python - python中的图像偏度和峰度

Question

是否有一个 python 包可以为我提供一种方法来确定图像的偏度和峰度？任何例子都会很棒。

非常感谢。

Answer 1

我假设您有一张显示某种峰值的图像，并且您对获取该峰值在 x 和 y 方向上的偏度和峰度（可能还有标准偏差和质心）很感兴趣。

我也想知道这个。奇怪的是，我没有发现任何 python 图像分析包都实现了这一点。OpenCV 有一个矩函数，我们应该能够从中得到偏度，但是矩只有 3 阶，而我们需要 4 阶才能得到峰度。

为了使事情变得更容易和更快，我认为在 x 和 y 方向上进行图像投影并从这些投影中找到统计数据在数学上等同于使用完整图像找到统计数据。在下面的代码中，我使用了这两种方法，并表明它们对于这个平滑示例是相同的。使用真实的、嘈杂的图像，我发现这两种方法也提供了相同的结果，但前提是您手动将图像数据转换为 float64（它导入为 float 32，并且“数字内容”导致结果略有不同。

下面是一个例子。您应该能够将“image_statistics()”函数剪切并粘贴到您自己的代码中。希望它适用于某人！:)

import numpy as np
import matplotlib.pyplot as plt
import time

plt.figure(figsize=(10,10))

ax1 = plt.subplot(221)
ax2 = plt.subplot(222)
ax4 = plt.subplot(224)

#Make some sample data as a sum of two elliptical gaussians:
x = range(200)
y = range(200)

X,Y = np.meshgrid(x,y)

def twoD_gaussian(X,Y,A=1,xo=100,yo=100,sx=20,sy=10):
    return A*np.exp(-(X-xo)**2/(2.*sx**2)-(Y-yo)**2/(2.*sy**2))

Z = twoD_gaussian(X,Y) + twoD_gaussian(X,Y,A=0.4,yo=75)

ax2.imshow(Z) #plot it

#calculate projections along the x and y axes for the plots
yp = np.sum(Z,axis=1)
xp = np.sum(Z,axis=0)

ax1.plot(yp,np.linspace(0,len(yp),len(yp)))
ax4.plot(np.linspace(0,len(xp),len(xp)),xp)

#Here is the business:
def image_statistics(Z):
    #Input: Z, a 2D array, hopefully containing some sort of peak
    #Output: cx,cy,sx,sy,skx,sky,kx,ky
    #cx and cy are the coordinates of the centroid
    #sx and sy are the stardard deviation in the x and y directions
    #skx and sky are the skewness in the x and y directions
    #kx and ky are the Kurtosis in the x and y directions
    #Note: this is not the excess kurtosis. For a normal distribution
    #you expect the kurtosis will be 3.0. Just subtract 3 to get the
    #excess kurtosis.
    import numpy as np

    h,w = np.shape(Z)

    x = range(w)
    y = range(h)


    #calculate projections along the x and y axes
    yp = np.sum(Z,axis=1)
    xp = np.sum(Z,axis=0)

    #centroid
    cx = np.sum(x*xp)/np.sum(xp)
    cy = np.sum(y*yp)/np.sum(yp)

    #standard deviation
    x2 = (x-cx)**2
    y2 = (y-cy)**2

    sx = np.sqrt( np.sum(x2*xp)/np.sum(xp) )
    sy = np.sqrt( np.sum(y2*yp)/np.sum(yp) )

    #skewness
    x3 = (x-cx)**3
    y3 = (y-cy)**3

    skx = np.sum(xp*x3)/(np.sum(xp) * sx**3)
    sky = np.sum(yp*y3)/(np.sum(yp) * sy**3)

    #Kurtosis
    x4 = (x-cx)**4
    y4 = (y-cy)**4
    kx = np.sum(xp*x4)/(np.sum(xp) * sx**4)
    ky = np.sum(yp*y4)/(np.sum(yp) * sy**4)


    return cx,cy,sx,sy,skx,sky,kx,ky

#We can check that the result is the same if we use the full 2D data array
def image_statistics_2D(Z):
    h,w = np.shape(Z)

    x = range(w)
    y = range(h)

    X,Y = np.meshgrid(x,y)

    #Centroid (mean)
    cx = np.sum(Z*X)/np.sum(Z)
    cy = np.sum(Z*Y)/np.sum(Z)

    ###Standard deviation
    x2 = (range(w) - cx)**2
    y2 = (range(h) - cy)**2

    X2,Y2 = np.meshgrid(x2,y2)

    #Find the variance
    vx = np.sum(Z*X2)/np.sum(Z)
    vy = np.sum(Z*Y2)/np.sum(Z)

    #SD is the sqrt of the variance
    sx,sy = np.sqrt(vx),np.sqrt(vy)

    ###Skewness
    x3 = (range(w) - cx)**3
    y3 = (range(h) - cy)**3

    X3,Y3 = np.meshgrid(x3,y3)

    #Find the thid central moment
    m3x = np.sum(Z*X3)/np.sum(Z)
    m3y = np.sum(Z*Y3)/np.sum(Z)

    #Skewness is the third central moment divided by SD cubed
    skx = m3x/sx**3
    sky = m3y/sy**3

    ###Kurtosis
    x4 = (range(w) - cx)**4
    y4 = (range(h) - cy)**4

    X4,Y4 = np.meshgrid(x4,y4)

    #Find the fourth central moment
    m4x = np.sum(Z*X4)/np.sum(Z)
    m4y = np.sum(Z*Y4)/np.sum(Z)

    #Kurtosis is the fourth central moment divided by SD to the fourth power
    kx = m4x/sx**4
    ky = m4y/sy**4

    return cx,cy,sx,sy,skx,sky,kx,ky


#Calculate the image statistics using the projection method
stats_pr = image_statistics(Z)

#Confirm that they are the same by using a 2D calculation
stats_2d = image_statistics_2D(Z)

names = ('Centroid x','Centroid y','StdDev x','StdDev y','Skewness x','Skewness y','Kurtosis x','Kurtosis y')

print 'Statistis\t1D\t2D'
for name,i1,i2 in zip(names, stats_2d,stats_pr):
    print '%s \t%.2f \t%.2f'%(name, i1,i2)

plt.show()

输出的屏幕截图，只是为了好玩：

还有一件事：根据您对图像所做的确切操作，您可能会考虑使用ImageJ进行图像分析——但要小心！矩插件将让您计算偏度、峰度等。ImageJ 在分析>>设置测量菜单中确实有“偏度”和“峰度”，但我认为这实际上找到了强度直方图的偏度和峰度（我被愚弄了一分钟）。