# Edge Detection 邊緣檢測

## Aims 宗旨​

• Intensity Images 強度圖像
• Edge Detection 邊緣檢測
• Convolution 卷積

## Intensity Images 強度圖像​

• An intensity image is a data matrix, whose values represent intensities within some range. 強度圖像是一個數據矩陣，其值表示某個範圍內的強度。
• represented as a single matrix, with each element of the matrix corresponding to one image pixel 強度圖像由一個矩陣表示，矩陣中的每個元素對應一個圖像像素。
• In matlab: To display an intensity image, use the imagesc ("image scale") function 用 imagesc 函數顯示強度圖像。

## Indexed Images 索引圖像​

• An indexed image consists of a data matrix, X, and a colormap matrix, map. 索引圖像由數據矩陣 X 和 colormap 矩陣 map 組成。
• map is an m-by-3 array of class double containing floating-point values in the range [0, 1]. map 是 double 類的 m×3 數組，包含 [0, 1] 範圍內的浮點值
• Each row of map specifies the red, green, and blue components of a single color. map 的每一行指定了一個單獨顏色的紅、綠、藍分量。

## Guess the image 猜圖​

• The image is a function mapping coordinates to intensity $f(x,y)$ 圖像是將坐標映射到強度 $f(x,y)$ 的函數

• The gradient of the intensity is a vector 強度的梯度是一個向量

$\vec{G}[f(x, y)]=\left[\begin{array}{c} G_x \\ G_y \end{array}\right]=\left[\begin{array}{c} \frac{d f}{d x} \\ \frac{d f}{d y} \end{array}\right]$

• We can think of the gradient as having an x and a y component 梯度有 x 和 y 兩個分量

• Our image is discrete with pixels indexed by i and j 我們的圖像是由 i 和 j 索引的像素

• We want to estimated in the same place 我們想在同一個地方估計

\begin{aligned} G_x & \cong f[i, j+1]-f[i, j] \\ G_y & \cong f[i, j]-f[i+1, j] \end{aligned}

• For each mask of weights you multiply the corresponding pixel by the weight and sum over all pixels 對於每個權重掩碼，您將相應的像素乘以權重並對所有像素求和

• Roberts 羅伯茨

• Sobel 索貝爾

• Sobel 索貝爾

## Convolution 卷積​

• Convolution is the computation of weighted sums of image pixels. 卷積是圖像像素的加權和計算。
• For each pixel [i,j] in the image, the value h[i,j] is calculated by translating the mask to pixel [i,j] and taking the weighted sum of pixels in neighbourhood of [i,j] 針對圖像中的每個像素 [i,j]，通過將掩碼平移到像素 [i,j] 並對 [i,j] 的鄰域像素進行加權求和，計算出 h[i,j] 的值。

## What do these filters do 這些過濾器有什麼作用​

• Steps:
• Take image 拍照
• Convolve mask with image for each direction 卷積遮罩與圖像
• Calculate derivatives Gx and Gy 計算導數 Gx 和 Gy
• Calculate magnitude = $M(\vec{G})= \sqrt{G_x^2 +G_y^2}$ 計算震級 = $M(\vec{G})= \sqrt{G_x^2 +G_y^2}$

## Filtering 過濾​

• We could detect edges by calculating the intensity change (gradient) across the image 我們可以通過計算圖像中的強度變化（梯度）來檢測邊緣
• We could implement this using the idea of filtering 我們可以使用過濾的概念來實現這一點

## Highly Directed Work 高度導向的工作​

• Gaussian (Canny) edge detection 高斯（Canny）邊緣檢測
• Second order operators 二階算子
• Thresholding 閾值