万户网站后台控制中心,怎么做网站推广平台,wordpress文章页禁止右键,制作网页时关于可以采用的图像文件格式正确的描述是SVD方法是模型降阶的一类重要方法#xff0c;本征正交分解#xff08;POD#xff09;和平衡截断#xff08;BT#xff09;都属于SVD类方法。 要想深入了解模型降阶技术#xff0c;我们可以先从SVD的应用入手#xff0c;做一个直观的了解。 1. SVD的定义和分类 我们想寻找…SVD方法是模型降阶的一类重要方法本征正交分解POD和平衡截断BT都属于SVD类方法。 要想深入了解模型降阶技术我们可以先从SVD的应用入手做一个直观的了解。 1. SVD的定义和分类 我们想寻找一个A的逼近Ak使得rank(Ak) k n且|A - Ak|最小。
下面的定理也称为Schmidt-Mirsky, Eckart-Young定理说明矩阵A的低秩逼近可以用SVD实现 2. SVD在图像压缩中的应用
原始图片, rank720 绘制其RGB的奇异值 压缩图片rank144: 压缩图片rank72 代码
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.image as imageA image.imread(svd-image-compression-img.jpg)# Each pixel (typically) consists of 3 bytes — for the red, green and blue components of the color, respectively.
# So, if we want to efficiently store the image, we need to somehow efficiently encode 3 matrices R, G and B
# for each color component, respectively.
# We can extract the 3 color component matrices as briefly mentioned above as follows:
# 0xff代表十进制数值255
R A[:,:,0] / 0xff
G A[:,:,1] / 0xff
B A[:,:,2] / 0xff# Now, we compute the SVD decomposition:
R_U, R_S, R_VT np.linalg.svd(R)
G_U, G_S, G_VT np.linalg.svd(G)
B_U, B_S, B_VT np.linalg.svd(B)# polt the singular values
xaxis np.arange(0, len(R_S))
plt.plot(xaxis, R_S, labelR_S)
plt.plot(xaxis, G_S, labelG_S)
plt.plot(xaxis, B_S, labelB_S)
plt.legend()relative_rank 0.1
max_rank int(relative_rank * min(R.shape[0], R.shape[1]))
print(max rank %d % max_rank) # 144def read_as_compressed(U, S, VT, k):Ak np.zeros((U.shape[0], VT.shape[1]))for i in range(k):U_i U[:,[i]]VT_i np.array([VT[i]])Ak S[i] * (U_i VT_i)return Ak## Actually, it is easier and more efficient to perform the same operation
## with a lower-rank matrix multiplication.
# def read_as_compressed(U, S, VT, k):
# return (U[:,:k] np.diag(S[:k])) VT[:k]R_compressed read_as_compressed(R_U, R_S, R_VT, max_rank)
G_compressed read_as_compressed(G_U, G_S, G_VT, max_rank)
B_compressed read_as_compressed(B_U, B_S, B_VT, max_rank)compressed_float np.dstack((R_compressed, G_compressed, B_compressed))
compressed (np.minimum(compressed_float, 1.0) * 0xff).astype(np.uint8)# Plot
plt.figure()
plt.imshow(A)plt.figure()
plt.imshow(compressed)image.imsave(compressed.jpg, compressed)参考资料 [A.C. Antoulas 2001] Approximation of large-scale dynamical systems: An overview [潘建瑜] 矩阵计算_讲义 Compressing images with singular value decomposition (SVD) | ZeroBone
