在python中计算图像的特征向量 [英] Compute eigenvectors of image in python

问题描述

我正在尝试将2D高斯拟合到图像中。噪音非常低，所以我的尝试是旋转图像，使两个主轴不共同变化，找出最大值并计算两个维度的标准偏差。选择的武器是python。

I'm trying to fit a 2D Gaussian to an image. Noise is very low, so my attempt was to rotate the image such that the two principal axes do not co-vary, figure out the maximum and just compute the standard deviation in both dimensions. Weapon of choice is python.

但是我一直在寻找图像的特征向量 - numpy.linalg.py 假设离散数据点。我想过将这个图像作为一个概率分布，对几千个点进行采样，然后根据该分布计算特征向量，但我确信必须有一种找到特征向量的方法（即，半主和半 - 直接来自该图像的高斯椭圆的短轴。任何想法？

However I got stuck at finding the eigenvectors of the image - numpy.linalg.py assumes discrete data points. I thought about taking this image to be a probability distribution, sampling a few thousand points and then computing the eigenvectors from that distribution, but I'm sure there must be a way of finding the eigenvectors (ie., semi-major and semi-minor axes of the gaussian ellipse) directly from that image. Any ideas?

非常感谢：）

推荐答案

只是一个请注意，有几种工具可以将高斯拟合到图像中。我唯一能想到的就是 scikits.learn ，这不是'完全是面向图像的，但我知道还有其他的。

Just a quick note, there are several tools to fit a gaussian to an image. The only thing I can think of off the top of my head is scikits.learn, which isn't completely image-oriented, but I know there are others.

准确地计算协方差矩阵的特征向量，这在计算上是非常昂贵的。您必须将图像的每个像素（或大型随机样本）与x，y点相关联。

To calculate the eigenvectors of the covariance matrix exactly as you had in mind is very computationally expensive. You have to associate each pixel (or a large-ish random sample) of image with an x,y point.

基本上，您执行以下操作：

Basically, you do something like:

    import numpy as np
    # grid is your image data, here...
    grid = np.random.random((10,10))

    nrows, ncols = grid.shape
    i,j = np.mgrid[:nrows, :ncols]
    coords = np.vstack((i.reshape(-1), j.reshape(-1), grid.reshape(-1))).T
    cov = np.cov(coords)
    eigvals, eigvecs = np.linalg.eigh(cov)

你可以利用它是一个定期采样的图像和相反，计算它的时刻（或惯性轴）。这对于大型图像来说要快得多。

You can instead make use of the fact that it's a regularly-sampled image and compute it's moments (or "intertial axes") instead. This will be considerably faster for large images.

作为一个简单的例子，（我正在使用我的一个以前的答案，万一你觉得它很有用......）

As a quick example, (I'm using a part of one of my previous answers, in case you find it useful...)

import numpy as np
import matplotlib.pyplot as plt

def main():
    data = generate_data()
    xbar, ybar, cov = intertial_axis(data)

    fig, ax = plt.subplots()
    ax.imshow(data)
    plot_bars(xbar, ybar, cov, ax)
    plt.show()

def generate_data():
    data = np.zeros((200, 200), dtype=np.float)
    cov = np.array([[200, 100], [100, 200]])
    ij = np.random.multivariate_normal((100,100), cov, int(1e5))
    for i,j in ij:
        data[int(i), int(j)] += 1
    return data 

def raw_moment(data, iord, jord):
    nrows, ncols = data.shape
    y, x = np.mgrid[:nrows, :ncols]
    data = data * x**iord * y**jord
    return data.sum()

def intertial_axis(data):
    """Calculate the x-mean, y-mean, and cov matrix of an image."""
    data_sum = data.sum()
    m10 = raw_moment(data, 1, 0)
    m01 = raw_moment(data, 0, 1)
    x_bar = m10 / data_sum
    y_bar = m01 / data_sum
    u11 = (raw_moment(data, 1, 1) - x_bar * m01) / data_sum
    u20 = (raw_moment(data, 2, 0) - x_bar * m10) / data_sum
    u02 = (raw_moment(data, 0, 2) - y_bar * m01) / data_sum
    cov = np.array([[u20, u11], [u11, u02]])
    return x_bar, y_bar, cov

def plot_bars(x_bar, y_bar, cov, ax):
    """Plot bars with a length of 2 stddev along the principal axes."""
    def make_lines(eigvals, eigvecs, mean, i):
        """Make lines a length of 2 stddev."""
        std = np.sqrt(eigvals[i])
        vec = 2 * std * eigvecs[:,i] / np.hypot(*eigvecs[:,i])
        x, y = np.vstack((mean-vec, mean, mean+vec)).T
        return x, y
    mean = np.array([x_bar, y_bar])
    eigvals, eigvecs = np.linalg.eigh(cov)
    ax.plot(*make_lines(eigvals, eigvecs, mean, 0), marker='o', color='white')
    ax.plot(*make_lines(eigvals, eigvecs, mean, -1), marker='o', color='red')
    ax.axis('image')

if __name__ == '__main__':
    main()

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