Edge Detection

1 Feature detection

• Containing vast information

SO it’s important to determine

1. WHERE

   • concentrate on a part and ignore others
   • e.g. Object recognition: Ignore background

2. WHAT

   • Feature can be located
     • edge
     • feature points

2 Edge detection

2.1 Feature

1. Brightness (value) changes rapidly
2. Differentiation (近傍ピクセルとの微分処理 )
3. Important feature for object recognition
4. Weak to noise(Because it is differentiation)

2.2 Kinds

2.3 Differentiation

• Grandient

  $$\nabla I=(\frac{\partial I}{\partial x}, \frac{\partial I}{\partial y})$$
  • Represents the direction and the speed of the change in brightness

• Laplacian

  $$\nabla^2 I=(\frac{\partial^2 I}{\partial x^2}, \frac{\partial^2 I}{\partial y^2})$$

边缘就是明暗剧烈变化的地方，所以我们可以通过一次微分的极值或二次微分的变曲点确定edge

3 edge operator

为了计算微分，我们通过edge operator——即一种filter来实现

3.1 一次微分

对于一个形同$[[I_{i,j+1}, I_{i+1,j+1}],[I_{i,j}, I_{i+1,j}]]$的2*2 window内的像素，我们可以将微分转换为

$$\frac{\partial I}{\partial x} \approx \frac{1}{2\varepsilon}((I_{i+1,j+1}-I_{i,j+1})+(I_{i+1,j}-I_{i,j}))$$ $$\frac{\partial I}{\partial y} \approx \frac{1}{2\varepsilon}((I_{i,j+1}-I_{i,j})+(I_{i+1,j+1}-I_{i+1,j}))$$

表现为filter即为

分别求出两个方向的edge后合并才能生成总的edge图

• Location more precise
• Weak to noise
• Low detection power
  $$对噪声的优化很弱的后果...处处都是边缘！$$

3.2 二次微分

$$\frac{\partial^2 I}{\partial x^2} \approx \frac{1}{\varepsilon^2}(I_{i-1,j}-2I_{i,j}+I_{i+1,j})$$ $$\frac{\partial^2 I}{\partial y^2} \approx \frac{1}{\varepsilon^2}(I_{i,j-1}-2I_{i,j}+I_{i,j+1})$$

3.3 算法

• Roberts
• Prewitt
  • 先ほどのカーネルではノイズの影響が非常に強く出てしまうので、平滑化処理を加えた形

• Sobel

  • Location imprecise
  • Robust to noise
  • High detection power

• LoG
  

  • 用于平滑化
  • 近似于DOG（倒不如说有时候用DOG会更方便所以可以近似）
• Canny
  • 通过将一系列的检出算法合并的强力检出工具
    1. Blur image I with 2D Gaussian
    2. Find the edge‐normal direction at each pixel:
    3. Calculate the strength of the edge
    4. Find the maximal strength in the edge‐normal direction as the zero‐crossing in that direction (this step is called non‐maximum suppression)
  • 改变参数$\sigma$可以提取各种各样不同的特征

4. Corner detection

除了通过微分求出练成线的轮廓线，我们还可以通过求物体“角点”确认物体的位置

4.1 Susan

1. window作为一个圆形对图片遍历，圈内与圆心相同亮度的部分称为USAN
2. 若USAN的面积小于某个阈值即认为这个点是角点（就是在圆内占比极低）

   想象一个比例为1:3的饼状统计图…那1/4是不是就是个正方形的角？

3. USAN面积排行
   1. 圆内全部为相同亮度（即在一个物体内部时）面积最大
   2. 圆内只有一半左右为相同亮度（即边缘）面积次大
   3. 圆内只有小部分为相同亮度时（即角）面积最小
      

4.2 harris

Detect a point where the sum of square changes
of the image is largest when shifted slightly

• Points easily distinguishable from nearby points