Directional image processing using wavelet pairs

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 low-frequency approximations and on the extraction of highfrequency details. The DWT thus yields a dyadic representation that divides the spectrum into lowand high-frequency 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 dual-tree 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 dual-tree M-band 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 extended2 the dyadic case (M = 2) results of Kingsbury, Selesnick and Baraniuk3 to the M-band case. This extension to Figure 1. Two-dimensional wavelet spatial representation for M = 3. The negatively and positively oriented shapes capture different directions and frequencies.