{
 "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\n",
    "\n",
    "try:\n",
    "    from skimage.metrics import structural_similarity as ssim\n",
    "    HAS_SKIMAGE = True\n",
    "except ImportError:\n",
    "    ssim = None\n",
    "    HAS_SKIMAGE = False\n",
    "\n",
    "print(f\"scikit-image available: {HAS_SKIMAGE}\")"
   ]
  },
  {
   "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",
    "    if not path.exists():\n",
    "        raise FileNotFoundError(f\"Could not find image file: {path}\")\n",
    "\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.tight_layout()\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",
    "    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)\n",
    "\n",
    "\n",
    "def image_ssim(A, B, max_val=255.0):\n",
    "    \"\"\"\n",
    "    Structural similarity index.\n",
    "\n",
    "    For RGB images, compute SSIM channel-by-channel and average.\n",
    "    Returns None when scikit-image is unavailable.\n",
    "    \"\"\"\n",
    "    if not HAS_SKIMAGE:\n",
    "        return None\n",
    "\n",
    "    A = A.astype(np.float64)\n",
    "    B = B.astype(np.float64)\n",
    "\n",
    "    if A.ndim == 2:\n",
    "        return float(ssim(A, B, data_range=max_val))\n",
    "\n",
    "    if A.ndim == 3:\n",
    "        vals = [ssim(A[:, :, c], B[:, :, c], data_range=max_val) for c in range(A.shape[2])]\n",
    "        return float(np.mean(vals))\n",
    "\n",
    "    raise ValueError(\"Images must be either 2D (grayscale) or 3D (RGB).\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fe1d4932",
   "metadata": {},
   "source": [
    "## Choose an image"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "42bafca5",
   "metadata": {},
   "outputs": [],
   "source": [
    "from pathlib import Path\n",
    "\n",
    "MODE = \"rgb\"  # use \"gray\" for grayscale, \"rgb\" for color\n",
    "\n",
    "candidate_paths = [\n",
    "    Path(\"../images/bella.jpg\"),\n",
    "    Path(\"images/bella.jpg\"),\n",
    "    Path(\"bella.jpg\"),\n",
    "]\n",
    "\n",
    "IMAGE_PATH = None\n",
    "for p in candidate_paths:\n",
    "    if p.exists():\n",
    "        IMAGE_PATH = p\n",
    "        break\n",
    "\n",
    "if IMAGE_PATH is None:\n",
    "    raise FileNotFoundError(\n",
    "        \"Could not find bella.jpg. Put it in ../images/, images/, or the notebook folder.\"\n",
    "    )\n",
    "\n",
    "img = load_image(IMAGE_PATH, mode=MODE)\n",
    "print(\"Using image:\", IMAGE_PATH)\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 = 40\n",
    "seed = 0\n",
    "\n",
    "img_noisy = add_gaussian_noise(img, sigma=sigma, seed=seed)\n",
    "\n",
    "noisy_output_path = IMAGE_PATH.with_name(f\"{IMAGE_PATH.stem}_noisy.png\")\n",
    "Image.fromarray(np.clip(img_noisy, 0, 255).astype(np.uint8)).save(noisy_output_path)\n",
    "print(\"Saved noisy image to:\", noisy_output_path)\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",
    "comparison_output_path = IMAGE_PATH.with_name(f\"{IMAGE_PATH.stem}_truncated_svd_multiple_ks.png\")\n",
    "print(\"Saved comparison figure to:\", comparison_output_path)\n",
    "plt.savefig(comparison_output_path, bbox_inches=\"tight\")\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": "f2fe6fe2",
   "metadata": {},
   "source": [
    "\n",
    "## Efficient search over many values of $k$\n",
    "\n",
    "A naive implementation would recompute the SVD from scratch for every candidate value of $k$.\n",
    "That is extremely expensive: every reconstruction would require a fresh factorization of each\n",
    "channel of the noisy image.\n",
    "\n",
    "A much better approach is:\n",
    "\n",
    "1. compute the SVD **once** for each channel;\n",
    "2. reuse those factors for every candidate $k$;\n",
    "3. compare reconstructions using MSE, PSNR, and optionally SSIM.\n",
    "\n",
    "This is also a nice numerical linear algebra point: all rank-$k$ truncated reconstructions come\n",
    "from the **same** singular value decomposition.\n",
    "\n",
    "We compare two variants:\n",
    "\n",
    "- **plain truncated SVD**, applied directly to each channel;\n",
    "- **centered truncated SVD**, where we subtract each channel's column mean before factorizing and\n",
    "  add it back after reconstruction.\n",
    "\n",
    "The centered version sometimes improves reconstruction slightly because the low-rank approximation\n",
    "spends less effort representing the mean structure.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "4277a913",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "def precompute_svd_image(img):\n",
    "    \"\"\"Precompute plain SVD factors for each channel.\"\"\"\n",
    "    if img.ndim == 2:\n",
    "        A = img.astype(np.float64)\n",
    "        U, s, Vt = np.linalg.svd(A, full_matrices=False)\n",
    "        return [(U, s, Vt)]\n",
    "\n",
    "    cache = []\n",
    "    for c in range(img.shape[2]):\n",
    "        A = img[:, :, c].astype(np.float64)\n",
    "        U, s, Vt = np.linalg.svd(A, full_matrices=False)\n",
    "        cache.append((U, s, Vt))\n",
    "    return cache\n",
    "\n",
    "\n",
    "def precompute_centered_svd_image(img):\n",
    "    \"\"\"Precompute centered SVD factors for each channel.\"\"\"\n",
    "    if img.ndim == 2:\n",
    "        A = img.astype(np.float64)\n",
    "        col_mean = A.mean(axis=0, keepdims=True)\n",
    "        A_centered = A - col_mean\n",
    "        U, s, Vt = np.linalg.svd(A_centered, full_matrices=False)\n",
    "        return [(U, s, Vt, col_mean)]\n",
    "\n",
    "    cache = []\n",
    "    for c in range(img.shape[2]):\n",
    "        A = img[:, :, c].astype(np.float64)\n",
    "        col_mean = A.mean(axis=0, keepdims=True)\n",
    "        A_centered = A - col_mean\n",
    "        U, s, Vt = np.linalg.svd(A_centered, full_matrices=False)\n",
    "        cache.append((U, s, Vt, col_mean))\n",
    "    return cache\n",
    "\n",
    "\n",
    "def reconstruct_from_svd_cache(cache, k):\n",
    "    \"\"\"Reconstruct from precomputed plain SVD factors.\"\"\"\n",
    "    channels = []\n",
    "    for U, s, Vt in cache:\n",
    "        kk = min(k, len(s))\n",
    "        recon = (U[:, :kk] * s[:kk]) @ Vt[:kk, :]\n",
    "        channels.append(np.clip(recon, 0, 255))\n",
    "\n",
    "    if len(channels) == 1:\n",
    "        return channels[0]\n",
    "    return np.stack(channels, axis=2)\n",
    "\n",
    "\n",
    "def reconstruct_from_centered_svd_cache(cache, k):\n",
    "    \"\"\"Reconstruct from precomputed centered SVD factors.\"\"\"\n",
    "    channels = []\n",
    "    for U, s, Vt, col_mean in cache:\n",
    "        kk = min(k, len(s))\n",
    "        recon = (U[:, :kk] * s[:kk]) @ Vt[:kk, :] + col_mean\n",
    "        channels.append(np.clip(recon, 0, 255))\n",
    "\n",
    "    if len(channels) == 1:\n",
    "        return channels[0]\n",
    "    return np.stack(channels, axis=2)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8d1dacbb",
   "metadata": {},
   "source": [
    "\n",
    "## Scoring reconstructions\n",
    "\n",
    "We first compute a **baseline** by comparing the noisy image to the clean one. Then we score\n",
    "rank-$k$ reconstructions. A smaller MSE and a larger PSNR indicate better fidelity to the clean\n",
    "image. If `scikit-image` is available, we also compute SSIM.\n",
    "\n",
    "A useful conceptual warning is important here:\n",
    "\n",
    "> The best low-rank approximation in a matrix norm does **not** necessarily produce the image that\n",
    "> looks best to a human observer.\n",
    "\n",
    "Why? Because human perception cares about things like edges, texture, and local contrast, while\n",
    "MSE and PSNR are purely pixelwise. A reconstruction can score well numerically and still look too\n",
    "smooth, too blurry, or otherwise unnatural.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "c48c94cc",
   "metadata": {},
   "outputs": [],
   "source": [
    "baseline_mse = mse(img, img_noisy)\n",
    "baseline_psnr = psnr(img, img_noisy)\n",
    "\n",
    "print(f\"Baseline noisy vs clean:\")\n",
    "print(f\"  MSE : {baseline_mse:.2f}\")\n",
    "print(f\"  PSNR: {baseline_psnr:.2f}\")\n",
    "\n",
    "if HAS_SKIMAGE:\n",
    "    baseline_ssim = image_ssim(img, img_noisy)\n",
    "    print(f\"  SSIM: {baseline_ssim:.4f}\")\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "231330d1",
   "metadata": {},
   "source": [
    "\n",
    "## Automatic search over many values of $k$\n",
    "\n",
    "Because all rank-$k$ reconstructions come from the same SVD, we precompute the factorizations\n",
    "once and then search efficiently over candidate values of $k$.\n",
    "\n",
    "For very large images this can still be somewhat expensive, so for exploratory work a coarser grid\n",
    "such as `range(5, 151, 5)` is often sufficient. Once a promising region is found, one can refine\n",
    "the search around that region.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "8bae249d",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "candidate_ks = list(range(1, 151, 5))\n",
    "\n",
    "plain_cache = precompute_svd_image(img_noisy)\n",
    "centered_cache = precompute_centered_svd_image(img_noisy)\n",
    "\n",
    "plain_scores = []\n",
    "centered_scores = []\n",
    "\n",
    "for k in candidate_ks:\n",
    "    plain = reconstruct_from_svd_cache(plain_cache, k)\n",
    "    centered = reconstruct_from_centered_svd_cache(centered_cache, k)\n",
    "\n",
    "    plain_row = (k, mse(img, plain), psnr(img, plain))\n",
    "    centered_row = (k, mse(img, centered), psnr(img, centered))\n",
    "\n",
    "    if HAS_SKIMAGE:\n",
    "        plain_row = plain_row + (image_ssim(img, plain),)\n",
    "        centered_row = centered_row + (image_ssim(img, centered),)\n",
    "\n",
    "    plain_scores.append(plain_row)\n",
    "    centered_scores.append(centered_row)\n",
    "\n",
    "best_plain_by_mse = min(plain_scores, key=lambda x: x[1])\n",
    "best_plain_by_psnr = max(plain_scores, key=lambda x: x[2])\n",
    "best_centered_by_mse = min(centered_scores, key=lambda x: x[1])\n",
    "best_centered_by_psnr = max(centered_scores, key=lambda x: x[2])\n",
    "\n",
    "print(\"Plain SVD:\")\n",
    "print(\"  Best by MSE :\", best_plain_by_mse)\n",
    "print(\"  Best by PSNR:\", best_plain_by_psnr)\n",
    "\n",
    "print(\"Centered SVD:\")\n",
    "print(\"  Best by MSE :\", best_centered_by_mse)\n",
    "print(\"  Best by PSNR:\", best_centered_by_psnr)\n",
    "\n",
    "if HAS_SKIMAGE:\n",
    "    best_plain_by_ssim = max(plain_scores, key=lambda x: x[3])\n",
    "    best_centered_by_ssim = max(centered_scores, key=lambda x: x[3])\n",
    "\n",
    "    print(\"Plain SVD:\")\n",
    "    print(\"  Best by SSIM:\", best_plain_by_ssim)\n",
    "    print(\"Centered SVD:\")\n",
    "    print(\"  Best by SSIM:\", best_centered_by_ssim)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "166d0877",
   "metadata": {},
   "source": [
    "\n",
    "## Metric curves versus $k$\n",
    "\n",
    "Plotting the metrics as functions of $k$ is often more informative than looking only at the\n",
    "single best value. Frequently the metric is nearly flat across a whole range of ranks, in which\n",
    "case several nearby values of $k$ have very similar numerical performance.\n",
    "\n",
    "That is exactly the situation where visual inspection matters most: among a cluster of nearly tied\n",
    "candidates, the one that looks nicest to the eye may not be the exact numerical winner.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "0e1000de",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "plain_ks = [row[0] for row in plain_scores]\n",
    "plain_mses = [row[1] for row in plain_scores]\n",
    "plain_psnrs = [row[2] for row in plain_scores]\n",
    "\n",
    "centered_ks = [row[0] for row in centered_scores]\n",
    "centered_mses = [row[1] for row in centered_scores]\n",
    "centered_psnrs = [row[2] for row in centered_scores]\n",
    "\n",
    "plt.figure(figsize=(8, 4))\n",
    "plt.plot(plain_ks, plain_mses, label=\"Plain SVD\")\n",
    "plt.plot(centered_ks, centered_mses, label=\"Centered SVD\")\n",
    "plt.xlabel(\"k\")\n",
    "plt.ylabel(\"MSE\")\n",
    "plt.title(\"MSE versus rank k\")\n",
    "plt.legend()\n",
    "plt.tight_layout()\n",
    "plt.show()\n",
    "\n",
    "plt.figure(figsize=(8, 4))\n",
    "plt.plot(plain_ks, plain_psnrs, label=\"Plain SVD\")\n",
    "plt.plot(centered_ks, centered_psnrs, label=\"Centered SVD\")\n",
    "plt.xlabel(\"k\")\n",
    "plt.ylabel(\"PSNR\")\n",
    "plt.title(\"PSNR versus rank k\")\n",
    "plt.legend()\n",
    "plt.tight_layout()\n",
    "plt.show()\n",
    "\n",
    "if HAS_SKIMAGE:\n",
    "    plain_ssims = [row[3] for row in plain_scores]\n",
    "    centered_ssims = [row[3] for row in centered_scores]\n",
    "\n",
    "    plt.figure(figsize=(8, 4))\n",
    "    plt.plot(plain_ks, plain_ssims, label=\"Plain SVD\")\n",
    "    plt.plot(centered_ks, centered_ssims, label=\"Centered SVD\")\n",
    "    plt.xlabel(\"k\")\n",
    "    plt.ylabel(\"SSIM\")\n",
    "    plt.title(\"SSIM versus rank k\")\n",
    "    plt.legend()\n",
    "    plt.tight_layout()\n",
    "    plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d008c548",
   "metadata": {},
   "source": [
    "\n",
    "## Visual comparison near the best ranks\n",
    "\n",
    "Finally, we inspect a few reconstructions around the automatically selected ranks. This is important\n",
    "because the reconstruction that is optimal in Frobenius norm, MSE, or PSNR is not guaranteed to be\n",
    "the reconstruction a human would actually prefer.\n",
    "\n",
    "Low-rank approximation is mathematically optimal for a precise matrix objective, but photographic\n",
    "quality is influenced by far more than that. Fine textures, fur, sharp edges, and local contrast\n",
    "can all matter a great deal perceptually, and some of those are exactly the kinds of features that\n",
    "get smoothed away by aggressive truncation.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "cb0c7d57",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "# Pick a few candidate ranks around the PSNR-optimal values\n",
    "plain_best_k = best_plain_by_psnr[0]\n",
    "centered_best_k = best_centered_by_psnr[0]\n",
    "\n",
    "plain_inspect_ks = sorted(set(k for k in [plain_best_k - 10, plain_best_k - 5, plain_best_k, plain_best_k + 5, plain_best_k + 10] if k >= 1))\n",
    "centered_inspect_ks = sorted(set(k for k in [centered_best_k - 10, centered_best_k - 5, centered_best_k, centered_best_k + 5, centered_best_k + 10] if k >= 1))\n",
    "\n",
    "print(\"Plain SVD ranks to inspect   :\", plain_inspect_ks)\n",
    "print(\"Centered SVD ranks to inspect:\", centered_inspect_ks)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "aab1aff3",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "import math\n",
    "\n",
    "# Build a gallery: original, noisy, then several plain and centered reconstructions\n",
    "gallery_images = [img, img_noisy]\n",
    "gallery_titles = [\"Original\", f\"Noisy (sigma={sigma})\"]\n",
    "\n",
    "for k in plain_inspect_ks:\n",
    "    gallery_images.append(reconstruct_from_svd_cache(plain_cache, k))\n",
    "    gallery_titles.append(f\"Plain SVD, k={k}\")\n",
    "\n",
    "for k in centered_inspect_ks:\n",
    "    gallery_images.append(reconstruct_from_centered_svd_cache(centered_cache, k))\n",
    "    gallery_titles.append(f\"Centered SVD, k={k}\")\n",
    "\n",
    "ncols = 2\n",
    "nrows = math.ceil(len(gallery_images) / ncols)\n",
    "\n",
    "fig, axes = plt.subplots(nrows, ncols, figsize=(6 * ncols, 4 * nrows))\n",
    "axes = np.array(axes).reshape(-1)\n",
    "\n",
    "for ax, im, title in zip(axes, gallery_images, gallery_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",
    "    ax.set_title(title)\n",
    "    ax.axis(\"off\")\n",
    "\n",
    "for ax in axes[len(gallery_images):]:\n",
    "    ax.axis(\"off\")\n",
    "\n",
    "plt.tight_layout()\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2433f279",
   "metadata": {},
   "source": [
    "\n",
    "## Remarks and possible extensions\n",
    "\n",
    "- Truncated SVD provides the best rank-$k$ approximation in Frobenius norm, but that does **not**\n",
    "  automatically mean it gives the most visually pleasing denoised image.\n",
    "- For real photographs, low-rank methods often smooth away texture and local detail along with the\n",
    "  noise.\n",
    "- The visually best image may lie near the metric optimum rather than exactly at it.\n",
    "- One can compare this method with more perceptual denoisers such as wavelet methods, bilateral\n",
    "  filtering, non-local means, or modern learned denoisers.\n",
    "- A useful next step would be to compare how the preferred $k$ changes as the noise level\n",
    "  $\\sigma$ increases.\n"
   ]
  }
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