Multiscale Image Filtering via Hierarchical Kernel Approximation
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Solution Overview
Problem
Existing image processing methods for applying large linear translation-invariant filters are computationally expensive, particularly due to the high cost of convolution operations, and lack translation invariance, which affects accuracy in tasks like gradient field integration and scattered data interpolation.
Innovation Solution
A multiscale scheme that uses numerically optimized small kernels to approximate the effect of large filters, allowing for efficient computation in linear time by repeatedly performing convolutions with fixed-width kernels while downsampling and upsampling images, tailored to directly reproduce the effect of specific LTI operators.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If convolution with large kernels is used for image processing, then filtering accuracy is improved, but computational cost increases to O(n2)
Solution Approach 1:
The patent segments the large kernel convolution operation into multiple smaller kernel convolutions performed at different scales. By decomposing the filtering task into hierarchical levels with progressively smaller kernels, the method achieves O(n) computational complexity while maintaining accurate approximation of the original large kernel filtering effect.
2Productivity
If FFT-based methods are used for periodic domains, then computational cost reduces to O(nlogn), but translation invariance is lost due to subsampling operations
Solution Approach 1:
The patent applies local quality by using fixed-width kernels at each scale that maintain consistent filtering properties regardless of position in the image. This local consistency ensures translation invariance is preserved, as the same kernel operations are applied uniformly across all regions and scales, unlike subsampling methods that introduce position-dependent artifacts.
3Productivity
If separable kernel approximation is used, then computational cost reduces to O(kn), but accuracy deteriorates for non-separable kernels
Solution Approach 1:
The patent transitions from 2D kernel approximation to a multi-scale hierarchical approach, adding the scale dimension to the problem. By operating at multiple resolutions and combining results, the method achieves accurate approximation of non-separable kernels without being constrained to separable forms, effectively solving the accuracy-efficiency trade-off through dimensional expansion.
Data Source
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AI summary
A computer implemented method for A computer implemented method for applying a numerical approximation of a convolution of image I as represented by hierarchical signals a l of level l with filter f, said method including the steps of: generating a forward transformation by applying a convolution between a l and kernel hI or each level of hierarchy l and by down-sampling the result of convolved a l and hI; generating a backward transformation by applying, for each level of hierarchy l, a convolution between kernel h 2 and an up-sampled forward transformation and combining the result with a convolution of al with kernel g; and combining the forward transformation with the backward transformation, to yield â 0 being an approximation of a convolution between a and f, wherein kernels hI, hI, and g are optimized kernels of filter f.