Utilizing the structure of a redundant dictionary comprised of wavelets and curvelets with compressed sensing

The discrete curvelet transform decomposes an image into a set of fundamental components that are distinguished by direction and size and a low-frequency representation. The curvelet representation of a natural image is approximately sparse; thus, it is useful for compressed sensing. However, with natural images, the low-frequency portion is seldom sparse. This manuscript presents a method to modify the redundant sparsifying transformation comprised of the wavelet and curvelet transforms to take advantage of this fact for compressed sensing image reconstruction. Instead of relying on sparsity for this low-frequency estimate, the Nyquist-Shannon sampling theorem specifies a rectangular region centered on the 0 frequency to be collected, which is used to generate a blurry estimate. A basis pursuit denoising problem is solved to determine the details with a modified sparsifying transformation. Improvements in quality are shown on magnetic resonance and optical images. (c) 2022 SPIE and IS&T

Automated summary

The discrete curvelet transform in compressed sensing image reconstruction can take advantage of the Nyquist-Shannon sampling theorem by modifying the redundant sparsifying transformation to generate a blurry estimate, resulting in improved quality.