Contourlet Transforms for Feature DetectionWei-shi TsaiMay 9, 2008AbstractThis project will involve the exploration of a directional extensionof multidimensional wavelet transforms, called “contourlets”, to per-form pattern recognition. First, the general concept of a directionalextension vs. a regular multidimensional wavelet transform will bediscussed and explain the reasoning behind the directional extension.Then, a comparison will be done using sample images between thecontourlet transform and other edge detection methods for featuredetection.1IntroductionTraditionally, feature detection/extraction was done with a variety of meth-ods, such as Laplacian operators, gradient operators, the Laplacian of Gaus-sians, difference of Gaussians, Canny detectors, or anisotropic diffusion.However, wavelet transforms have come into light as a means of featuredetection. [1]Typically, feature detection/extraction is a preliminary step in machinelearning and machine vision applications. However, there is no perfect edgedetector or feature extraction algorithm; one edge detector may work verywell in one application, while the exact same algorithm may fail in otherapplications. Therefore, it is useful to have as many types of algorithmsavailable for evaluation.This paper aims to explore an edge detection algorithm using contourlettransforms. We will attempt to give a brief overview of the contourlet trans-form, use it for edge detection, and compare it against other edge detectionalgorithms.BackgroundWavelets are classified as a linear transform that is capable of displayingthe transformed output at multiple resolutions depending on the point oftime/space and at the desired frequency. In contrast to the short-time Fouriertransform (STFT), the resolution changes depending on the frequency that2is to be examined and at what time or spatial area is to be examined. [2]In the 1-D case, wavelets are used for signal processing by the virtuethat wavelets can store more frequency information with less coefficients andreconstruction is only limited by the coefficients that are available. Waveletscan be naively extended to the 2-D case by means of separable functions,but there is limited directional information stored in a regular 2-D wavelettransform. Because of the seperability limitations, only a horizontal, vertical,and 45 degree component can be easily determined. Incidentally, edges canbe seen easily, but directional information about the edge is not known.Because of this, it takes more coefficients to do a proper reconstruction ofthe edges. [3]Typically, a separable 2-D wavelet transform provides:multiresolution, which is the ability to visualize the transform withvarying resolution from coarse to finelocalization, which is the ability of the basis elements to be localized inboth the spacial and frequency domainscritical sampling, which is the ability for the basis elements to havelittle redundancy.However, it is not capable of providing:directionality, which is having basis elements defined in a variety ofdirections3anisotropy, which is having basis elements defined in various aspectratios and shapes. [4]There are many directional extensions of the 2-D wavelet transform thatcould be potentially examined that also possess directionality and anisotropy.The contourlet transform is a discrete extension of the curvelet transform thataims to c apture curves instead of points, and provides for directionality andanisotropy. Figure 1 shows the general concept of capturing curves. [5]Figure 1: Conceptual visualization of curvelets/contourlets.Contourlets are implemented by using a filter bank that decouples the multi-scale and the directional decompositions. In Figure 2, Do and Vetterli showa conceptual filter bank setup that shows this decoupling. We can see that amultiscale decomposition is done by a Laplacian pyramid, then a directionaldecomposition is done using a directional filter bank. This transform is suit-able for applications involving edge detection with a high curve content. [4]4Figure 2: Filter bank for contourlet transform.Using Contourlets for Edge DetectionOur approach involves taking the contourlet transform of test grayscale im-ages. Code for the contourlet transform is available through the author’s website. [6] The code for the contourlet transform is flexible enough to also dothe regular 2-D separable wavelet transform. The edge detection algorithmis as follows:1. Take the contourlet transform of the image.2. Choose a scale factor to use, and truncate all other coefficients.3. Invert the transform.4. Threshold using the mean of the pixel values of the image.In addition, there is built in code in MATLAB’s Image Processing Toolbox fordoing edge detection using the Prewitt gradient operator, the Sobol gradient5operator, and Canny’s method. These are also run on the test grayscaleimage for comparison purposes. A MATLAB script is used to automate thisprocess.ResultsThe results shown here are not objective results. It is more practical toevaluate the objective performance of a feature extraction algorithm by howthe features extracted contribute to a particular application. This type ofanalysis is beyond the scope of this paper and the reader is referred to paperson machine learning and machine vision.Figure 3 shows the Lena image. Subjectively, the contourlet detectorcaptures edges here rather well, and does better than Canny’s algorithm orwith regular wavelets. We use scale factor 1.Figure 4 shows the Elaine image. Subjectively, the contourlet detectordoes worse than Canny’s method because it has trouble with some of theedges in the background. Again, we use scale factor 1.ConclusionIn this paper, we have shown that the contourlet transform can be used foredge detection. However, it is not perfect and it is not expected to be perfect.We only performed a subjective analysis because an objective analysis should6be done based on how the extracted features are used, such as in a computer-aided detection algorithm or in a machine vision algorithm. Further work ispossible to potentially exploit multiple scales for edge detection, reduce thenoise that is generated in the edge map, and extend the algorithm to colorimages.Figure 3: Lena image.7Figure 4: Elaine image.References[1] P. A. Mlsna and J. J. Rodriguez, “Gradient and laplacian edge detection,”in Handbook of Image and Video Processing, 2nd ed. Elsevier AcademicPress, 2005, ch. 4.13.[2] S. G. Mallat, “A