Fast Digital Curvelet Transform for Sparse Edge Representation
Find Innovative SolutionsGenerate Solutions
Solution Overview
Problem
Existing signal processing technologies, such as wavelets, are inadequate for efficiently representing and analyzing objects with intermediate dimensional structures like edges in images, leading to suboptimal sparsity and accuracy in data representation and reconstruction.
Innovation Solution
The development of fast digital curvelet transforms, which provide a multiscale pyramid with many directions and positions at each length scale, offering optimally sparse representations of objects with edges through a method involving Fourier transforms, resampling, shearing, and inverse Fast Fourier Transforms, resulting in O(n^2 log n) complexity for 2D and O(n^3 log n) for 3D arrays.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If wavelet decomposition is used for data representation, then the transform can be computed efficiently, but the sparsity and accuracy for representing objects with edges is suboptimal
Solution Approach 1:
The transform is segmented into multiple stages: initial Fourier transform, dyadic annulus decomposition, trapezoidal region subdivision, and iterative resampling-shearing-FFT cycles. Each stage processes specific frequency components separately, enabling efficient computation while achieving optimal sparsity for edge representations.
Solution Approach 2:
The patent introduces directional information by subdividing frequency space into trapezoidal regions corresponding to different orientations. This adds an angular dimension to the traditional multiscale decomposition, enabling the transform to capture both scale and direction, which is crucial for representing edges and intermediate dimensional structures.
2Adaptability or versatility
If traditional multiscale methods are used, then the analysis is simpler, but they are ill-suited for detecting and organizing intermediate dimensional structures like edges
Solution Approach 1:
The transform applies different processing characteristics to different regions of frequency space. Each trapezoidal region is processed with resampling and shearing operations tailored to its specific orientation and scale, enabling the transform to adapt locally to the geometric features present in different frequency bands.
Solution Approach 2:
The transform dynamically adjusts the number and orientation of trapezoidal regions at each scale level. The decomposition adapts to the data by creating a multiscale pyramid with many directions and positions at each length scale, allowing the transform structure to flexibly accommodate intermediate dimensional structures.
3Loss of information
If curvelet transform with many directions and positions is implemented, then optimally sparse representations are achieved, but the computational complexity increases to O(n^2 log n) for 2D and O(n^3 log n) for 3D arrays
Solution Approach 1:
The patent performs preliminary Fourier transformation and frequency space decomposition before the main transform iterations. By pre-organizing the frequency data into dyadic annuli and trapezoidal regions, the subsequent resampling and shearing operations can proceed more efficiently, reducing overall computational burden.
Solution Approach 2:
The patent replaces direct spatial-domain convolution operations with frequency-domain multiplications and FFT-based computations. This substitution leverages the efficiency of Fast Fourier Transforms to compute the complex multi-directional projections required for curvelet transforms, significantly improving computational efficiency.
Data Source
AI summary
Fast digital implementations of the second generation curvelet transform for use in data processing are disclosed. One such digital transformation is based on unequally-spaced fast Fourier transforms (USFFT) while another is based on the wrapping of specially selected Fourier samples. Both digital transformations return a table of digital curvelet coefficients indexed by a scale parameter, an orientation parameter, and a spatial location parameter. Both implementations are fast in the sense that they run in about O(n2 log n) flops for n by n Cartesian arrays or about O(N log N) flops for Cartesian arrays of size N=n3; in addition, they are also invertible, with rapid inversion algorithms of about the same complexity.