Image Pyramid

今天我们介绍图像处理邻域中比较常用的一种方法，image pyramid, 也叫图像金字塔。就是将图像进行一层一层的下采样，图像金字塔是为了构建图像的多尺度，让模型能够更好的适应图像的尺度变化，图像金字塔可以广泛应用于图像识别，目标检测，还有光流配准，块匹配都能看到它的身影。图像金字塔主要有两种，一种是高斯金字塔，gaussian pyramid，另外一种是拉普拉斯金字塔，Laplacian Pyramids。

(3)G0=IG1=Down(G0∗F)G2=Down(G1∗F)⋅⋅⋅GN=Down(GN−1∗F)" role="presentation">G0=IG1=Down(G0∗F)G2=Down(G1∗F)⋅⋅⋅GN=Down(GN−1∗F)(3)(3)G0=IG1=Down(G0∗F)G2=Down(G1∗F)⋅⋅⋅GN=Down(GN−1∗F)

Gk" role="presentation" style="position: relative;">GkGk 表示的每一层金字塔中的图像，F" role="presentation" style="position: relative;">FF 表示高斯卷积核，∗" role="presentation" style="position: relative;">∗∗ 表示卷积操作，Down" role="presentation" style="position: relative;">DownDown 表示下采样，上面的表达式，就可以构建一个图像金字塔。这个在 Open-CV 中有现成的函数，下面给出一段代码，看看高斯金字塔的构建：

    import numpy as np
    import matplotlib.pyplot as plt

    A = cv2.imread('D:/Python_Code/Test_img/2.jpg')
    row, col, dpt = A.shape
    pyr_level = 4
    # generate Gaussian pyramid for A
    G = A.copy()
    gpA = [G]
    for i in range(pyr_level):
        G = cv2.pyrDown(G)
        gpA.append(G)

    G = np.zeros([row, col, dpt], dtype='uint8')

    rowX2 = row // 2
    colX2 = col // 2
    G[:rowX2, :colX2, :] = gpA[1]
    rowX4 = rowX2 // 2
    colX4 = colX2 // 2
    G[rowX2:rowX2+rowX4, colX2:colX2+colX4, :] = gpA[2]
    G[:rowX4, colX2:colX2+colX4, :] = gpA[2]
    rowX8 = rowX4 // 2
    colX8 = colX4 // 2
    G[rowX2+rowX4:rowX2+rowX4+rowX8, colX2+colX4:colX2+colX4+colX8,          :] = gpA[3]
    G[ :rowX8, colX2+colX4:colX2+colX4+colX8, :] = gpA[3]
    cv2.imshow("gau_pyr", G)

下面给出一个效果图：

下面看看，拉普拉斯金字塔，拉普拉斯金字塔其实是根据高斯金字塔计算得来的：

(4)L0=G0−Up(G1∗F)L1=G1−Up(G2∗F)L2=G2−Up(G3∗F)⋅⋅⋅LN−1=GN−1−Up(GN∗F)LN=GN" role="presentation">L0=G0−Up(G1∗F)L1=G1−Up(G2∗F)L2=G2−Up(G3∗F)⋅⋅⋅LN−1=GN−1−Up(GN∗F)LN=GN(4)(4)L0=G0−Up(G1∗F)L1=G1−Up(G2∗F)L2=G2−Up(G3∗F)⋅⋅⋅LN−1=GN−1−Up(GN∗F)LN=GN

利用拉普拉斯金字塔，可以实现图像的重建，根据上面的表达式，我们可以得到：

(6)GN−1≈LN−1+Up(LN)GN−2≈LN−2+Up(GN−1)⋅⋅⋅G1≈L1+Up(G2)G0≈L0+Up(G1)" role="presentation">GN−1≈LN−1+Up(LN)GN−2≈LN−2+Up(GN−1)⋅⋅⋅G1≈L1+Up(G2)G0≈L0+Up(G1)(6)(6)GN−1≈LN−1+Up(LN)GN−2≈LN−2+Up(GN−1)⋅⋅⋅G1≈L1+Up(G2)G0≈L0+Up(G1)

也就是说，把拉普拉斯金字塔层层上采样，再累加，就可以重建出最初的图像。下面给出一段代码：

    import cv2
    import numpy as np
    A = cv2.imread('D:/Python_Code/Test_img/2.jpg')

    pyr_level = 4
    # generate Gaussian pyramid for A
    G = A.copy()
    gpA = [G]
    for i in range(pyr_level):
        G = cv2.pyrDown(G)
        gpA.append(G)

    # generate Laplacian Pyramid for A
    lpA = [gpA[pyr_level -1 ]]
    for i in range(pyr_level - 1,0,-1):
        GE = cv2.pyrUp(gpA[i])
        L = cv2.subtract(gpA[i-1],GE)
        lpA.append(L)

    # Now add left and right halves of images in each level
    LS = []
    for la,lb in zip(lpA,lpB):
        rows,cols,dpt = la.shape
        ls = la
        LS.append(ls)

    # now reconstruct
    ls_ = LS[0]
    for i in range(1,pyr_level):
        ls_ = cv2.pyrUp(ls_)
        ls_ = cv2.add(ls_, LS[i])

     cv2.imwrite('Pyramid_blending2.jpg',ls_)

原图：

重建后的图：