Image Denoising via Higher Order SVD (HOSVD)

This repository contains 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.

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Methodology

The algorithm works by:

1. Partitioning images into overlapping 8×8 patches
2. Grouping similar patches using spiral window search
3. Constructing 3D tensors for each patch group
4. Applying HOSVD decomposition
5. Thresholding core tensor values (hard threshold at τ = σ√(s²K))
6. Reconstructing denoised patches
7. 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

Repository Structure

├── notebooks/              # Jupyter notebooks for workflow
├── src/                    # Core implementation
│   ├── denoiser.py
│   ├── patch_grouping.py
│   ├── aggregation.py
│   ├── offsets.py
│   ├── metrics.py
│   └── visualization.py
├── results/
│   ├── figures/            # Visual results
│   ├── final_results.csv   # Complete metric data
│   └── mean_by_sigma.csv   # Averaged results by noise level
└── requirements.txt

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Installation

git clone https://github.com/pedram-ep/hosvd-image-denoising.git
cd image-denoising-hosvd

python3 -m venv venv
source venv/bin/activate
pip install -r requirements.txt

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References

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