Abstract
The empirical wavelet transform is an adaptive multi-resolution analysis tool based on the idea of building filters on a data-driven partition of the Fourier domain. However, existing 2D extensions are constrained by the shape of the detected partitioning. In this paper, we provide theoretical results that permits us to build 2D empirical wavelet filters based on an arbitrary partitioning of the frequency domain. We also propose an algorithm to detect such partitioning from an image spectrum by combining a scale-space representation to estimate the position of dominant harmonic modes and a watershed transform to find the boundaries of the different supports making the expected partition. This whole process allows us to define the empirical watershed wavelet transform. We illustrate the effectiveness and the advantages of such adaptive transform, first visually on toy images, and next on both unsupervised texture segmentation and image deconvolution applications.
Highlights
Within the field of image processing and computer vision, multi-resolution analysis is a vastly applicable tool
We propose an algorithm, based on scale-space representations and the watershed transform, to automatically detect a partition corresponding to dominant harmonic modes
We compare the deconvolution results obtained by using framelets [40], empirical curvelet option 1 and the empirical watershed wavelets
Summary
Within the field of image processing and computer vision, multi-resolution analysis is a vastly applicable tool. The use of multi-resolution analysis and compressive sensing theory has led to state-of-the-art denoising and deconvolution techniques. The use of multi-resolution analysis has shown to be quite effective in texture analysis. Wavelets have been the standard tool for multi-resolution analysis, and their effectiveness is matched by their popularity. Wavelets are prescriptive in their construction in the sense that the wavelet filters are built independently of the data. Adaptive (i.e., data-driven) methods have shown many promising improvements in applications across all fields of science. We review the most popular adaptive methods developed in the last two decades
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