如何通过对第一列中的行求和并将其他列归零来从另一个构建稀疏矩阵？

Question

在我阅读 scikit-image 图书馆网站上 random walkerAlgorithm 的代码后，我尝试从头开始实现 Omega、Laplacian 和 A 矩阵在数学上定义如下：
其中CI::I表示邻域，即4个相连的邻域(i-1,i+1,j-1,j+1)，eij是两个像素点之间的边，除了它们的强度 alpha 和 beta 之间的差异是任意标量。
我已经在下面的附加代码中实现了拉普拉斯算子和 Omega，但是我不知道如何实现矩阵 A，因为我不知道如何为稀疏矩阵的切片赋值。

import numpy as np
import time
from scipy import sparse


def make_graph_edges(image):
    if(len(image.shape)==2):
        n_x, n_y = image.shape
        vertices = np.arange(n_x * n_y ).reshape((n_x, n_y))
        edges_horizontal = np.vstack(( vertices[:, :-1].ravel(), vertices[:, 1:].ravel()))   # X *(Y-1)
        edges_vertical   = np.vstack(( vertices[   :-1].ravel(), vertices[1:   ].ravel()))   #(X-1)* Y  
        edges = np.hstack((edges_horizontal, edges_vertical))
    return edges

权重函数：

def compute_weights(image,mask,alpha, beta, eps=1.e-6):
    intra_gradients = np.concatenate([np.diff(image, axis=ax).ravel()
     for ax in [1, 0] ], axis=0) ** 2            # gradient ^2
    # 5-Connected
    inter_gradients = np.concatenate([np.diff(mask, axis=ax).ravel()
    for ax in [1, 0] ], axis=0)**2 
    # inter_gradients = np.concatenate((inter_gradients,(mask-image).ravel()),axis=0)**2  # gradient ^2
    # print('inter_gradients shape',inter_gradients.shape)
    #----------------------------------------
    # 1-Connected
    # inter_gradients = (image - mask)**2
    #----------------------------------------
    # Normalize gradients
    intra_gradients = (intra_gradients - np.amin(intra_gradients))/(np.amax(intra_gradients)- np.amin(intra_gradients))
    inter_gradients = (inter_gradients - np.amin(inter_gradients))/(np.amax(inter_gradients)- np.amin(inter_gradients))
    #------------------------------------------------------
    intra_scale_factor  = -beta  / (10 * image.std())
    intra_weights = np.exp(intra_scale_factor * intra_gradients)
    intra_weights += eps
    #------------------------------------------------------
    inter_scale_factor  = -alpha / (10 * image.std())
    inter_weights = np.exp(inter_scale_factor * inter_gradients)
    inter_weights += eps
    #------------------------------------------------------
    return -intra_weights, inter_weights

构建矩阵：

def build_matrices(image, mask, alpha=90, beta=130): 
   edges_2D = make_graph_edges(image)
   intra_weights,inter_weights=compute_weights(image=image,mask=mask,alpha=alpha ,beta=beta, eps=1.e-6 )
    #================
    # Matrix Laplace
    #================    
    # Build the sparse linear system
    pixel_nb  = edges_2D.shape[1]  # N = n_x * (n_y - 1) * +  (n_x - 1) * n_y
    print('Edges Shape: ',edges_2D.shape,'intra-Weights shape: ',intra_weights.shape)
    i_indices = edges_2D.ravel()   # Src - Dest
    print('i',i_indices.shape)
    j_indices = edges_2D[::-1].ravel() # Same list in reverse order ( Dest - Src)
    print('j',j_indices.shape)    
    stacked_intra = np.hstack((intra_weights, intra_weights)) # weights (S-->D, D-->S) are same because graph is undirected
    lap = sparse.coo_matrix((2*stacked_intra, (i_indices, j_indices)), shape=(pixel_nb, pixel_nb))
    lap.setdiag(-2*np.ravel(lap.sum(axis=0)))
    print('Lap',lap.shape)
    Laplace = lap.tocsr()
    #================
    # Matrix Omega
    #================
    # Build the sparse linear system   
    stacked_inter = np.hstack((inter_weights, inter_weights)) # weights (S-->D, D-->S) are same because graph is undirected
    Omeg = sparse.coo_matrix((2*stacked_inter, (i_indices, j_indices)), shape=(pixel_nb, pixel_nb))
    Omeg.setdiag(2*np.ravel((image-mask)**2))
    print('Omeg',Omeg.shape)
    Omega = Omeg.tocsr()
    #================
    # Matrix A
    #================     
    # Build the sparse linear system  
    Mat_A = 0
    return Laplace, Omega, Mat_A

Answer 1

答案是：

weights = Omega.copy() 
firstColumn  = weights.sum(axis=1)/2
otherColumns = sparse.csr_matrix((weights.shape[0],weights.shape[1]-1))
Mat_A = sparse.hstack((firstColumn, otherColumns))

如何通过对第一列中的行求和并将其他列归零来从另一个构建稀疏矩阵？

How to build Sparse Matrix from Another by summation of rows in the first column, and zeroing the other columns?

image-processing

scipy

sparse-matrix

python-3.x