Compressed Sensing Signal Reconstruction via Sparse Matrix Projection
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Solution Overview
Problem
Current methods for reconstructing digital signals, images, or spectra require a large number of measurements, which is time-consuming and costly, and there is a need for a system that can achieve acceptable-quality reconstructions with fewer measurements.
Innovation Solution
The method employs Compressed Sensing (CS) matrices and algorithms to approximate digital signals using significantly fewer measurements than traditional schemes, allowing for reduced-dimensionality measurements while promoting sparsity in the reconstruction process.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If traditional measurement schemes are used to reconstruct digital signals, images, or spectra, then measurement precision is maintained, but the number of measurements required increases, leading to increased time consumption and cost
Solution Approach 1:
The patent applies compressed sensing theory to transform the measurement problem from a high-dimensional space (requiring m measurements for an m-dimensional signal) to a lower-dimensional space (requiring only n measurements where n < m). By representing the signal in a sparse basis and using random projection matrices, the invention reduces the measurement dimensionality while preserving reconstruction quality, directly resolving the contradiction between measurement precision and time consumption
Solution Approach 2:
The invention changes the fundamental parameter of measurement quantity from m (traditional) to n (compressed), where n is significantly less than m. This parameter change is enabled by exploiting the sparsity of signals in certain bases and using probabilistic measurement matrices, allowing acceptable-quality reconstructions with far fewer measurements, thereby reducing measurement time while maintaining precision
2Measurement precision
If traditional measurement schemes are used to reconstruct digital signals, images, or spectra, then measurement precision is maintained, but the cost of data capture increases
Solution Approach 1:
The patent reduces the quantity of measurements from m to n (where n < m) by changing the measurement approach to compressed sensing. This dimensional reduction directly lowers the cost of data capture while maintaining reconstruction quality, as fewer measurements are required to achieve the same level of precision
Solution Approach 2:
The invention changes the measurement parameter from traditional full sampling (m measurements) to compressed sampling (n measurements). This parameter change enables cost reduction while preserving measurement precision, as the compressed sensing framework allows acceptable-quality reconstructions with significantly fewer measurements
3Loss of time
If fewer measurements are made to reduce time and cost, then measurement time and cost decrease, but reconstruction quality may deteriorate
Solution Approach 1:
The invention changes the measurement parameter from traditional full sampling to compressed sensing sampling, enabling fewer measurements (n < m) while maintaining reconstruction quality. This parameter change resolves the contradiction by showing that reduced measurement time does not necessarily lead to deteriorated quality when compressed sensing algorithms are used
Solution Approach 2:
The patent replaces traditional mechanical sampling systems with a compressed sensing framework that uses random projection matrices and sparse representation. This substitution allows fewer measurements to achieve the same reconstruction quality, resolving the contradiction between measurement time and quality by changing the fundamental measurement mechanism
Data Source
AI summary
Method and apparatus for compressed sensing yields acceptable quality reconstructions of an object from reduced numbers of measurements. A component x of a signal or image is represented as a vector having m entries. Measurements y, comprising a vector with n entries, where n is less than m, are made. An approximate reconstruction of the m-vector x is made from y. Special measurement matrices allow measurements y=Ax+z, where y is the measured m-vector, x the desired n-vector and z an m-vector representing noise. “A” is an n by m matrix, i.e. an array with fewer rows than columns. “A” enables delivery of an approximate reconstruction, x#, of x. An embodiment discloses approximate reconstruction of x from the reduced-dimensionality measurement y. Given y, and the matrix A, x# of x is possible. This embodiment is driven by the goal of promoting the approximate sparsity of x#.