Shrinkage Fields (image Restoration)

Shrinkage fields is a random field-based machine learning technique that aims to perform high quality image restoration (denoising and deblurring) using low computational overhead.

Method

The restored image $x$ is predicted from a corrupted observation $y$ after training on a set of sample images $S$.

A shrinkage (mapping) function $_\left(v\right)=_^_\exp \left(-\frac^\right)$ is directly modeled as a linear combination of radial basis function kernels, where $\gamma$ is the shared precision parameter, $\mu$ denotes the (equidistant) kernel positions, and M is the number of Gaussian kernels.

Because the shrinkage function is directly modeled, the optimization procedure is reduced to a single quadratic minimization per iteration, denoted as the prediction of a shrinkage field $_\left(\text\right)=^\left\lbrack \frac\right\rbrack =^\eta$ where $\mathcal$ denotes the discrete Fourier transform and $F_x$ is the 2D convolution $\text\otimes \text$ with point spread function filter, $\breve$ is an optical transfer function defined as the discrete Fourier transform of $\text$, and $^$ is the complex conjugate of $\breve$.

$_$ is learned as $_=_\left(_\right)$ for each iteration $t$ with the initial case $_=y$, this forms a cascade of Gaussian conditional random fields (or cascade of shrinkage fields (CSF)). Loss-minimization is used to learn the model parameters $_=_^$.

The learning objective function is defined as $J\left(_\right)=_^l\left(_^;_^\right)$, where $l$ is a differentiable loss function which is greedily minimized using training data $_^$ and $_^$.

Performance

Preliminary tests by the author suggest that RTF₅

obtains slightly better denoising performance than $_^$, followed by $_^$, $_^$, $_^$, and BM3D.

BM3D denoising speed falls between that of $_^$ and $_^$, RTF being an order of magnitude slower.

Advantages

* Results are comparable to those obtained by BM3D (reference in state of the art denoising since its inception in 2007)
* Minimal runtime compared to other high-performance methods (potentially applicable within embedded devices)
* Parallelizable (e.g.: possible GPU implementation)
* Predictability: $O(D \log D)$ runtime where $D$ is the number of pixels
* Fast training even with CPU

Implementations

* A reference implementation has been written in MATLAB and released under the BSD 2-Clause license
shrinkage-fields

See also

* Random field
* Discrete Fourier transform
* Convolution
* Noise reduction
* Deblurring

References

*

{{Noise, state=uncollapsed
Image noise reduction techniques