Patch-based Image Denoising via Higher Order SVD

Wed, 01 Jan 2025 00:00:00 +0000

A near complete implementation of the method in the paper “Image Denoising using the Higher Order Singular Value Decomposition”. This project uses the Python language and vast libraries to apply this algorithm for grayscale images, and evaluates it by applying the method on the dataset “CBSD68” and calculating different image-difference scores.

This project a Python implementation of patch-based image denoising using Higher Order Singular Value Decomposition (HOSVD), based on the method introduced in:

“Image Denoising using the Higher Order Singular Value Decomposition”
Ajit Rajwade, Anand Rangarajan, and Arunava Banerjee.
IEEE Transactions on Pattern Analysis and Machine Intelligence, 2013.

Methodology

The algorithm works by:

Partitioning images into overlapping 8×8 patches
Grouping similar patches using spiral window search
Constructing 3D tensors for each patch group
Applying HOSVD decomposition
Thresholding core tensor values (hard threshold at τ = σ√(s²K))
Reconstructing denoised patches
Aggregating patches with averaging

Note: Implementation is optimized for grayscale images and omits the Wiener filtering step from the original paper

Evaluation Metrics

We used seven metrics to evaluate denoising performance on the CBSD68 dataset:

Metric	Formula	Description	Ideal Value
MSE	$\frac{1}{MN}\sum_{i=0}^{M-1}\sum_{j=0}^{N-1}[I(i,j)-K(i,j)]^2$	Mean Squared Error	Lower is better
RMSE	$\sqrt{\text{MSE}}$	Root Mean Squared Error	Lower is better
MAE	$\frac{1}{MN}\sum_{i=0}^{M-1}\sum_{j=0}^{N-1}\\|I(i,j)-K(i,j)\\|$	Mean Absolute Error	Lower is better
PSNR	$10 \cdot \log_{10}\left(\frac{L^2_{\max}}{\text{MSE}}\right)$	Peak Signal-to-Noise Ratio	Higher is better
SSIM	$\frac{(2\mu_x\mu_y + c_1)(2\sigma_{xy} + c_2)}{(\mu_x^2 + \mu_y^2 + c_1)(\sigma_x^2 + \sigma_y^2 + c_2)}$	Structural Similarity Index	Closer to 1
NRMSE	$\frac{\sqrt{\frac{1}{N}\sum_{i=1}^N(x_i - y_i)^2}}{\text{median}(\\|x\\|)}$	Normalized RMSE	Lower is better
UQI	$\frac{4\sigma_{xy}\bar{x}\bar{y}}{(\sigma_x^2 + \sigma_y^2)(\bar{x}^2 + \bar{y}^2)}$	Universal Quality Index	Closer to 1

References

Rajwade, A., Rangarajan, A., & Banerjee, A. (2013). “Image Denoising using the Higher Order Singular Value Decomposition”.
Feschet, F. (2019). “Implementation of a denoising algorithm based on High‑Order Singular Value Decomposition of tensors”
Guo, J., Chen, H., Shen, Z., and Wang, Z. (2022). “Image denoising based on global image similar patches searching and HOSVD to patches tensor”
Wang, Z., and Bovik, A. C. (2002). “A universal image quality index”
Wang, Z., Bovik, A. C., Sheikh, H. R., and Simoncelli, E. P., “Image quality assessment: From error visibility to structural similarity”
CBSD68‑dataset: Color BSD68 dataset for image denoising benchmarks. Available at:

Image Processing |

Patch-based Image Denoising via Higher Order SVD

Methodology

Evaluation Metrics

References