仿射和弹性变换（affine and elastic transform）的python实现

仿射变换：

相当于对于图像做了一个平移、旋转、放缩、剪切、对称。与刚体变换相同的是，可以保持线点之间的平行和共线关系。即，原来平行的直线变化后还是平行的。但是和刚体变换不同的是线段之间的长度会发生变化。

仿射变换是指在几何中，一个向量空间进行一次线性变换并接上一个平移，变换为另一个向量空间。在有限维的情况，每个仿射变换可以由一个矩阵A和一个向量b给出，它可以写作A和一个附加的列b。一个仿射变换对应于一个矩阵和一个向量的乘法，而仿射变换的复合对应于普通的矩阵乘法，只要加入一个额外的行到矩阵的底下，这一行全部是0除了最右边是一个1。
设有图像A， 大小：M*N， 对于 任意像素（x1, y1）其变换为：

在python中用opencv实现：
由于变换矩阵的确定太过复杂，所以会采用3点去定法。即给出3个点在原图和变换后图像的位置，返回一个变换矩阵。pts1和pts2是3个点的list

M = cv2.getAffineTransform(pts1, pts2) 

随机的仿射变换

    shape = image.shape
    shape_size = shape[:2]
    # Random affine
    center_square = np.float32(shape_size) // 2
    square_size = min(shape_size) // 3
    pts1 = np.float32([center_square + square_size, [center_square[0]+square_size, center_square[1]-square_size], center_square - square_size])
    pts2 = pts1 + random_state.uniform(-alpha_affine, alpha_affine, size=pts1.shape).astype(np.float32)
    M = cv2.getAffineTransform(pts1, pts2)
    image = cv2.warpAffine(image, M, shape_size[::-1], borderMode=cv2.BORDER_REFLECT_101)  

弹性变化：

弹性变换算法(Elastic Distortion)最先是由Patrice等人在2003年的ICDAR上发表的《Best Practices for Convolutional Neural Networks Applied to Visual Document Analysis》提出的，最开始应用在mnist手写体数字识别数据集中，发现对原图像进行弹性变换的操作扩充样本以后，对于手写体数字的识别效果有明显的提升。此后成为一种很普遍的扩充字符样本图像的方式。参考链接

弹性变化是对像素点各个维度产生(-1，1)区间的随机标准偏差，并用高斯滤波（0，sigma）对各维度的偏差矩阵进行滤波，最后用放大系数alpha控制偏差范围。因而由A(x,y)得到的A’(x+delta_x,y+delta_y)。A‘的值通过在原图像差值得到，A’的值充当原来A位置上的值。一般来说，alpha越小，sigma越大，产生的偏差越小，和原图越接近。
代码实现：

import numpy as np
import pandas as pd
import cv2
from scipy.ndimage.interpolation import map_coordinates
from scipy.ndimage.filters import gaussian_filter
import matplotlib.pyplot as plt
def Elastic_transform(image, alpha, sigma):
    shape = image.shape
    shape_size = shape[:2]
    dx = gaussian_filter((random_state.rand(*shape) * 2 - 1), sigma) * alpha
    dy = gaussian_filter((random_state.rand(*shape) * 2 - 1), sigma) * alpha
    dz = np.zeros_like(dx)

    x, y, z = np.meshgrid(np.arange(shape[1]), np.arange(shape[0]), np.arange(shape[2]))
    indices = np.reshape(y+dy, (-1, 1)), np.reshape(x+dx, (-1, 1)), np.reshape(z, (-1, 1))

    return map_coordinates(image, indices, order=1, mode='reflect').reshape(shape)

Reference：
[Simard2003] Simard, Steinkraus and Platt, “Best Practices for Convolutional Neural Networks applied to Visual Document Analysis”, in Proc. of the International Conference on Document Analysis and Recognition, 2003.

随机仿射弹性变换实现：

def elastic_transform(image, alpha, sigma, alpha_affine, random_state=None):
    """Elastic deformation of images as described in [Simard2003]_ (with modifications).
    .. [Simard2003] Simard, Steinkraus and Platt, "Best Practices for
         Convolutional Neural Networks applied to Visual Document Analysis", in
         Proc. of the International Conference on Document Analysis and
         Recognition, 2003.
     Based on https://gist.github.com/erniejunior/601cdf56d2b424757de5
    """
    if random_state is None:
        random_state = np.random.RandomState(None)

    shape = image.shape
    shape_size = shape[:2]

    # Random affine
    center_square = np.float32(shape_size) // 2
    square_size = min(shape_size) // 3
    pts1 = np.float32([center_square + square_size, [center_square[0]+square_size, center_square[1]-square_size], center_square - square_size])
    pts2 = pts1 + random_state.uniform(-alpha_affine, alpha_affine, size=pts1.shape).astype(np.float32)
    M = cv2.getAffineTransform(pts1, pts2)
    image = cv2.warpAffine(image, M, shape_size[::-1], borderMode=cv2.BORDER_REFLECT_101)

    dx = gaussian_filter((random_state.rand(*shape) * 2 - 1), sigma) * alpha
    dy = gaussian_filter((random_state.rand(*shape) * 2 - 1), sigma) * alpha
    dz = np.zeros_like(dx)

    x, y, z = np.meshgrid(np.arange(shape[1]), np.arange(shape[0]), np.arange(shape[2]))
    indices = np.reshape(y+dy, (-1, 1)), np.reshape(x+dx, (-1, 1)), np.reshape(z, (-1, 1))

    return map_coordinates(image, indices, order=1, mode='reflect').reshape(shape)

代码参考kaggle data augmentation

随机仿射弹性变换效果