriately over their region of influence. It is not exactly clear to the author what the best method is for this step. 4 Detecting edges Using the analogue to the one dimensional theory presented in [1], the set of points in the original image at which an edge occurs should be contained in the set of local maxima of the absolute value of the wavelet coefficients at the finest level of detail. In one dimension, this concept is relatively clear, because there is only one direction along which the coefficients could be maximed. In two dimensions, there are immediately several directions to consider. Also, in one dimension, there is only one wavelet coefficient at each point. In two dimensions, though, there are three (coming from the horizontal, vertical, and so-called diagonal wavelets). The result of these changes from 1D to 2D is that a significant amount of ambiguity is introduced. There are now two main questions. At a given spatial point, what quantity should be maximized? And in which direction should it be a maximum? In our examples, we have implemented various algorithms, each one answering the above questions differently. 2

3 5 Results Figure 1 is the is the picture we started with. Figure 2 is a plot of the edges, as found by Method 1: a point is labeled an edge if it is a local maximum of the absolute value of at least one of the coefficients in the corresponding direction of the coefficient. Figure 3 Is a plot of the edges, as found by Method 2: a point is labeled an edge if it is a local maximum of the quantity M(x, y) = V (x, y) + H(x, y) + D(x, y), where the V, H, and D are the vertical, horizontal, and diagonal coefficients, respectively. Figure 4 is a plot of the edges, as found by Method 3: at each point, an important direction is chosen according to the magnitudes of the three coefficients at that point. Then a point is a labeled an edge if it is a local maximum of M along the important direction. Notice that all of these methods seem to spread spread the edges out somewhat. Ideally these edges would be sharp, especially when the edges in the image seem to be very distinct, as is the case on the cameraman s back. The fact that our pictures have this problem could be either a) a problem with the upsampling technique, b) a problem with the particular choices we have made, or c) an intrinsic problem of the method of using the 2D wavelet coefficients. A deeper problem seems to be the failure of these methods to detect the certain edges entirely (e.g. the silo in the background or the vertical pole hanging directly below the camera), while at the same time detecting an abundance of edges in the noisy areas such as the grass. 6 Conclusions As the word suggests, this exploration leaves many questions open. Is there a truly two dimensional analogue of Mallat s theorem in one dimension? If so, what is the correct approach? Would this significantly improve upon the naive method of performing one dimensional decompositions twice and combining their results? We continue to search for answers to these questions in our ongoing work, with the hope that using a shearlet decomposition rather than a two dimensional wavelet decomposition may help. The main reason 3

4 50 Figure 1: Cameraman to hope for an improvement via shearlets is that the direction information at each is much finer. References [1] S. Mallat and W.L. Hwang, Singularity detection and processing with wavelets, IEEE Trans. Info. Theory, vol. 38, no. 2, pp , March

5 50 Figure 2: Cameraman with Method 1 50 Figure 3: Cameraman with Method 2 5

6 50 Figure 4: Cameraman with Method 3 6

Edge Detection by Multi-Dimensional Wavelets

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