Otsu‘s Thresholding的工作原理「建议收藏」

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Otsu’s Thresholding的工作原理

在双峰图像（像素的直方图只包含两个峰值），一个好的阈值将在这两个值的中间。Otsu算法就是从图像的像素直方图中确定了一个最佳的全局阈值 $t$，可以最小化weighted within-class variance加权类内方差。

$${\sigma }_w^2(t)=q_1(t){\sigma }_1^2(t)+q_2(t){\sigma }_2^2(t)$$
权重：
$$q_1(t)=\sum _{i=1}^{t}P(i)\&q_2(t)=\sum _{i=t+1}^{I}P(i)$$
均值：
$${\mu }_1(t)=\sum _{i=1}^t\frac{iP(i)}{q_1(t)}\&{\mu }_2(t)=\sum _{i=t+1}^I\frac{iP(i)}{q_2(t)}$$
方差：
$${\sigma }_1^2(t)=\sum _{i=1}^t[i-{\mu }_1(t){]}^2\frac{P(i)}{q_1(t)}\&{\sigma }_2^2(t)=\sum _{i=t+1}^I[i-{\mu }_2(t){]}^2\frac{P(i)}{q_2(t)}$$

import cv2 as cv
import numpy as np
import seaborn as sns
import matplotlib.pyplot as plt

img = cv.imread('./images/kakaka.jpg')
img = cv.cvtColor(img, cv.COLOR_BGR2RGB)
img_gray = cv.cvtColor(img, cv.COLOR_RGB2GRAY)
#img_blur = cv.GaussianBlur(img_gray, (5, 5), 0)

Numpy Otsu’s thresholding

hist_cv = cv.calcHist([img_gray],[0],None,[256],[0,256]) # （256，1）
hist_np = np.bincount(img_gray.ravel(), minlength=256)   # （256，）
hist_norm = hist_np/hist_np.sum()
Q = hist_norm.cumsum()
bins = np.arange(256)
fn_min = np.inf
thresh = -1

for i in range(1, 256):
    p1, p2 = np.hsplit(hist_norm, [i])
    q1, q2 = Q[i], 1.0 - Q[i]    # Q[255] = 1.0
    if q1 < 1.e-6 or q2 < 1.e-6:
        continue
    b1, b2 = np.hsplit(bins, [i])
    # means and variances
    m1, m2 = np.sum(p1*b1)/q1, np.sum(p2*b2)/q2
    v1, v2 = np.sum(((b1 - m1)**2)*p1)/q1, np.sum(((b2 - m2)**2)*p2)/q2
    # weighted within-class variances
    fn = v1*q1 + v2*q2
    if fn < fn_min:
        fn_min = fn
        thresh = i
print("threshold t : %d"%thresh)

threshold t : 115

mask = np.zeros_like(img_gray)
mask[np.where(img_gray >= 115)] = 1
plt.imshow(img*mask.reshape(245, 250, 1))

OpenCV otsu’s thresholding

ret, otsu = cv.threshold(img_gray, 0, 255, cv.THRESH_BINARY+cv.THRESH_OTSU)
print("threshold t : %d"%ret)

threshold t : 114

ROI

img_roi = cv.bitwise_and(img, img ,mask = otsu)