【线代&NumPy】第十二章 - 奇异值课后练习（附加代码记录）

import numpy as np
import matplotlib.pyplot as plt
import imageio as im

src = im.imread('flower3.jpg')
A = src[:,:,0]

def reconstruct(U, Sigma, VT, n_elements):
    Sigma = Sigma[:, :n_elements]
    VT = VT[:n_elements, :]
    B = np.matmul(U, np.matmul(Sigma, VT))
    return B

plt.rcParams.update({'xtick.major.width': 0,
                     'xtick.labelsize': 0,
                     'ytick.major.width': 0,
                     'ytick.labelsize': 0,
                     'axes.linewidth': 0})

U, s, VT = np.linalg.svd(A)
m, n = A.shape
Sigma = np.zeros((m, n))
k = np.size(s)
Sigma[:k, :k] = np.diag(s)

plt.subplot(3,3,1)
plt.title('original')
plt.imshow(A)

plt.subplot(3,3,2)
plt.title('rank 1')
B = reconstruct(U, Sigma, VT, 1)
plt.imshow(B, cmap = 'gray')
plt.subplot(3,3,3)

plt.title('upto rank 5')
B = reconstruct(U, Sigma, VT, 5)
plt.imshow(B, cmap = 'gray')
plt.subplot(3,3,4)

plt.title('upto rank 20')
B = reconstruct(U, Sigma, VT, 20)
plt.imshow(B, cmap = 'gray')
plt.subplot(3,3,5)

plt.title('upto rank 30')
B = reconstruct(U, Sigma, VT, 30)
plt.imshow(B, cmap = 'gray')
plt.subplot(3,3,6)

plt.title('upto rank 50')
B = reconstruct(U, Sigma, VT, 50)
plt.imshow(B, cmap = 'gray')
plt.subplot(3,3,7)

plt.title('upto rank 100')
B = reconstruct(U, Sigma, VT, 100)
plt.imshow(B, cmap = 'gray')
plt.subplot(3,3,8)

plt.title('upto rank 150')
B = reconstruct(U, Sigma, VT, 150)
plt.imshow(B, cmap = 'gray')
plt.subplot(3,3,9)

plt.title('upto rank 200')
B = reconstruct(U, Sigma, VT, 200)
plt.imshow(B, cmap = 'gray')
plt.show()

import cv2
import numpy as np
import matplotlib.pyplot as plt

def compute_xform(corners1, corners2):
    A = [ ]
    for i in range(4):
        x1, y1 = corners1[i]
        x2, y2 = corners2[i]
        A.append([x1, y1, 1, 0, 0, 0, -x2*x1, -x2*y1, -x2])
        A.append([0, 0, 0, x1, y1, 1, -y2*x1, -y2*y1, -y2])

    A = np.asarray(A)
    U, S, V = np.linalg.svd(A)
    xform = V[-1, :]
    xform = np.reshape(xform, (3, 3))

    return xform

def transform_image(xform, image1, corners1, corners2):
    warped = cv2.warpPerspective(image1, xform, (500, 500))

    h1, w1 = image1.shape[:2] 
    h2, w2 = warped.shape[:2] 

    out_image = np.zeros((max(h1, h2), w1+w2, 3), dtype=np.uint8)
    out_image[:h1, :w1, :3] = image1
    out_image[:h2, w1:w1+w2, :3] = warped

    for i in range(4):
        c1 = corners1[i]
        c2 = (corners2[i][0] + w1, corners2[i][1])
        cv2.circle(out_image, c1, radius = 2, color = (0, 255, 0),
                   thickness = 2)
        cv2.circle(out_image, c2, radius = 2, color = (30, 255, 255),
                   thickness = 2)
        cv2.line(out_image, c1, c2, color = (0, 0, 255), thickness = 1)

    cv2.imshow('original and transformed images', out_image)
    cv2.waitKey(0)
        
def main():
    img_path1 = 'cameraman1.png'
    img1 = cv2.imread(img_path1, cv2.IMREAD_COLOR) 
    corners1 = [(122, 51), (26, 300), (454, 131), (330, 414)]
    corners2 = [(50, 50), (50, 450), (450, 50), (450, 450)]
    xform = compute_xform(corners1, corners2)
    print(xform) 
    transform_image(xform, img1, corners1, corners2) 
    
if __name__ == '__main__':
    main()

参考文献

Introduction to Linear Algebra, International 4 th Edition by Gilbert Strang, Wellesley Cambridge Press.

百度百科[EB/OL]. []. https://baike.baidu.com/

本篇完。