Reconstructing Reflection Coefficients via L1-Norm Minimization
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Solution Overview
Problem
Current imaging methods for determining the internal structure of specimens often smooth out discrete structures, leading to inaccurate representations of the internal structure due to assumptions about smoothness, and are sensitive to errors in measurement series, which can destabilize the reconstruction process.
Innovation Solution
A method for reconstructing the spatial distribution of reflection coefficients by parameterizing the specimen into volume elements, creating route matrices for different measurement configurations, and merging measurement series and matrices to solve for reflection coefficients, allowing for sparse sampling that violates Nyquist's theorem, enabling sharper structure representation and robust error handling.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Stability of the object's composition
If smoothness assumption is applied according to Occam's principle, then the solution stability is improved, but the sharpness of discrete structures is lost
Solution Approach 1:
The patent changes the boundary condition from a smoothness constraint to a sparsity constraint on the reflection coefficient vector. By minimizing the L1-norm of the reflection coefficient vector instead of assuming smoothness, the method enables sharp discrete structures to be reconstructed while maintaining solution stability through the sparsity prior.
Solution Approach 2:
The patent applies local quality by allowing different regions of the specimen to have different properties: most regions have zero reflection coefficient (homogeneous) while specific localized regions have non-zero reflection coefficients representing discrete structures. This sparse representation captures sharp local features without requiring global smoothness.
2Measurement precision
If complex filtering is applied to measurement series, then the measurement precision is improved, but the device complexity increases
Solution Approach 1:
The patent replaces complex physical filtering equipment with computational signal processing. By using L1-norm minimization and sparse reconstruction algorithms, the method achieves accurate measurement results through software-based error handling and signal processing rather than requiring complex hardware filtering systems.
Solution Approach 2:
The patent introduces an intermediary computational model (sparse reconstruction algorithm) that mediates between the raw measurements and the final image. This computational intermediary handles errors and noise in the measurement series without requiring complex physical filtering, simplifying the overall system while maintaining precision.
3Manufacturing precision
If sufficient measurement series are recorded to ensure accuracy, then the reconstruction quality is improved, but the measurement time increases
Solution Approach 1:
The patent applies partial action by using fewer measurement series than traditionally required, compensated by the sparsity constraint. The L1-norm minimization allows accurate reconstruction with incomplete or reduced measurement data, reducing measurement time while maintaining or improving reconstruction quality through the power of sparse optimization.
Applied Scientific Principles
This section explains which scientific principles are used to turn an abstract innovation direction into a practical engineering solution.
Function Achieved in This Case
This approach allows for the reconstruction of sharply defined internal structures with simpler and cost-effective equipment, reducing the need for complex filtering and achieving accurate three-dimensional distribution of reflection coefficients, even when measurements violate Shannon's sampling theorem.
Implementation Method 1
wave-mechanical interactions, such as reflections or scattering, of the sample body can be measured
Implementation Method 2
wave-mechanical interactions, such as reflections or scattering, of the sample body can be measured
Data Source
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AI summary
One aspect of the invention relates to a method for reconstructing the spatial distribution of a reflection coefficient for waves in a sample body, comprising the following steps: parameterizing the sample body by means of N volume elements; determining M different measurement configurations of an emitting device and an associated receiving device; defining the number T of measurement points; setting up M path matrices G[1], wherein the possible paths of reflected or scattered waves are encoded in the i-th path matrix; registering M series of measurements, wherein an excitation signal a[i] is fed to the emitting device in the i-th series of measurements, and the associated receiving device (5) registers a series of measurements x[i]; calculating a predictive differential vector Δx = x- G-s [n], which contains the difference between the elements of the M registered series of measurements x[i] and the predictable series of measurements G[i] tj-s[n] j; minimizing a norm of the predictive differential vector || Δx ||; and providing the reflection coefficients s[n]. The invention also relates to an apparatus for carrying out the method.