## Revision f680a6e5868e2543549eccb094f0daafd46f06ad (click the page title to view the current version)

# Edge Lecture

## Changes from f680a6e5868e2543549eccb094f0daafd46f06ad to d694765af9eb41df185817f2e18dad4938eb88f3

--- title: Lecture: Edge Detection categories: lecture --- # Differentiation - The Canny edge detector + $\nabla I = [ I_x, I_y ]$ is the gradient vector. + It has length and direction in each pixel in the image. + Length $||\nabla I(x,y)||^2= \nabla I^T\nabla I$ + Select points which satisfy two criteria + Local optimum *along the direction of the gradient* + Larger than a chosen threshold $\tau$. + Sometimes we use a soft and a hard threshold, where points between the two thresholds are selected if they are adjacent to other selected points. + We can calculate $\nabla I$ with either Sobel or the derivative of a Gaussian. # Connected Components 1. Start with a singleton set (single pixel). 2. Use a mask, typically a $3\times3$ patch and centre it at each pixel already selected. 3. Pixels in the mask *and* the original image are added to the set. 4. Iterate until no pixels are added. # Line Fitting 1. Consider each connected component by itself; each one is a set of pixels with $(x,y)$ co-ordinates. 2. Calculate the centre, that is the mean $(\bar x,\bar y)$. 2. Calculate pixel positions relative tp the, i.e. $(\tilde x_i,\tilde y_i)$ where $\tilde x_i=x_i-\bar x$ and $\tilde y_i=y_i-\bar y$. 3. Consider the matrix $$D= \begin{bmatrix} \sum_i \tilde x_i^2 & \sum_i \tilde x_i\tilde y_i \\ \sum_i \tilde x_i\tilde y_i & \sum_i \tilde y_i^2 & \end{bmatrix} $$ 4. Suppose as an example that the $\tilde y_i$ are (approximately) zero and that there are many large $\tilde x_i$. + the matrix has one zero eigenvector and one eigenvector in the direction of $x$. + the points form an edge in the $x$ direction 5. If the edge is rotated both $\tilde x_i$ and $\tilde y_i$ are non-zero, but the eigenvectors behave the same. + One zero eigenvector + One eigenvector in the direction of the edge. 6. If the line is not straight, this is not perfect and the smaller eigenvalue is also non-zero. 7. The line is described by $y\sin\theta-x\cos\theta=\rho$, where $\rho$ is the distance between the line and the origin. **TODO** drawing See drawing in [Hough Tutorial](https://docs.opencv.org/3.4/d9/db0/tutorial_hough_lines.html) + Note the two points on the line at $$\big(0,\frac{r}{\sin\theta}\big) \quad\text{and}\quad \big(\frac{r}{\cos\theta}\big)$$ + Writing $y=\alpha x + \beta$, we obviously have $$\beta=\frac{r}{\sin\theta}$$ + We can solve for $\alpha$ to get $$\beta=-\frac{r}{\cos\theta}$$ # Hough