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Pawel Sarkowicz
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{
"cells": [
{
"cell_type": "markdown",
"id": "6ae6c7f8",
"metadata": {},
"source": [
"# Spectral Image Denoising via Truncated SVD\n",
"\n",
"This notebook extracts the image denoising project into a standalone workflow and extends it from **grayscale images** to **actual color images**.\n",
"\n",
"The core idea is the same as in the original write-up: if an image matrix has singular value decomposition\n",
"$$\n",
"A = U \\Sigma V^T,\n",
"$$\n",
"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",
"\n",
"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."
]
},
{
"cell_type": "markdown",
"id": "31a665c9",
"metadata": {},
"source": [
"## Outline\n",
"\n",
"1. Load an image from disk\n",
"2. Convert it to grayscale or keep it in RGB\n",
"3. Add synthetic Gaussian noise\n",
"4. Compute a truncated SVD reconstruction\n",
"5. Compare the original, noisy, and denoised images\n",
"6. Measure quality using MSE and PSNR\n",
"\n",
"This notebook is written so that you can use **your own image files** directly."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "88584c56",
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"from PIL import Image\n",
"from pathlib import Path"
]
},
{
"cell_type": "markdown",
"id": "30e96441",
"metadata": {},
"source": [
"## A note on color images\n",
"\n",
"For a grayscale image, SVD applies directly to a single matrix. 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": "f275cbc9",
"metadata": {},
"source": [
"## Helper functions\n",
"\n",
"We begin with some utilities for:\n",
"- loading images,\n",
"- adding Gaussian noise,\n",
"- reconstructing rank-$k$ approximations,\n",
"- computing image-quality metrics."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "21adfcaf",
"metadata": {},
"outputs": [],
"source": [
"def load_image(path, mode=\"rgb\"):\n",
" \"\"\"\n",
" Load an image from disk.\n",
"\n",
" Parameters\n",
" ----------\n",
" path : str or Path\n",
" Path to the image file.\n",
" mode : {\"rgb\", \"gray\"}\n",
" Whether to load the image as RGB or grayscale.\n",
"\n",
" Returns\n",
" -------\n",
" np.ndarray\n",
" Float image array scaled to [0, 255].\n",
" Shape is (H, W, 3) for RGB and (H, W) for grayscale.\n",
" \"\"\"\n",
" path = Path(path)\n",
" img = Image.open(path)\n",
"\n",
" if mode.lower() in {\"gray\", \"grayscale\", \"l\"}:\n",
" img = img.convert(\"L\")\n",
" else:\n",
" img = img.convert(\"RGB\")\n",
"\n",
" return np.asarray(img, dtype=np.float64)\n",
"\n",
"\n",
"def show_image(img, title=None):\n",
" \"\"\"Display a grayscale or RGB image.\"\"\"\n",
" plt.figure(figsize=(6, 6))\n",
" if img.ndim == 2:\n",
" plt.imshow(np.clip(img, 0, 255), cmap=\"gray\", vmin=0, vmax=255)\n",
" else:\n",
" plt.imshow(np.clip(img, 0, 255).astype(np.uint8))\n",
" if title is not None:\n",
" plt.title(title)\n",
" plt.axis(\"off\")\n",
" plt.show()\n",
"\n",
"\n",
"def add_gaussian_noise(img, sigma=25, seed=0):\n",
" \"\"\"\n",
" Add Gaussian noise to an image.\n",
"\n",
" Parameters\n",
" ----------\n",
" img : np.ndarray\n",
" Image array in [0, 255].\n",
" sigma : float\n",
" Standard deviation of the noise.\n",
" seed : int\n",
" Random seed for reproducibility.\n",
"\n",
" Returns\n",
" -------\n",
" np.ndarray\n",
" Noisy image clipped to [0, 255].\n",
" \"\"\"\n",
" rng = np.random.default_rng(seed)\n",
" noisy = img + rng.normal(loc=0.0, scale=sigma, size=img.shape)\n",
" return np.clip(noisy, 0, 255)\n",
"\n",
"\n",
"def truncated_svd_matrix(A, k):\n",
" \"\"\"\n",
" Return the rank-k truncated SVD approximation of a 2D matrix A.\n",
" \"\"\"\n",
" U, s, Vt = np.linalg.svd(A, full_matrices=False)\n",
" k = min(k, len(s))\n",
" return (U[:, :k] * s[:k]) @ Vt[:k, :]\n",
"\n",
"\n",
"def truncated_svd_image(img, k):\n",
" \"\"\"\n",
" Apply truncated SVD to a grayscale or RGB image.\n",
"\n",
" For RGB images, truncated SVD is applied channel-by-channel.\n",
"\n",
" Parameters\n",
" ----------\n",
" img : np.ndarray\n",
" Shape (H, W) for grayscale or (H, W, 3) for RGB.\n",
" k : int\n",
" Truncation rank.\n",
"\n",
" Returns\n",
" -------\n",
" np.ndarray\n",
" Reconstructed image clipped to [0, 255].\n",
" \"\"\"\n",
" if img.ndim == 2:\n",
" recon = truncated_svd_matrix(img, k)\n",
" return np.clip(recon, 0, 255)\n",
"\n",
" if img.ndim == 3:\n",
" channels = []\n",
" for c in range(img.shape[2]):\n",
" channel_recon = truncated_svd_matrix(img[:, :, c], k)\n",
" channels.append(channel_recon)\n",
" recon = np.stack(channels, axis=2)\n",
" return np.clip(recon, 0, 255)\n",
"\n",
" raise ValueError(\"Image must be either 2D (grayscale) or 3D (RGB).\")\n",
"\n",
"\n",
"def mse(A, B):\n",
" \"\"\"Mean squared error between two images.\"\"\"\n",
" return np.mean((A.astype(np.float64) - B.astype(np.float64)) ** 2)\n",
"\n",
"\n",
"def psnr(A, B, max_val=255.0):\n",
" \"\"\"Peak signal-to-noise ratio in decibels.\"\"\"\n",
" err = mse(A, B)\n",
" if err == 0:\n",
" return np.inf\n",
" return 10 * np.log10((max_val ** 2) / err)"
]
},
{
"cell_type": "markdown",
"id": "fe1d4932",
"metadata": {},
"source": [
"## Choose an image"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "42bafca5",
"metadata": {},
"outputs": [],
"source": [
"IMAGE_PATH = \"../images/bella.jpg\" \n",
"MODE = \"rgb\" # use \"gray\" for grayscale, \"rgb\" for color\n",
"\n",
"img = load_image(IMAGE_PATH, mode=MODE)\n",
"print(\"Image shape:\", img.shape)\n",
"show_image(img, title=f\"Original image ({MODE})\")"
]
},
{
"cell_type": "markdown",
"id": "05e52222",
"metadata": {},
"source": [
"## Add synthetic Gaussian noise\n",
"\n",
"We add noise so that the denoising effect is visible and measurable."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "528e69b3",
"metadata": {},
"outputs": [],
"source": [
"sigma = 25\n",
"seed = 0\n",
"\n",
"img_noisy = add_gaussian_noise(img, sigma=sigma, seed=seed)\n",
"\n",
"img_noisy.save('../images/bella_noisy.jpg')\n",
"\n",
"show_image(img_noisy, title=f\"Noisy image (sigma={sigma})\")"
]
},
{
"cell_type": "markdown",
"id": "1bbcc1d8",
"metadata": {},
"source": [
"## Visualizing rank-$k$ reconstructions\n",
"\n",
"For small $k$, the reconstruction captures only coarse structure.\n",
"As $k$ increases, more detail returns. For denoising, there is often a useful middle ground:\n",
"enough singular values to preserve structure, but not so many that we reintroduce noise."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "563df53a",
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"import math\n",
"\n",
"ks = [5, 20, 50, 100]\n",
"\n",
"# Collect all images + titles\n",
"images = []\n",
"titles = []\n",
"\n",
"# Original\n",
"images.append(img)\n",
"titles.append(\"Original\")\n",
"\n",
"# Noisy\n",
"images.append(img_noisy)\n",
"titles.append(\"Noisy\")\n",
"\n",
"# Reconstructions\n",
"for k in ks:\n",
" recon = truncated_svd_image(img_noisy, k)\n",
" images.append(recon)\n",
" titles.append(f\"k = {k}\")\n",
"\n",
"# Grid setup\n",
"ncols = 2\n",
"nrows = math.ceil(len(images) / ncols)\n",
"\n",
"fig, axes = plt.subplots(nrows, ncols, figsize=(6 * ncols, 4 * nrows))\n",
"axes = axes.flatten() # easier indexing\n",
"\n",
"# Plot everything\n",
"for i, (ax, im, title) in enumerate(zip(axes, images, titles)):\n",
" if im.ndim == 2:\n",
" ax.imshow(im, cmap=\"gray\", vmin=0, vmax=255)\n",
" else:\n",
" ax.imshow(np.clip(im, 0, 255).astype(np.uint8))\n",
" \n",
" ax.set_title(title)\n",
" ax.axis(\"off\")\n",
"\n",
"# Hide any unused axes\n",
"for j in range(len(images), len(axes)):\n",
" axes[j].axis(\"off\")\n",
"\n",
"plt.tight_layout()\n",
"plt.savefig('../images/bella_truncated_svd_multiplie_ks.png')\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"id": "309579fa",
"metadata": {},
"source": [
"## Quantitative evaluation\n",
"\n",
"We compare each reconstruction against the **clean original image**, not against the noisy one.\n",
"A good denoising rank should typically:\n",
"- reduce MSE relative to the noisy image,\n",
"- increase PSNR relative to the noisy image."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "56ce07ee",
"metadata": {},
"outputs": [],
"source": [
"baseline_mse = mse(img, img_noisy)\n",
"baseline_psnr = psnr(img, img_noisy)\n",
"\n",
"print(f\"Noisy image baseline -> MSE: {baseline_mse:.2f}, PSNR: {baseline_psnr:.2f} dB\")\n",
"\n",
"results = []\n",
"for k in ks:\n",
" recon = truncated_svd_image(img_noisy, k)\n",
" results.append((k, mse(img, recon), psnr(img, recon)))\n",
"\n",
"print(\"\\nRank-k reconstructions:\")\n",
"for k, m, p in results:\n",
" print(f\"k = {k:3d} | MSE = {m:10.2f} | PSNR = {p:6.2f} dB\")"
]
},
{
"cell_type": "markdown",
"id": "de9c3f3c",
"metadata": {},
"source": [
"## Automatic search over many values of $k$\n",
"\n",
"This is often useful because the best denoising rank is image-dependent."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "e097dcf4",
"metadata": {},
"outputs": [],
"source": [
"candidate_ks = list(range(1, 151, 5))\n",
"\n",
"scores = []\n",
"for k in candidate_ks:\n",
" recon = truncated_svd_image(img_noisy, k)\n",
" scores.append((k, mse(img, recon), psnr(img, recon)))\n",
"\n",
"best_by_mse = min(scores, key=lambda x: x[1])\n",
"best_by_psnr = max(scores, key=lambda x: x[2])\n",
"\n",
"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",
"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",
"axes[2].set_title(f\"Best reconstruction (k={best_k})\")\n",
"\n",
"for ax in axes:\n",
" ax.axis(\"off\")\n",
"\n",
"plt.tight_layout()\n",
"plt.savefig('../images/bella_best_truncated.png')\n",
"plt.show()"
]
},
{
"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."
]
}
],
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