Wavelets and curvelets restore images

For more than 100 years, mathematicians have used the Fourier transform to decompose arbitrary functions into a basis function set of sine and cosine.

Jan 1st, 2002

For more than 100 years, mathematicians have used the Fourier transform to decompose arbitrary functions into a basis function set of sine and cosine. With this capability, the Fourier transform finds many uses in determining the frequency and power spectrums of time-varying signals. Unfortunately, by using sine and cosine functions as its basis functions, the Fourier transform assumes that the signal being decomposed is both periodic and of infinite length. Therefore, it cannot easily represent sharp transient signals that are not periodic.

In image processing, this limitation has led to the use of wavelet transforms as an alternative means to analyze, compress, and enhance images. Indeed, the inclusion of wavelet-based compression in the latest JPEG-2000 standard attests to the success of this relatively new mathematical discipline.

Despite the success of wavelet transforms in some imaging applications, many images contain features that are not conducive to wavelet operations. In noisy images that contain elongated patterns or structures, for example, wavelet transforms may not be capable of detecting these images.

To overcome this limitation, Jean-Luc Starck of the Astrophysics Service of CEA/Saclay (Gif-sur-Yvette, France) and his colleagues at Stanford University (Stanford, CA) and the California Institute of Technology (Pasadena, CA) have developed two new multiscale representation systems called ridgelets and curvelets. When combined with traditional wavelet methods, these functions have been used to substantially improve noise removal within images.

Ridgelets and curvelets are mathematical functions that exhibit very high directional sensitivity. Essentially, as Starck points out, ridgelet-based analysis can be understood as wavelet analysis in the Radon domain (see figure). The Radon transform changes two-dimensional images that contains lines into a domain of possible line parameters, where each line in the image yields a peak positioned at the corresponding line parameter. The ridgelet transform then applies a one-dimensional wavelet transform to the slices of the Radon transform where the angular variable is constant and time is varying.

In practice, the ridgelet transform can only be used to find lines that are the size of the image. To detect line segments, the image must be decomposed into a set of blocks and the ridgelet transform applied to each block. Although wavelets are not good at restoring long edges within images, curvelets are also challenged when images have small features within them. However, if curvelet and wavelet transforms are combined, they become more effective in restoring noisy images. To demonstrate this, a noisy image was filtered by a wavelet transform, a curvelet transform, and a combined transform. Results showed that by using the combined transform, the resultant image was free of many of the image artifacts associated with typical image-filtering methods.

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