Morphological operations—convex hull

Image morphology is an important concept in the field of image processing, and convex hull (Convex Hull) is one of the commonly used operations. It can be explained mathematically as a closed region that encloses the smallest convex polygon of a given set of points.

The principle of convex hull:

The convex hull is a convex polygon that surrounds a given set of points, ensuring that all vertices of this convex polygon are in the given set of points. By finding this convex polygon, we are able to describe the overall appearance of the original shape. Mathematically, for a given set of points $P$ , the convex hull refers to a convex hull containing $The smallest convex polygon of all points in P$ . The formation of convex hull can use convex hull algorithm, such as Graham scanning method, Jarvis stepping method or QuickHull, etc.

Function and applicable scenarios:

Object recognition and analysis: In computer vision, convex hulls are often used for object recognition and analysis, and are especially useful for boundary detection and shape analysis.
Image segmentation: is used for boundary extraction and object detection in image segmentation.
Pattern Recognition: is used for shape recognition and contour analysis.

The corresponding mathematical formula (based on point set $\{P_1, P_2, \ldots, P_n\}$ Example):

provided $n$ A set of points $\{P_1, P_2 , \ldots, P_n\}$ ，凸壳 $C H$ can be represented by the following mathematics:

$\text{ConvexHull}(P)$

Code example (using OpenCV library in Python):

import cv2
import numpy as np

# 创建一个示例图像，假设有一些点集
points = np.array([[50, 50], [150, 50], [100, 150], [50, 150], [200, 200], [250, 150]])

# 创建一个空白图像
img = np.zeros((300, 300, 3), dtype=np.uint8)

# 在图像上绘制点
for point in points:
    cv2.circle(img, tuple(point), 5, (255, 255, 255), -1)

# 计算凸壳
hull = cv2.convexHull(points)

# 绘制凸壳
cv2.drawContours(img, [hull], 0, (0, 255, 0), 2)

# 显示结果图像
cv2.imshow('Convex Hull', img)
cv2.waitKey(0)
cv2.destroyAllWindows()