Question

是否有一个python包可以为我提供一种计算图像偏斜度和峰度的方法？任何一个例子都会很棒。

非常感谢。

Answer 1

我假设您有一个显示某种峰值的图像，并且您有兴趣在x和y方向上获得该峰值的偏斜和峰度（可能是标准偏差和质心）。

我也想知道这件事。奇怪的是，我没有在任何python图像分析包中找到它。 OpenCV有一个moments function，我们应该能够从这些中得到偏斜，但是这些时刻只能达到第3阶，我们需要第4阶才能获得峰度。

为了使事情变得更容易和更快，我认为在x和y方向上投影图像并从这些投影中查找统计数据在数学上等同于使用完整图像查找统计数据。在下面的代码中，我使用了两种方法，并表明它们对于这个平滑的示例是相同的。使用真实，嘈杂的图像，我发现这两种方法也提供了相同的结果，但只有当您手动将图像数据转换为float64（它导入为float 32，并且“数字填充”导致结果略有不同时）。

以下是一个例子。您应该能够将“image_statistics（）”函数剪切并粘贴到您自己的代码中。希望它对某人有用！：）

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：

screen shot of output

还有一件事：根据您对图像的确切操作，您可能会考虑使用ImageJ进行图像分析 - 但请注意！ moments plugin将让你计算偏度，峰度等.ImageJ在Analyze＆gt;＆gt;设置测量菜单中确实有“偏斜”和“峰度”，但我认为这实际上找到了偏斜度和峰度强度直方图（我被愚弄了一分钟）。

图像偏斜＆amp; python中的峰度

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