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Pawel Sarkowicz
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whole_project_as_readme.md
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{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "6ae6c7f8",
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"metadata": {},
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"source": [
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"# Spectral Image Denoising via Truncated SVD\n",
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"\n",
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"This notebook extracts the image denoising project into a standalone workflow and extends it from **grayscale images** to **actual color images**.\n",
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"\n",
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"The core idea is the same as in the original write-up: if an image matrix has singular value decomposition\n",
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"\\[\n",
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"A = U \\Sigma V^T,\n",
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"\\]\n",
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"then the best rank-$k$ approximation to \\(A\\) in Frobenius norm is obtained by truncating the SVD. This is the **Eckart–Young–Mirsky theorem**.\n",
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"\n",
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"For a grayscale image, the image is a single matrix. For an RGB image, we treat the image as **three matrices**, one for each channel, and apply truncated SVD to each channel separately."
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]
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},
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{
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"cell_type": "markdown",
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"id": "31a665c9",
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"metadata": {},
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"source": [
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"## Outline\n",
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"\n",
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"1. Load an image from disk\n",
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"2. Convert it to grayscale or keep it in RGB\n",
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"3. Add synthetic Gaussian noise\n",
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"4. Compute a truncated SVD reconstruction\n",
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"5. Compare the original, noisy, and denoised images\n",
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"6. Measure quality using MSE and PSNR\n",
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"\n",
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"This notebook is written so that you can use **your own image files** directly."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "88584c56",
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"metadata": {},
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"outputs": [],
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"source": [
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"import numpy as np\n",
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"import matplotlib.pyplot as plt\n",
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"from PIL import Image\n",
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"from pathlib import Path"
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]
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},
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{
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"cell_type": "markdown",
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"id": "f275cbc9",
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"metadata": {},
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"source": [
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"## Helper functions\n",
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"\n",
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"We begin with some utilities for:\n",
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"- loading images,\n",
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"- adding Gaussian noise,\n",
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"- reconstructing rank-$k$ approximations,\n",
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"- computing image-quality metrics."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "21adfcaf",
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"metadata": {},
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"outputs": [],
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"source": [
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"def load_image(path, mode=\"rgb\"):\n",
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" \"\"\"\n",
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" Load an image from disk.\n",
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"\n",
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" Parameters\n",
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" ----------\n",
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" path : str or Path\n",
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" Path to the image file.\n",
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" mode : {\"rgb\", \"gray\"}\n",
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" Whether to load the image as RGB or grayscale.\n",
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"\n",
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" Returns\n",
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" -------\n",
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" np.ndarray\n",
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" Float image array scaled to [0, 255].\n",
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" Shape is (H, W, 3) for RGB and (H, W) for grayscale.\n",
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" \"\"\"\n",
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" path = Path(path)\n",
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" img = Image.open(path)\n",
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"\n",
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" if mode.lower() in {\"gray\", \"grayscale\", \"l\"}:\n",
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" img = img.convert(\"L\")\n",
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" else:\n",
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" img = img.convert(\"RGB\")\n",
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"\n",
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" return np.asarray(img, dtype=np.float64)\n",
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"\n",
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"\n",
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"def show_image(img, title=None):\n",
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" \"\"\"Display a grayscale or RGB image.\"\"\"\n",
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" plt.figure(figsize=(6, 6))\n",
|
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" if img.ndim == 2:\n",
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" plt.imshow(np.clip(img, 0, 255), cmap=\"gray\", vmin=0, vmax=255)\n",
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" else:\n",
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" plt.imshow(np.clip(img, 0, 255).astype(np.uint8))\n",
|
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" if title is not None:\n",
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" plt.title(title)\n",
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" plt.axis(\"off\")\n",
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" plt.show()\n",
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"\n",
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"\n",
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"def add_gaussian_noise(img, sigma=25, seed=0):\n",
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" \"\"\"\n",
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" Add Gaussian noise to an image.\n",
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"\n",
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" Parameters\n",
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" ----------\n",
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" img : np.ndarray\n",
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" Image array in [0, 255].\n",
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" sigma : float\n",
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" Standard deviation of the noise.\n",
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" seed : int\n",
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" Random seed for reproducibility.\n",
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"\n",
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" Returns\n",
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" -------\n",
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" np.ndarray\n",
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" Noisy image clipped to [0, 255].\n",
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" \"\"\"\n",
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" rng = np.random.default_rng(seed)\n",
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" noisy = img + rng.normal(loc=0.0, scale=sigma, size=img.shape)\n",
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" return np.clip(noisy, 0, 255)\n",
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"\n",
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"\n",
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"def truncated_svd_matrix(A, k):\n",
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" \"\"\"\n",
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" Return the rank-k truncated SVD approximation of a 2D matrix A.\n",
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" \"\"\"\n",
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" U, s, Vt = np.linalg.svd(A, full_matrices=False)\n",
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" k = min(k, len(s))\n",
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" return (U[:, :k] * s[:k]) @ Vt[:k, :]\n",
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"\n",
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"\n",
|
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"def truncated_svd_image(img, k):\n",
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" \"\"\"\n",
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" Apply truncated SVD to a grayscale or RGB image.\n",
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"\n",
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" For RGB images, truncated SVD is applied channel-by-channel.\n",
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"\n",
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" Parameters\n",
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" ----------\n",
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" img : np.ndarray\n",
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" Shape (H, W) for grayscale or (H, W, 3) for RGB.\n",
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" k : int\n",
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" Truncation rank.\n",
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"\n",
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" Returns\n",
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" -------\n",
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" np.ndarray\n",
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" Reconstructed image clipped to [0, 255].\n",
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" \"\"\"\n",
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" if img.ndim == 2:\n",
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" recon = truncated_svd_matrix(img, k)\n",
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" return np.clip(recon, 0, 255)\n",
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"\n",
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" if img.ndim == 3:\n",
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" channels = []\n",
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" for c in range(img.shape[2]):\n",
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" channel_recon = truncated_svd_matrix(img[:, :, c], k)\n",
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" channels.append(channel_recon)\n",
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" recon = np.stack(channels, axis=2)\n",
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" return np.clip(recon, 0, 255)\n",
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"\n",
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" raise ValueError(\"Image must be either 2D (grayscale) or 3D (RGB).\")\n",
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"\n",
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"\n",
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"def mse(A, B):\n",
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" \"\"\"Mean squared error between two images.\"\"\"\n",
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" return np.mean((A.astype(np.float64) - B.astype(np.float64)) ** 2)\n",
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"\n",
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"\n",
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"def psnr(A, B, max_val=255.0):\n",
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" \"\"\"Peak signal-to-noise ratio in decibels.\"\"\"\n",
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" err = mse(A, B)\n",
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" if err == 0:\n",
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" return np.inf\n",
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" return 10 * np.log10((max_val ** 2) / err)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "fe1d4932",
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"metadata": {},
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"source": [
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"## Choose an image\n",
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"\n",
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"Set `IMAGE_PATH` to any real image on your machine.\n",
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"\n",
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"Examples:\n",
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"- `\"bella.jpg\"`\n",
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"- `\"my_photo.png\"`\n",
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"- `\"images/dog.jpeg\"`"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "42bafca5",
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"metadata": {},
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"outputs": [],
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"source": [
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"IMAGE_PATH = \"bella.jpg\" # change this to your actual file\n",
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"MODE = \"rgb\" # use \"gray\" for grayscale, \"rgb\" for color\n",
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"\n",
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"img = load_image(IMAGE_PATH, mode=MODE)\n",
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"print(\"Image shape:\", img.shape)\n",
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"show_image(img, title=f\"Original image ({MODE})\")"
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]
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},
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{
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"cell_type": "markdown",
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"id": "05e52222",
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"metadata": {},
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"source": [
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"## Add synthetic Gaussian noise\n",
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"\n",
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"We add noise so that the denoising effect is visible and measurable."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "528e69b3",
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||||
"metadata": {},
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||||
"outputs": [],
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"source": [
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"sigma = 25\n",
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"seed = 0\n",
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"\n",
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"img_noisy = add_gaussian_noise(img, sigma=sigma, seed=seed)\n",
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"show_image(img_noisy, title=f\"Noisy image (sigma={sigma})\")"
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]
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},
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{
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"cell_type": "markdown",
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"id": "1bbcc1d8",
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"metadata": {},
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"source": [
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"## Visualizing rank-$k$ reconstructions\n",
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"\n",
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"For small \\(k\\), the reconstruction captures only coarse structure.\n",
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"As \\(k\\) increases, more detail returns. For denoising, there is often a useful middle ground:\n",
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"enough singular values to preserve structure, but not so many that we reintroduce noise."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "563df53a",
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"metadata": {},
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"outputs": [],
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"source": [
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"ks = [5, 20, 50, 100]\n",
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"\n",
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"ncols = len(ks) + 2\n",
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"fig, axes = plt.subplots(1, ncols, figsize=(4 * ncols, 4))\n",
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"\n",
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"# Original\n",
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"if img.ndim == 2:\n",
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" axes[0].imshow(img, cmap=\"gray\", vmin=0, vmax=255)\n",
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"else:\n",
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" axes[0].imshow(np.clip(img, 0, 255).astype(np.uint8))\n",
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"axes[0].set_title(\"Original\")\n",
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"axes[0].axis(\"off\")\n",
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"\n",
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"# Noisy\n",
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"if img_noisy.ndim == 2:\n",
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" axes[1].imshow(img_noisy, cmap=\"gray\", vmin=0, vmax=255)\n",
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"else:\n",
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" axes[1].imshow(np.clip(img_noisy, 0, 255).astype(np.uint8))\n",
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"axes[1].set_title(\"Noisy\")\n",
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"axes[1].axis(\"off\")\n",
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"\n",
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"# Reconstructions\n",
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"for ax, k in zip(axes[2:], ks):\n",
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" recon = truncated_svd_image(img_noisy, k)\n",
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" if recon.ndim == 2:\n",
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" ax.imshow(recon, cmap=\"gray\", vmin=0, vmax=255)\n",
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" else:\n",
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" ax.imshow(np.clip(recon, 0, 255).astype(np.uint8))\n",
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" ax.set_title(f\"k = {k}\")\n",
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" ax.axis(\"off\")\n",
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"\n",
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"plt.tight_layout()\n",
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"plt.show()"
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]
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},
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{
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"cell_type": "markdown",
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"id": "309579fa",
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"metadata": {},
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"source": [
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"## Quantitative evaluation\n",
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"\n",
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"We compare each reconstruction against the **clean original image**, not against the noisy one.\n",
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"A good denoising rank should typically:\n",
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"- reduce MSE relative to the noisy image,\n",
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"- increase PSNR relative to the noisy image."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "56ce07ee",
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"metadata": {},
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"outputs": [],
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"source": [
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"baseline_mse = mse(img, img_noisy)\n",
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"baseline_psnr = psnr(img, img_noisy)\n",
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"\n",
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"print(f\"Noisy image baseline -> MSE: {baseline_mse:.2f}, PSNR: {baseline_psnr:.2f} dB\")\n",
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"\n",
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"results = []\n",
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"for k in ks:\n",
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" recon = truncated_svd_image(img_noisy, k)\n",
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" results.append((k, mse(img, recon), psnr(img, recon)))\n",
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"\n",
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"print(\"\\nRank-k reconstructions:\")\n",
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"for k, m, p in results:\n",
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||||
" print(f\"k = {k:3d} | MSE = {m:10.2f} | PSNR = {p:6.2f} dB\")"
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]
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},
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||||
{
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||||
"cell_type": "markdown",
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"id": "de9c3f3c",
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"metadata": {},
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"source": [
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||||
"## Automatic search over many values of \\(k\\)\n",
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||||
"\n",
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||||
"This is often useful because the best denoising rank is image-dependent."
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||||
]
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||||
},
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||||
{
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||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"id": "e097dcf4",
|
||||
"metadata": {},
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||||
"outputs": [],
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||||
"source": [
|
||||
"candidate_ks = list(range(1, 151, 5))\n",
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||||
"\n",
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||||
"scores = []\n",
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||||
"for k in candidate_ks:\n",
|
||||
" recon = truncated_svd_image(img_noisy, k)\n",
|
||||
" scores.append((k, mse(img, recon), psnr(img, recon)))\n",
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||||
"\n",
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||||
"best_by_mse = min(scores, key=lambda x: x[1])\n",
|
||||
"best_by_psnr = max(scores, key=lambda x: x[2])\n",
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||||
"\n",
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||||
"print(\"Best by MSE :\", best_by_mse)\n",
|
||||
"print(\"Best by PSNR:\", best_by_psnr)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"id": "4b9dc5c7",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"best_k = best_by_psnr[0]\n",
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||||
"best_recon = truncated_svd_image(img_noisy, best_k)\n",
|
||||
"\n",
|
||||
"fig, axes = plt.subplots(1, 3, figsize=(15, 5))\n",
|
||||
"\n",
|
||||
"if img.ndim == 2:\n",
|
||||
" axes[0].imshow(img, cmap=\"gray\", vmin=0, vmax=255)\n",
|
||||
" axes[1].imshow(img_noisy, cmap=\"gray\", vmin=0, vmax=255)\n",
|
||||
" axes[2].imshow(best_recon, cmap=\"gray\", vmin=0, vmax=255)\n",
|
||||
"else:\n",
|
||||
" axes[0].imshow(np.clip(img, 0, 255).astype(np.uint8))\n",
|
||||
" axes[1].imshow(np.clip(img_noisy, 0, 255).astype(np.uint8))\n",
|
||||
" axes[2].imshow(np.clip(best_recon, 0, 255).astype(np.uint8))\n",
|
||||
"\n",
|
||||
"axes[0].set_title(\"Original\")\n",
|
||||
"axes[1].set_title(\"Noisy\")\n",
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||||
"axes[2].set_title(f\"Best reconstruction (k={best_k})\")\n",
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||||
"\n",
|
||||
"for ax in axes:\n",
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||||
" ax.axis(\"off\")\n",
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||||
"\n",
|
||||
"plt.tight_layout()\n",
|
||||
"plt.show()"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "87d08518",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## A note on color images\n",
|
||||
"\n",
|
||||
"For a grayscale image, SVD applies directly to a single matrix.\n",
|
||||
"\n",
|
||||
"For a color image \\(A \\in \\mathbb{R}^{n \\times p \\times 3}\\), we write\n",
|
||||
"\\[\n",
|
||||
"A = (A_R, A_G, A_B),\n",
|
||||
"\\]\n",
|
||||
"where each channel is an \\(n \\times p\\) matrix. We then compute a rank-\\(k\\) approximation for each channel:\n",
|
||||
"\\[\n",
|
||||
"A_R \\approx (A_R)_k,\\qquad\n",
|
||||
"A_G \\approx (A_G)_k,\\qquad\n",
|
||||
"A_B \\approx (A_B)_k,\n",
|
||||
"\\]\n",
|
||||
"and stack them back together.\n",
|
||||
"\n",
|
||||
"This is the most direct extension of the grayscale method, and it works well as a first linear-algebraic treatment of color denoising."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "1a3acfe5",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Remarks and possible extensions\n",
|
||||
"\n",
|
||||
"- The same rank \\(k\\) was used for every color channel. You could instead choose different ranks per channel.\n",
|
||||
"- You can test this on photographs, scanned documents, or screenshots.\n",
|
||||
"- Truncated SVD is excellent for illustrating low-rank structure, but it is not the only denoising method.\n",
|
||||
"- A more advanced next step would be to compare SVD denoising against:\n",
|
||||
" - Gaussian blur,\n",
|
||||
" - median filtering,\n",
|
||||
" - wavelet denoising,\n",
|
||||
" - non-local means,\n",
|
||||
" - autoencoder-based denoising.\n",
|
||||
"\n",
|
||||
"For this notebook, though, the point is to keep the method squarely grounded in linear algebra."
|
||||
]
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"kernelspec": {
|
||||
"display_name": "Python 3",
|
||||
"language": "python",
|
||||
"name": "python3"
|
||||
},
|
||||
"language_info": {
|
||||
"name": "python",
|
||||
"version": "3.x"
|
||||
}
|
||||
},
|
||||
"nbformat": 4,
|
||||
"nbformat_minor": 5
|
||||
}
|
||||
Reference in New Issue
Block a user