Patch-based Image Denoising via Higher Order SVD
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: GitHub
