对图像进行SVD和PCA降维，可用于压缩或者图像数据增强(python版)

不懂原理的同学请参考：

https://blog.csdn.net/qq_43337858/article/details/102738352?utm_medium=distribute.pc_relevant.none-task-blog-BlogCommendFromMachineLearnPai2-6.control&depth_1-utm_source=distribute.pc_relevant.none-task-blog-BlogCommendFromMachineLearnPai2-6.control

def svd(img, topk_percent=0.7):
    """
    使用svd对图片降维，可作为一种数据增强手段
    每列作为一个向量，先构建方阵，再求特征值 特征向量，取前N个主成分，再重构图像
    :param img: 输入图像
    :param topk_percent: 图像恢复率，
    :return: img after svd
    """
    img_src = img[...]
    if len(img.shape) == 3:
        img_src = cv2.cvtColor(img_src, cv2.COLOR_BGR2GRAY)

    h, w = img_src.shape
    data = np.asarray(img_src, np.double)
    # 以下两种方式都可以
    # method 1
    U, s, V = np.linalg.svd(data)
    K = round(len(s) * topk_percent)
    S = np.diag(s)
    major_data = np.dot(U[:, :K], np.dot(S[:K, :K], V[:K, :]))
    # # method 2
    # feat_values, feat_vectors = np.linalg.eig(np.dot(data.T, data))
    # feat_index = np.argsort(np.sqrt(feat_values), axis=0)[::-1]
    # S = np.diag(feat_values)
    # V = feat_vectors[:, feat_index]
    # S_inv = np.asmatrix(S).I
    # V_inv = np.asmatrix(V).I
    # U = np.dot(np.dot(data, V), S_inv)
    # K = round(S.shape[0] * topk_percent)
    # major_data = np.dot(np.dot(U[:, :K], S[:K, :K]), V_inv[:K, :])

    rebuild_img = np.asarray(major_data, np.uint8)

    cv2.imshow('1', rebuild_img)
    cv2.waitKey(0)
    return rebuild_img


def pca(img, topk_percent=0.7):
    """
    使用pca对图片降维，可作为一种数据增强手段
    每列作为一个向量，先0均值化，再求协方差矩阵的特征值和特征向量，取前N个主成分，再重构图像
    :param img: 输入图像
    :param topk_percent: 图像恢复率，
    :return: img after pca
    """
    img_src = img[...]
    if len(img.shape) == 3:
        img_src = cv2.cvtColor(img_src, cv2.COLOR_BGR2GRAY)

    print(img_src.shape)
    h, w = img_src.shape
    data = np.asarray(img_src, np.double)
    # 计算每列的mean
    _mean = np.mean(data, axis=0)
    data -= _mean
    # 以 列为变量计算方式，计算协方差矩阵
    data_cov = np.cov(data, rowvar=False)
    feat_values, feat_vectors = np.linalg.eig(data_cov)
    feat_index = np.argsort(np.sqrt(feat_values), axis=0)[::-1]
    V = feat_vectors[:, feat_index]
    K = round(len(feat_values) * topk_percent)# 重建图像
    major_data = np.dot(np.dot(data, V[:, :K]), V[:, :K].T) + _mean
    rebuild_img = np.asarray(major_data, np.uint8)

    cv2.imshow('1', rebuild_img)
    cv2.waitKey(0)
    return rebuild_img