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mlq移动最小二乘方法

news 来源：原创 2024/5/9 18:27:55

Image Deformation Using Moving Least Squares

  1. 移动最小二乘的移动是指， 所有traindata 对 每个grid point的影响（weight）是不一样的。
  2. 文中介绍三种变换： 仿射，相似，刚体变换
  3. 对应的代码： https://github.com/Jarvis73/Moving-Least-Squares

https://github.com/search?q=Moving+Least+Squares

  1. 问题：

当有一些点聚在一起，会产生大的影响，即使 偏移很小。？
因此点应该均匀？

  1. 对于起始点固定的情况比较友好，可以提前计算出A，然后每次目标变化的时候A也是不变的。

  2. 注意边界
    clip 应该到 h-1, w-1更好，而不是设为0
    在这里插入图片描述

    其次不取整，采用cv2.remap更好：下面通过光流warp 图像

    def flow_warp(im1, im2, flow):
 """
 :param im1: pre image
 :param im2: next image
 :param flow: optical flow ,im1 to im2, h, w, 2,  0:(x),1:(y)
 :return:
 """
 h, w, c = im1.shape
 hh = np.arange(0, h)
 ww = np.arange(0, w)
 xx, yy = np.meshgrid(ww, hh)

 # tx = xx - flow[..., 0]
 # ty = yy - flow[..., 1]
 tx = flow[..., 0]
 ty = flow[..., 1]
 mask = ((tx < 0) + (ty < 0) + (tx > w - 1) + (ty > h - 1)) > 0
 # print(mask.shape, mask.dtype, np.sum(mask))
 # out = interp_linear(image, tx, ty)
 # warped im1, should compare with im2
 out = cv2.remap(im1, tx.astype(np.float32), ty.astype(np.float32), interpolation=cv2.INTER_LINEAR)
 out[mask] = 0
 return out

  3. 画网格：

import cv2
import numpy as np

import matplotlib.pyplot as plt
from PIL import Image

from matplotlib.collections import LineCollection

def plot_grid(x,y, ax=None, **kwargs):

    ax = ax or plt.gca()

    segs1 = np.stack((x,y), axis=2)

    segs2 = segs1.transpose(1,0,2)

    ax.add_collection(LineCollection(segs1, **kwargs))

    ax.add_collection(LineCollection(segs2, **kwargs))

    ax.autoscale()

    ax.invert_yaxis()  # y轴反向

if __name__ == "__main__":
    import os

    os.environ["KMP_DUPLICATE_LIB_OK"] = "TRUE"
    image = np.array(Image.open("images/toy.jpg"))


    plt.figure()
    ax = plt.subplot(121)
    ax.imshow(cv2.resize(image, (100, 100)))
    # h, w, c = 100, 100, 3
    # hh, ww = np.arange(h), np.arange(w)
    # xx, yy = np.meshgrid(ww, hh)
    f = lambda x, y: (x + 6 * np.exp(-x ** 2 - y ** 2), y)
    grid_x, grid_y = np.meshgrid(np.linspace(0, 100, 40), np.linspace(0, 100, 40))
    plot_grid(grid_x, grid_y, color="lightgrey")
    distx, disty = f(grid_x, grid_y)
    plot_grid(distx, disty, color="C0")
    ax.invert_yaxis()  # y轴反向
    plt.show()
  1. affine 我的实现
import time

import cv2
import numpy as np
import torch
from PIL import Image
from matplotlib import pyplot as plt
from torchvision.utils import flow_to_image

from img_utils import mls_affine_deformation
from plot_grid import plot_grid





def mls_affine():
    image = np.array(Image.open("images/toy.jpg"))

    height, width, _ = image.shape
    gridX = np.arange(width, dtype=np.int16)
    gridY = np.arange(height, dtype=np.int16)
    xx, yy = np.meshgrid(gridX, gridY)
    # print(xx[:4,:4], yy[:4,:4])
    # affine = mls_affine_deformation(vy, vx, p, q, alpha=1)

    eps = 1e-8
    alpha = 2
    q = np.array([
        [155, 30], [155, 125], [155, 225],
        [235, 100], [235, 160], [295, 85], [293, 180]
    ])
    p = np.array([
        [211, 42], [155, 125], [100, 235],
        [235, 80], [235, 140], [295, 85], [295, 180]
    ])
    n = len(p)
    q = q.astype(np.float32).reshape(n, 1, 1, 2)
    p = p.astype(np.float32).reshape(n, 1, 1, 2)
    v = np.vstack((yy.reshape(1, height, width), xx.reshape(1, height, width))).astype(np.float32)
    # v = np.dstack((yy, xx)).transpose(2, 0, 1) # python速度提升的方法， 第一个是对输入明确类型， 第二个是尽量不要用dstack

    v = v.transpose(1, 2, 0).reshape(-1, height, width, 2)
    print(p.shape, v.shape)
    w = 1 / (np.sum((p.reshape(-1, 1, 1, 2) - v) ** alpha, axis=-1) + eps) # n, h, w
    print(w.shape)
    w_norm = w / np.sum(w, axis=0, keepdims=True)   # n, h, w
    p_star = np.sum(p.reshape(-1, 1, 1, 2) * w_norm.reshape(-1, height, width, 1), axis=0) # 1, h, w, 2
    q_star = np.sum(q.reshape(-1, 1, 1, 2) * w_norm.reshape(-1, height, width, 1), axis=0) # 1, h, w, 2

    p_hat = p.reshape(-1, 1, 1, 2) - p_star  # n, h, w, 2
    q_hat = q.reshape(-1, 1, 1, 2) - q_star  # n, h, w, 2

    A_1row = np.sum(p_hat[..., 0][..., None] * p_hat * w_norm[..., None], axis=0, keepdims=True)  # 1, h, w, 2
    A_2row = np.sum(p_hat[..., 1][..., None] * p_hat * w_norm[..., None], axis=0, keepdims=True)  # 1, h, w, 2

    b_1row = np.sum(p_hat[..., 0][..., None] * q_hat * w_norm[..., None], axis=0, keepdims=True)  # 1, h, w, 2
    b_2row = np.sum(p_hat[..., 1][..., None] * q_hat * w_norm[..., None], axis=0, keepdims=True)  # 1, h, w, 2

    A = np.transpose(np.hstack((A_1row, A_2row)), [1, 2, 0, 3]).reshape(height, width, 2, 2)
    b = np.transpose(np.hstack((b_1row, b_2row)), [1, 2, 0, 3]).reshape(height, width, 2, 2)
    print(A.shape, b.shape,  A[:1,:1, :, :])

    # a11 = A[..., 0, 0]
    # a12 = A[..., 0, 1]
    # a21 = A[..., 1, 0]
    # a22 = A[..., 1, 1]
    # A_det = a11 * a22 - a12 * a21
    # A_inv = np.dstack((a22, -a12, -a21, a11)) / A_det[..., None]
    # A_inv = A_inv.reshape(height, width, 2, 2)
    A_inv = np.linalg.inv(A)
    M = np.einsum("ijmk,ijkn->ijmn", A_inv, b)

    x = v.reshape(height, width, 1, 2) - p_star.reshape(height, width, 1, 2)
    flow = np.einsum("ijmk,ijkn->ijmn", x, M) + q_star.reshape(height, width, 1, 2)

    flow = np.transpose(flow.reshape(height, width, 2), (2, 0, 1))
    flow[flow < 0] = 0
    flow[0][flow[0] > height - 1] = height - 1
    flow[1][flow[1] > width - 1] = width - 1
    return flow


def mls_affine2(xx, yy, p, q):
    start = time.time()
    height, width = xx.shape

    eps = 1e-8
    alpha = 2
    # start = time.time()
    n = len(p)
    q = q.astype(np.float32).reshape(n, 2, 1, 1)
    p = p.astype(np.float32).reshape(n, 2, 1, 1)
    v = np.vstack((yy.reshape(1, height, width), xx.reshape(1, height, width))).astype(np.float32)
    # v = np.dstack((yy, xx)).transpose(2, 0, 1) # python速度提升的方法， 第一个是对输入明确类型， 第二个是尽量不要用dstack

    p = p.transpose(0, 2, 3, 1)
    q = q.transpose(0, 2, 3, 1)
    v = v.transpose(1, 2, 0).reshape(-1, height, width, 2)
    print(p.shape, v.shape)
    w = 1 / (np.sum((p.reshape(-1, 1, 1, 2) - v) ** alpha, axis=-1) + eps) # n, h, w
    print(w.shape)
    end = time.time()
    print('run time:', end-start)
    w_norm = w / np.sum(w, axis=0, keepdims=True)   # n, h, w
    p_star = np.sum(p.reshape(-1, 1, 1, 2) * w_norm.reshape(-1, height, width, 1), axis=0) # 1, h, w, 2
    q_star = np.sum(q.reshape(-1, 1, 1, 2) * w_norm.reshape(-1, height, width, 1), axis=0) # 1, h, w, 2

    p_hat = p.reshape(-1, 1, 1, 2) - p_star  # n, h, w, 2
    q_hat = q.reshape(-1, 1, 1, 2) - q_star  # n, h, w, 2
    end = time.time()
    print('run time1:', end - start)
    A_1row = np.sum(p_hat[..., 0][..., None] * p_hat * w_norm[..., None], axis=0, keepdims=True)  # 1, h, w, 2
    A_2row = np.sum(p_hat[..., 1][..., None] * p_hat * w_norm[..., None], axis=0, keepdims=True)  # 1, h, w, 2

    b_1row = np.sum(p_hat[..., 0][..., None] * q_hat * w_norm[..., None], axis=0, keepdims=True)  # 1, h, w, 2
    b_2row = np.sum(p_hat[..., 1][..., None] * q_hat * w_norm[..., None], axis=0, keepdims=True)  # 1, h, w, 2
    end = time.time()
    print('run time2:', end - start)
    A = np.transpose(np.hstack((A_1row, A_2row)), [1, 2, 0, 3]).reshape(height, width, 2, 2)
    b = np.transpose(np.hstack((b_1row, b_2row)), [1, 2, 0, 3]).reshape(height, width, 2, 2)
    print(A.shape, b.shape,  A[:1,:1, :, :])
    end = time.time()
    print('run time3:', end - start)
    # a11 = A[..., 0, 0]
    # a12 = A[..., 0, 1]
    # a21 = A[..., 1, 0]
    # a22 = A[..., 1, 1]
    # A_det = a11 * a22 - a12 * a21
    # A_inv = np.dstack((a22, -a12, -a21, a11)) / A_det[..., None]
    # A_inv = A_inv.reshape(height, width, 2, 2)
    A_inv = np.linalg.inv(A)
    M = np.einsum("ijmk,ijkn->ijmn", A_inv, b)

    x = v.reshape(height, width, 1, 2) - p_star.reshape(height, width, 1, 2)
    flow = np.einsum("ijmk,ijkn->ijmn", x, M) + q_star.reshape(height, width, 1, 2)
    end = time.time()
    print('run time4:', end - start)
    flow = np.transpose(flow.reshape(height, width, 2), (2, 0, 1))
    flow[flow < 0] = 0
    flow[0][flow[0] > height - 1] = height - 1
    flow[1][flow[1] > width - 1] = width - 1
    end = time.time()
    print('run time5:', end - start)
    return flow.astype(np.int16)

def flow_warp(im1, im2, flow):
    """
    :param im1: pre image
    :param im2: next image
    :param flow: optical flow ,im1 to im2, h, w, 2,  0:(x),1:(y)
    :return:
    """
    h, w, c = im1.shape
    hh = np.arange(0, h)
    ww = np.arange(0, w)
    xx, yy = np.meshgrid(ww, hh)

    # tx = xx - flow[..., 0]
    # ty = yy - flow[..., 1]
    tx = flow[..., 0]
    ty = flow[..., 1]
    mask = ((tx < 0) + (ty < 0) + (tx > w - 1) + (ty > h - 1)) > 0
    # print(mask.shape, mask.dtype, np.sum(mask))
    # out = interp_linear(image, tx, ty)
    # warped im1, should compare with im2
    out = cv2.remap(im1, tx.astype(np.float32), ty.astype(np.float32), interpolation=cv2.INTER_LINEAR)
    out[mask] = 0
    return out

def mls_affine_my(image, p, q):
    '''
    其实给定p和 q， 完整的flow就已经确定
    Parameters
    ----------
    image： h, w, c
    p : n*2, (x, y)
    q : n*2, (x, y)
    Returns
    -------
    return flow:p->q, shape like image,  h*w*2: (x, y)
    '''
    eps = 1e-8
    height, width, _ = image.shape
    gridX = np.arange(width, dtype=np.int16)
    gridY = np.arange(height, dtype=np.int16)
    xx, yy = np.meshgrid(gridX, gridY)

    n = len(p)
    q = q.astype(np.float32).reshape(n, 1, 1, 2)
    p = p.astype(np.float32).reshape(n, 1, 1, 2)
    v = np.dstack((xx, yy))

    w = 1.0 / (np.sum((p - v) ** 2, axis=-1) + eps) ** 1
    w_norm = w / np.sum(w, axis=0, keepdims=True)

    p_star = np.sum(p * w_norm.reshape(-1, height, width, 1), axis=0)  # 1, h, w, 2
    q_star = np.sum(q * w_norm.reshape(-1, height, width, 1), axis=0)  # 1, h, w, 2

    p_hat = p - p_star  # n, h, w, 2
    q_hat = q - q_star  # n, h, w, 2
    print(p_hat.shape, q_hat.shape, w_norm.shape)

    A_1row = np.sum(p_hat[..., 0][..., None] * p_hat * w_norm[..., None], axis=0)  # 1, h, w, 2
    A_2row = np.sum(p_hat[..., 1][..., None] * p_hat * w_norm[..., None], axis=0)  # 1, h, w, 2

    b_1row = np.sum(p_hat[..., 0][..., None] * q_hat * w_norm[..., None], axis=0)  # 1, h, w, 2
    b_2row = np.sum(p_hat[..., 1][..., None] * q_hat * w_norm[..., None], axis=0)  # 1, h, w, 2

    A = np.dstack((A_1row, A_2row)).reshape(height, width, 2, 2)
    b = np.dstack((b_1row, b_2row)).reshape(height, width, 2, 2)
    print(A.shape, b.shape, A[:1, :1, :, :])

    # a11 = A[..., 0, 0]
    # a12 = A[..., 0, 1]
    # a21 = A[..., 1, 0]
    # a22 = A[..., 1, 1]
    # A_det = a11 * a22 - a12 * a21
    # A_inv = np.dstack((a22, -a12, -a21, a11)) / A_det[..., None]
    # A_inv = A_inv.reshape(height, width, 2, 2)
    A_inv = np.linalg.inv(A)
    M = np.einsum("ijmk,ijkn->ijmn", A_inv, b)

    x = v.reshape(height, width, 1, 2) - p_star.reshape(height, width, 1, 2)
    y = np.einsum("ijmk,ijkn->ijmn", x, M) + q_star.reshape(height, width, 1, 2)

    flow = y.reshape(height, width, 2) - v

    print(flow.shape, flow.dtype)

    return flow

def load_flow_to_numpy(path):
    with open(path, 'rb') as f:
        magic = np.fromfile(f, np.float32, count=1)
        assert (202021.25 == magic), 'Magic number incorrect. Invalid .flo file'
        w = np.fromfile(f, np.int32, count=1)[0]
        h = np.fromfile(f, np.int32, count=1)[0]
        data = np.fromfile(f, np.float32, count=2 * w * h)
    data2D = np.reshape(data, (h, w, 2))
    # print(data2D[:10,:10,0])
    return data2D
def flow_to_image_torch(flow):
    flow = torch.from_numpy(np.transpose(flow, [2, 0, 1]).astype(np.float32))
    flow_im = flow_to_image(flow)
    img = np.transpose(flow_im.numpy(), [1, 2, 0])

    #print(img.shape)
    return img

"""
箭头图
"""
def draw_flow(im, flow, step=40, norm=1):
    # 在间隔分开的像素采样点处绘制光流
    h, w = im.shape[:2]
    y, x = np.mgrid[step/2:h:step, step/2:w:step].reshape(2, -1).astype(int)

    if norm:
        fx, fy = flow[y, x].T / flow[y, x].max() * step // 2.00001
    else:
        fx, fy = flow[y, x].T  # / flow[y, x].max() * step // 2
    # 创建线的终点
    lines = np.vstack([x, y, x+fx, y+fy]).T.reshape(-1, 2, 2)
    lines = np.int32(lines)
    # 创建图像并绘制
    vis = im  #cv2.cvtColor(im, cv2.COLOR_GRAY2BGR)
    for (x1, y1), (x2, y2) in lines:
      cv2.line(vis, (x1, y1), (x2, y2), (0, 255, 0), 2)
      cv2.circle(vis, (x1, y1), 2, (0, 0, 255), -1)
    return vis
if __name__ == "__main__":
    import os

    os.environ["KMP_DUPLICATE_LIB_OK"] = "TRUE"
    image = np.array(Image.open("images/toy.jpg"))
    #
    # flow = mls_affine()
    # c, h, w = flow.shape
    # flow = np.transpose(flow, (1, 2, 0))[..., ::-1]
    # out = flow_warp(image, image, flow)
    # plt.figure()
    # plt.imshow(np.hstack((image, out)))
    # plt.show()
    # print('dd')


    q = np.array([
        [155, 30], [155, 125], [155, 225],
        [235, 100], [235, 160], [295, 85], [293, 180]
    ])
    p = np.array([
        [211, 42], [155, 125], [100, 235],
        [235, 80], [235, 140], [295, 85], [295, 180]
    ])
    q = q.reshape(-1, 2)[..., ::-1]
    p = p.reshape(-1, 2)[..., ::-1]

    h, w, c = image.shape
    hh = range(h)
    ww = range(w)
    xx, yy = np.meshgrid(ww, hh)

    flow = mls_affine_my(image, p, q)




    flow_im = flow_to_image_torch(flow)
    flow_im2 = draw_flow(image, flow, step=20, norm=1)
    plt.figure()
    plt.subplot(121)
    plt.imshow(flow_im)
    plt.subplot(122)
    plt.imshow(flow_im2)
    plt.show()

    xy = np.dstack((xx, yy)).astype(np.float32)
    newxy = xy + flow
    out = flow_warp(image, image, newxy)
    plt.figure()
    plt.imshow(np.hstack((image, out)))
    plt.show()

    plt.figure()
    xx = xx[::20, ::20]
    yy = yy[::20, ::20]
    plot_grid(xx, yy, color="lightgrey")

    newxy = xy - flow
    xx = newxy[..., 0]
    yy = newxy[..., 1]
    xx = xx[::20, ::20]
    yy = yy[::20, ::20]
    plot_grid(xx, yy, color="C0")
    plt.show()

    # aug2 = np.ones_like(image)
    # aug2[yy, xx] = image[tuple(flow[..., ::-1])]
    # plt.figure()
    # plt.imshow(np.hstack((image, aug2)))
    # plt.show()

    # 我的版本 和 他不一样？
    xx, yy = np.meshgrid(range(5), range(5))
    p1 = np.array([[1, 1], [3, 0.9], [3, 1.1], [3, 1], [3.1, 1], [2.9, 1], [3,1], [3,1], [3,1]])
    p2 = np.array([[1, 1], [4, 0.9], [4, 1.1], [4, 1], [4.1, 1], [3.9, 1], [4,1], [4,1], [4,1]]) # 0.5

    # p1 = np.array([[1, 1], [3, 0.9], [3, 1.1], [3, 1]])
    # p2 = np.array([[1, 1], [4, 0.9], [4, 1.1], [4, 1]]) # 0.5

    # p1 = np.array([[1, 1], [3, 0.9], [3, 1.1]]) # 0.5
    # p2 = np.array([[1, 1], [4, 0.9], [4, 1.1]])

    # p1 = np.array([[1, 1], [3.0, 0.9]])
    # p2 = np.array([[1, 1], [4.0, 1.1]])  # -1.04504478
    flow = mls_affine_my(xx[..., None], p1, p2)

    print(flow[1, 2, 0], flow[..., 0])
    flow = mls_affine_deformation(xx, yy, p2[...,::-1], p1[...,::-1])# 2,h,w        (r,c)
    print(flow.shape)
    flow = np.transpose(flow, (1,2,0))[..., ::-1] - np.dstack((xx, yy))
    print('ss2:', flow[1, 2, 0], flow[..., 0])

正确性待验证，包括仓库里的代码正确性也不保证。

  1. 关于weight 是否有更好的方式，来应对其他应用

    更好的weight计算方法，比如考虑图像内容差异。双边weight等。

  2. 就是一些特殊情况，边界情况，求逆有问题的情况等需要考虑

  3. 最好阅读原文Image Deformation Using Moving Least Squares，其他一些blog也可以参考

    https://blog.csdn.net/u011426016/article/details/125243631
    https://blog.csdn.net/DIAJEY/article/details/114322764

  4. 可以用在曲面拟合上， 3Dlut拟合， 光流图，深度图拟合优化等
    比如光流：

    import numpy as np
import scipy.optimize
import torch
from scipy import sparse
from matplotlib import pyplot as plt
from scipy.optimize import minimize
from scipy.sparse import csc_matrix
from torchvision.utils import flow_to_image
from tqdm import tqdm

from movelq import mls_affine_deformation
def generate_data(pic=False):

    s = np.random.uniform(0, 1, [24, 2])
    t = np.random.uniform(0, 1, [24])
    print(s)
    print(t, t* np.arange(24))
    t = t* np.arange(24)

    monotonic_e = False
    if monotonic_e:
        s = np.sort(s, axis=0)
        t = np.sort(t, axis=0)
        t[:6] = t[6:12][::-1]
    # 拟合spline
    z1 = np.polyfit(s[..., 0], t, 4)
    p1 = np.poly1d(z1)

    z2 = np.polyfit(s[..., 1], t, 4)
    p2 = np.poly1d(z2)

    xx = np.linspace(0, 1, 1000)
    z1 = p1(xx)
    z2 = p2(xx)

    if pic:
        # 画散点图
        fig = plt.figure()
        ax = fig.gca(projection='3d')
        ax.scatter(s[..., 0], s[..., 1], t, label="random points")

        ax.plot(xx, np.zeros_like(xx), z1, label=' x spline')
        ax.plot(np.zeros_like(xx), xx, z2, label=' y spline')

        ax.set_xlabel('X label')  # 画出坐标轴
        ax.set_ylabel('Y label')
        ax.set_zlabel('Z label')
        ax.legend()
        plt.show()

    # merge spline
    h = xx
    w = xx
    ww, hh = np.meshgrid(w, h)

    z = p1(ww) + p2(hh)

    z = 100 * (z - z.min()) / z.max()
    if pic:
        # 画散点图
        fig = plt.figure()
        ax = fig.gca(projection='3d')
        ax.scatter(s[..., 0], s[..., 1], t, label="random points")

        ax.plot(xx, np.zeros_like(xx), z1 , label=' x spline')
        ax.plot(np.zeros_like(xx), xx, z2 , label=' y spline')

        ax.plot_surface(ww, hh, z, cmap='rainbow')
        ax.set_xlabel('X label')  # 画出坐标轴
        ax.set_ylabel('Y label')
        ax.set_zlabel('Z label')
        ax.legend()
        plt.show()
    return ww, hh, z

def flow_to_image_torch(flow):
    flow = torch.from_numpy(np.transpose(flow, [2, 0, 1]).astype(np.float32))
    flow_im = flow_to_image(flow)
    img = np.transpose(flow_im.numpy(), [1, 2, 0])

    #print(img.shape)
    return img

def generate_flow():
    def load_flow_to_numpy(path):
        with open(path, 'rb') as f:
            magic = np.fromfile(f, np.float32, count=1)
            assert (202021.25 == magic), 'Magic number incorrect. Invalid .flo file'
            w = np.fromfile(f, np.int32, count=1)[0]
            h = np.fromfile(f, np.int32, count=1)[0]
            data = np.fromfile(f, np.float32, count=2 * w * h)
        data2D = np.reshape(data, (h, w, 2))
        # print(data2D[:10,:10,0])
        return data2D
    file = r'C:\Users\rp\source\repos\dis_flow\dis_flow\frame_0001.flo'
    flow_gt = load_flow_to_numpy(file)

    return flow_gt
if __name__ == "__main__":
    import os

    os.environ["KMP_DUPLICATE_LIB_OK"] = "TRUE"
    flow_gt = generate_flow()
    print(flow_gt.shape, flow_gt.dtype, flow_gt.min(), flow_gt.max())
    flow_gt_im = flow_to_image_torch(flow_gt)
    plt.figure()
    plt.imshow(flow_gt_im)
    #plt.show()

    # 计算 match point
    h, w, c = flow_gt.shape
    hh = np.arange(0, h)
    ww = np.arange(0, w)
    xx, yy = np.meshgrid(ww, hh)

    tx = xx + flow_gt[..., 0]
    ty = yy + flow_gt[..., 1]
    pp = np.hstack((yy.reshape(-1, 1), xx.reshape(-1, 1)))
    qq = np.hstack((ty.reshape(-1, 1), tx.reshape(-1, 1)))
    print(pp[100:105, :], qq[100:105, :])
    train_index = np.random.randint(0, len(pp), 2127)
    p = pp[train_index]
    q = qq[train_index]

    ret = mls_affine_deformation(xx, yy, p, q, alpha=1.0, eps=1e-8) # 2, h, w
    ret = ret.transpose(1, 2, 0)     # h, w, 2.  先r后c

    flow = np.dstack((yy, xx)) - ret
    flow = flow[..., ::-1]
    print('dd', flow_gt[100:105,100:105, 0], flow[100:105,100:105, 0])
    print(flow_gt[100:105, 100:105, 1], flow[100:105, 100:105, 1])
    pic = 1
    if pic:
        # 画散点图
        fig = plt.figure()
        ax1 = fig.gca(projection='3d')
        ax1.scatter(p[..., 0], p[..., 1], q[..., 0]-p[..., 0], )
        ax1.scatter(p[..., 0], p[..., 1], q[..., 1]-p[..., 1], )

        z = ret.reshape(-1, 2)[train_index]
        ax2 = fig.gca(projection='3d')
        ax2.scatter(p[..., 0], p[..., 1], p[..., 0] - z[..., 0], marker='+', c='r')
        ax2.scatter(p[..., 0], p[..., 1], p[..., 1] - z[..., 1], marker='+', c='k')

        plt.show()


    flow_im = flow_to_image_torch(flow.astype(np.float32))
    plt.figure()
    plt.imshow(flow_im)
    plt.show()

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