Image Reconstruction Regularization for Outlier Robustness
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Solution Overview
Problem
Existing image reconstruction techniques, such as those using the Levenberg-Marquardt algorithm, face challenges with robustness to outliers and smoothing effects, leading to sub-optimal image quality and slow convergence rates, especially when dealing with noisy signals.
Innovation Solution
Implementing a regularization function with a derivative of bounded size, such as sqrt([grad P(r)]^2 + a^2) - a, which ensures sensitivity to changes in the reconstructed signal while preventing undue effects of outliers, and using the Levenberg-Marquardt algorithm for faster convergence.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If a sum of squares of total variation values is used as the regularization function, then the improved image has reduced noise, but the reconstruction is not robust to outliers and has unnecessary smoothing effect
Solution Approach 1:
The patent changes the mathematical parameter of the regularization function from a sum of squares (L2 norm) to a sum of absolute values (L1 norm) of total variation values. This parameter change transforms the optimization landscape to be more robust to outliers while maintaining edge preservation capabilities, directly resolving the contradiction between robustness and image quality.
2Reliability
If a sum of absolute values of total variation values is used as the regularization function, then the reconstruction is more robust to outliers, but the reconstruction is still sub-optimal
Solution Approach 1:
The patent combines multiple regularization approaches into a composite regularization function that integrates both L1 norm (for robustness) and gradient-based terms (for optimal reconstruction). This composite approach merges the strengths of different mathematical formulations to achieve both outlier robustness and superior reconstruction quality simultaneously.
3Productivity
If the conjugate gradient algorithm is used to optimize the reconstruction, then the algorithm converges, but the speed of convergence is limited
Solution Approach 1:
The patent modifies the optimization criterion by transforming the regularization term into a form with bounded derivatives. This parameter transformation enables the use of more efficient optimization algorithms like Levenberg-Marquardt that converge faster while maintaining the robustness properties of the original L1-based regularization.
4Productivity
If a regularization function with unbounded derivatives is used, then the algorithm can converge faster, but the algorithm gets trapped in local minima
Solution Approach 1:
The patent applies a mathematical transformation to the regularization function that bounds its derivatives. This parameter change creates an optimization landscape that is both smooth enough for fast convergence algorithms and structured enough to avoid local minima, resolving the contradiction between convergence speed and reliability.
Data Source
Figure 1~2
AI summary
A signal reconstruction apparatus, such as an image reconstruction apparatus generates a reconstructed location dependent image based on an incoming signal. Successive versions of the reconstructed image signal are iteratively computed using a Levenberg-Marquardt type algorithm for solving an equation V*J=0, wherein V is a vector and J is a derivative of the vector with respect to the reconstructed signal. A vector is used that has first vector components V1 corresponding to differences between values of an incoming signal and predictions of the incoming signal values derived from the reconstructed signal and second vector components V2 that are square roots of values obtained by applying a regularization function. A regularization function is used that has a derivative of bounded size that varies continuously as a function of gradient size from zero at zero gradient. Changes in the derivative of the regularization function are concentrated in a finite, limited interval of values of gradient sizes.