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Electronic Imaging & Signal Processing
Directional image processing using wavelet pairs
Caroline Chaux, JeanChristophe Pesquet, and Laurent Duval
A novel transform based on Mband dualtree wavelets offers local, multiscale, and directional analysis.
3 October 2008, SPIE Newsroom. DOI: 10.1117/2.1200808.1242
The Fourier transform (FT) nicely represents a wiggling, sinelike data set with only a few peaks in the frequency domain. But while the FT is appropriate for simple stationary data like harmonic waves, complicated signals or images require more elaborate transforms. These transforms must catch fast local variations, detect blurred object contours, differentiate textures, and capture different size features. The standard discrete wavelet transform (DWT)^{1} has emerged as an appealing alternative to the FT. Its implementation relies on the iteration of lowfrequency approximations and on the extraction of highfrequency details. The DWT thus yields a dyadic representation that divides the spectrum into low and highfrequency bands. In addition, the transform is multiscale, capturing both big and small features. Some forms preserve orthonormality as well, an important property for filtering random noise. To detect features like edges or surfaces, researchers have developed many extensions to wavelets. These extensions often add redundancy to the transform, whereby data is converted into more coefficients than samples. Surprisingly, such redundancy helps retain key image features with a smaller subset of significant coefficients. With so many options, picking the appropriate transform for a class of signals can be confusing. Recently, however, the dualtree wavelet transform has reconciled representation power and simplicity. It is based on two sets of wavelets designed as a Hilbert (or quadrature) pair, similar to a sine and cosine signal. We developed an extension to this transform, the dualtree Mband wavelet decomposition, that possesses unique geometrical features. This decomposition approach provides a local, multiscale, directional analysis of images. Figure 1 shows the 2D wavelet spatial representation. The negatively and positively oriented shapes catch different directions and frequencies. We have extended^{2} the dyadic case (M=2) results of Kingsbury, Selesnick and Baraniuk^{3} to the Mband case. This extension to M≥2 improves the flexibility of the transform by decomposing n dimensional data into M^{n} trends (indexed in Figure 1 by m_{1} and m_{2} for n=2). It separates data into frequency bands and orientations at different scales. In addition, the dualtree decomposition typically has a redundancy of 2 in the real case and 4 in the complex case. The decomposition can be either orthogonal or biorthogonal.^{4} Figure 1. Twodimensional wavelet spatial representation for M=3. The negatively and positively oriented shapes capture different directions and frequencies. Our wavelet transform is composed of cascading Mband filter banks.^{5} To obtain the Mband trees, we must perform two Mband multiresolution analyses in parallel in the real case, or four in the complex case. To produce the redundancy, we combine a wavelet and its Hilbert transform.^{6} Successively applying the monodimensional transform on image rows and columns generates the separable, twodimensional version. The dualtree decompositions are almost shiftinvariant, with each trend keeping the same characteristics when the data is delayed. They also offer directional selectivity. These properties are highlighted in the textured Barbara image in Figure 2, which shows the decomposition coefficients for M=3 and two scales. Different subbands and the two sets of coefficients preferentially capture different directions. Each branch of the decomposition addresses a specific quadrant of the frequency domain shown in Figure 3. Figure 2. Barbara image ^{7} decomposition on two scales. Each subband and coefficient captures different directions. Figure 3. Direction selection in the 2D frequency plane for M=3. Each branch of the decomposition focuses on different quadrants in the frequency domain. To obtain the best results when processing realworld data, it is important to pay attention to the continuousdicrete domain transition. The wavelet transform, like many mathematical tools, is well defined in a continuous space formalism but is often used on sampled data. Since at least two transforms are applied to the same data set, we must use preprocessing to realign them with respect to the appropriate grid for each one. After preprocessing and decomposition, we take into account remaining correlations in the transform domain. These correlations exist since the decomposition is made redundant, and some data features appear at different scales (see Figure 2). Unlike many other geometrical transforms, the Hilbert relationship between the wavelets enables the modeling and explicit calculation of the coefficient correlations.^{8} This modeling is especially important for further processing in the wavelet domain. Our transform can be applied to more involved processing tasks, such as denoising or deconvolution. Wavelets are an appealing option for this work because they allow local modification of coefficients. The simplest approach, called thresholding or shrinkage,^{9} suppresses coefficients below a threshold. This method relies on the assumption that useful data features, which are well concentrated by the transform, will be kept, while small spread noise coefficients will be cancelled. With a precise knowledge of induced correlations, we can better preserve groups of meaningful neighboring coefficients. We have also proposed very efficient noise estimators using Mband dualtree wavelets.^{10} Figure 4 shows the effect of applying the transform. The top left satellite image has been corrupted by heavy Gaussian noise (top right), making the details barely distinguishable. Efficient noise detection combined with dualtree wavelets yields the bottom image. Most of the salient features have been blindly recovered, which would have been impossible with more standard filtering techniques. The transform may also prove useful for data reconstruction. Since the transform is redundant, we envision several reconstructions. We proposed^{2–4} using an optimal synthesis frame operator based on pseudoinverse techniques. Figure 4. Multispectral denoising. The original image (bottom) was corrupted by Gaussian noise (top right). Applying noise reduction and dualtree wavelets recovered the restored image (top left). The Mband dualtree wavelet transform has already proven its efficiency in several domains, including deconvolution,^{11} multispectral denoising^{10} and coherent noise removal in seismics.^{12} This approach will likely benefit other domains, including image compression^{13} and reconstruction. Future work will focus on new decompositions to analyze highdimensional data from nextgeneration imaging systems.
Caroline Chaux, JeanChristophe Pesquet Université ParisEst Institut Gaspard Monge, UMR CNRS 8049 MarnelaVallée, France Caroline Chaux received her PhD in signal and image processing in 2006 from the Université ParisEst, France. From 2006 to 2007, she worked at the National Institute for Research in Computer Science and Control in SophiaAntipolis, France. In 2007, she joined the Informatics Laboratory at the Université ParisEst. She received the Best Student Paper Award at the IEEE International Conference on Acoustics, Speech, and Signal Processing in 2005. JeanChristophe Pesquet received his PhD from the Université ParisSud (XI) in 1990, and in 1999 he earned a postdoctoral degree authorizing him to direct the research of others. From 1991 to 1999, he was a lecturer at the university, and a researcher at the Systems and Signals Laboratory at the French National Center for Scientific Research in GifsurYvette. He is currently a professor at the Université ParisEst and a researcher in the Informatics Laboratory. Laurent Duval French Insitute of Oil Rueil Malmaison, France In 2000, Laurent Duval received his PhD in signal processing from the Université ParisSud (XI) in Orsay, France. In April 2000, he joined the French Institute of Oil. He now applies signal processing to geoscience, materials characterization, chemical analysis and engine control. His interests include filter bank techniques as applied to signal detection, filtering and data compression.


