Aliasing-Free Convolutional Nonlinear Networks
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Solution Overview
Problem
Convolutional neural networks (CNNs) face significant challenges in achieving shift invariance due to aliasing issues induced by non-polynomial functions, particularly in applications like computational lithography where small image transformations lead to significant variations in device manufacturing, and existing solutions only partially mitigate these problems.
Innovation Solution
Implementing non-polynomial operations as implicitly defined functions, computed via iterative algorithms involving linear operations and pointwise multiplications, with proper spectrum control to avoid aliasing, and employing methods like symmetric summation and optimal image positioning to ensure shift invariance.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Adaptability or versatility
If non-polynomial functions are used in CNNs, then the network can perform complex nonlinear operations and achieve better pattern recognition performance, but aliasing is induced that causes loss of shift invariance and signal integrity
Solution Approach 1:
The patent segments the nonlinear operation into multiple polynomial approximation stages. Instead of applying a single non-polynomial function that causes aliasing, the network breaks down the computation into several polynomial operations with intermediate downsampling steps, each operating on a limited frequency band to avoid aliasing while collectively achieving the desired nonlinear transformation.
Solution Approach 2:
The patent applies preliminary low-pass filtering before each downsampling operation to remove high-frequency components that would cause aliasing. This preliminary action ensures that the signal bandwidth is restricted to satisfy the Nyquist criterion before sampling, preventing aliasing-induced loss of shift invariance while preserving the essential nonlinear characteristics through subsequent polynomial operations.
2Productivity
If straightforward downsampling is used in CNNs, then computational complexity is reduced and processing speed is improved, but significant aliasing is caused that degrades signal integrity
Solution Approach 1:
The patent applies preliminary low-pass filtering before downsampling to remove high-frequency components that would cause aliasing. This preliminary action ensures that the signal bandwidth is restricted to satisfy the Nyquist criterion before sampling, preventing aliasing while maintaining processing efficiency through controlled polynomial approximations.
Solution Approach 2:
The patent changes the bandwidth parameter of the signal by applying low-pass filtering with carefully selected cutoff frequencies before downsampling. This parameter change ensures that the signal satisfies the Nyquist criterion at each downsampling stage, preventing aliasing while maintaining computational efficiency through controlled polynomial approximations rather than complex anti-aliasing filters.
3Reliability
If polynomial approximation is used to avoid aliasing, then shift invariance is preserved, but computational complexity increases due to multiple iterative steps
Solution Approach 1:
The patent segments the complex nonlinear operation into multiple simpler polynomial approximation stages. Each stage operates on a restricted frequency band with lower computational requirements, and the cumulative effect of these segmented stages achieves the desired nonlinear transformation while preserving shift invariance and reducing overall computational complexity compared to direct non-polynomial operations.
Solution Approach 2:
The patent applies polynomial approximations of limited degree at each stage rather than attempting to capture the full nonlinear behavior in a single high-degree polynomial or non-polynomial function. This partial action approach achieves sufficient accuracy for pattern recognition tasks while significantly reducing computational complexity, as lower-degree polynomials require fewer operations and less memory.
Data Source
AI summary
Methods are disclosed for aliasing-free nonlinear signal processing, by implementing non-polynomial operations as implicitly defined functions that are computed iteratively using linear shift-invariant convolutions in conjunction with polynomial operations, where upsampling and/or downsampling of signals are employed to control their spectra and avoid aliasing completely. Techniques of system or image symmetrization are also disclosed to render a convolutional nonlinear network shift-invariant under an arbitrary spacetime shift.