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‎charpter19_SVD/louwill.jpg‎

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‎charpter19_SVD/svd.ipynb‎

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{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"#### SVD"
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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": 1,
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"metadata": {},
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"outputs": [],
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"source": [
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"import numpy as np"
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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": 2,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"array([[0, 1],\n",
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" [1, 1],\n",
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" [1, 0]])"
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]
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},
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"execution_count": 2,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"A = np.array([[0,1],[1,1],[1,0]])\n",
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"A"
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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": 3,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"(3, 3) (2,) (2, 2)\n"
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]
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}
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],
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"source": [
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"u, s, vt = np.linalg.svd(A, full_matrices=True)\n",
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"print(u.shape, s.shape, vt.shape)"
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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": 4,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"array([[-4.08248290e-01, 7.07106781e-01, 5.77350269e-01],\n",
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" [-8.16496581e-01, 5.55111512e-17, -5.77350269e-01],\n",
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" [-4.08248290e-01, -7.07106781e-01, 5.77350269e-01]])"
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]
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},
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"execution_count": 4,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"u"
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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": 5,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"array([1.73205081, 1. ])"
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]
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},
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"execution_count": 5,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"s"
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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": 6,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"array([[-0.70710678, -0.70710678],\n",
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" [-0.70710678, 0.70710678]])"
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]
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},
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"execution_count": 6,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"vt.T"
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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": 8,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"True"
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]
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},
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"execution_count": 8,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"np.allclose(A, np.dot(u[:,:2]*s, vt))"
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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": 10,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"array([[ 1.11022302e-16, 1.00000000e+00],\n",
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" [ 1.00000000e+00, 1.00000000e+00],\n",
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" [ 1.00000000e+00, -3.33066907e-16]])"
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]
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},
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"execution_count": 10,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"np.dot(u[:,:2]*s, vt)"
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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": 17,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"array([[1.73205081, 0. ],\n",
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" [0. , 1. ],\n",
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" [0. , 0. ]])"
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]
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},
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"execution_count": 17,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"s_ = np.zeros((3,2))\n",
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"for i in range(2):\n",
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" s_[i][i] = s[i]\n",
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"\n",
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"s_"
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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": 22,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"array([[ 1.11022302e-16, 1.00000000e+00],\n",
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" [ 1.00000000e+00, 1.00000000e+00],\n",
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" [ 1.00000000e+00, -3.33066907e-16]])"
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]
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},
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"execution_count": 22,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"np.dot(np.dot(u, s_), vt)"
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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": 7,
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"metadata": {},
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"outputs": [
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{
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"name": "stderr",
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"output_type": "stream",
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"text": [
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"100%|██████████████████████████████████████████████████████████████████████████████████| 50/50 [02:11<00:00, 2.63s/it]\n"
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]
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}
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],
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"source": [
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"import numpy as np\n",
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"import os\n",
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"from PIL import Image\n",
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"from tqdm import tqdm\n",
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"\n",
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"# 定义恢复函数，由分解后的矩阵恢复到原矩阵\n",
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"def restore(u, s, v, K): \n",
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" '''\n",
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" u:左奇异矩阵\n",
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" v:右奇异矩阵\n",
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" s:奇异值矩阵\n",
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" K:奇异值个数\n",
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" '''\n",
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" m, n = len(u), len(v[0])\n",
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" a = np.zeros((m, n))\n",
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" for k in range(K):\n",
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" uk = u[:, k].reshape(m, 1)\n",
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" vk = v[k].reshape(1, n)\n",
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" # 前k个奇异值的加总\n",
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" a += s[k] * np.dot(uk, vk) \n",
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" a = a.clip(0, 255)\n",
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" return np.rint(a).astype('uint8')\n",
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"\n",
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"A = np.array(Image.open(\"./louwill.jpg\", 'r'))\n",
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"# 对RGB图像进行奇异值分解\n",
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"u_r, s_r, v_r = np.linalg.svd(A[:, :, 0]) \n",
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"u_g, s_g, v_g = np.linalg.svd(A[:, :, 1])\n",
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"u_b, s_b, v_b = np.linalg.svd(A[:, :, 2])\n",
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"\n",
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"# 使用前50个奇异值\n",
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"K = 50 \n",
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"output_path = r'./svd_pic'\n",
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"# \n",
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"for k in tqdm(range(1, K+1)):\n",
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" R = restore(u_r, s_r, v_r, k)\n",
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" G = restore(u_g, s_g, v_g, k)\n",
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" B = restore(u_b, s_b, v_b, k)\n",
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" I = np.stack((R, G, B), axis=2) \n",
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" Image.fromarray(I).save('%s\\\\svd_%d.jpg' % (output_path, k))"
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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": 4,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"(959, 959, 3)"
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]
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},
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"execution_count": 4,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"A.shape"
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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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"metadata": {},
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"outputs": [],
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"source": []
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 3",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.7.3"
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},
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"toc": {
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"base_numbering": 1,
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"nav_menu": {},
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"number_sections": true,
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"sideBar": true,
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"skip_h1_title": false,
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"title_cell": "Table of Contents",
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"title_sidebar": "Contents",
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"toc_cell": false,
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"toc_position": {},
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"toc_section_display": true,
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"toc_window_display": false
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}
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},
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"nbformat": 4,
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"nbformat_minor": 2
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}

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