蟒蛇中的圖像偏度和峯度

Question

是否有一個python軟件包可以爲我提供一種方法來分辨圖像的偏度和峯度？任何例子都會很棒。

非常感謝。

Answer 1

我假設你有一個圖像顯示某種形式的峯值，並且你有興趣在x和y方向獲得該峯值的偏度和峯度（可能是標準偏差和質心）。

我在想這件事。奇怪的是，我沒有發現這實現到任何python圖像分析包。 OpenCV有一個moments function，我們應該能夠從這些中獲得偏斜，但是這些時刻只能達到三階，而我們需要四階才能得到峯度。

爲了使事情變得更容易和更快捷，我認爲在x和y方向上進行圖像投影並從這些投影中查找統計數字在數學上等同於使用完整圖像查找統計數據。在下面的代碼中，我使用這兩種方法，並顯示它們對於這個平滑示例是相同的。使用真實，嘈雜的圖像，我發現這兩種方法也提供了相同的結果，但只有在您手動將圖像數據轉換爲float64（它以浮點數32導入，「數值填充」導致結果略有不同時）。

下面是一個例子，你應該能夠剪切和粘貼「image_statistics（）」函數到你自己的代碼。我希望它的作品的人！:)輸出的

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() 

屏幕截圖，只是爲了好玩：

還有一件事：根據您對圖像的具體操作，您可能會考慮使用ImageJ進行圖像分析 - 但請注意！ moments plugin可以讓你計算偏度，峯度等。ImageJ在Analyze >> Set Measurements菜單中確實有「偏度」和「峯度」，但我認爲這實際上可以找到強度直方圖的偏度和峯度（我是騙了一分鐘）。