Curvelet transform with learning-based tiling

Compact signal and image representations are of crucial importance in a variety of application areas. Wavelet and wavelet-like transforms typically divide the frequency plane in a systematic non-adaptive approach. In this paper, we propose a learning-based method for adapting frequency domain tiling using the curvelet transform as the basis algorithm. The optimal tiling that better represents a single image or a given class of images is computed using denoising performance as the cost function. Simulated additive white Gaussian noise is removed from a given image using a thresholding algorithm. The curvelet tiling that generates maximal denoising performance as measured by PSNR or the logarithm of mean squared error (MSE) is considered optimal. The major adaptations considered are the number of scale decompositions, angular decompositions per scale/quadrant, and scale locations. A global optimization algorithm combining the three adaptations is proposed. Signal representations by adaptive curvelets are shown to outperform default curvelets in partial reconstruction error. Furthermore, adaptive curvelets are used in compressed sensing recovery of incomplete seismic datasets and face images. Visual and numerical improvements across a variety of images and different subsampling ratios are reported. Finally, adaptive curvelets denoising performance is tested on seismic datasets. Our results establish clear numerical and visual performance advantages over the default curvelet transform and the non-local means algorithm (NLM).