edge_detection.md

Edge Detection

Rapid changes in image intensity in small window caused by:

• Position
• Magnitude
• Orientation

Image Coordinate System

The x-coordinate is defined here as increasing in the "right"-direction, and the y-coordinate is defined as increasing in the "down"-direction.

Edge Detection Using First Derivative, Gradient Operator

First derivative edge detection provides both location and strength

The simplest approach is to use central differences:

while the gradient orientation can be estimated as:

• good localization

• noise sensitive

Roberts cross

Gradient of an image through discrete differentiation which is achieved by computing the sum of the squares of the differences between diagonally adjacent pixels.

we convolve the original image, with the following two kernels:

Let

be a point in the original image and

be a point in an image formed by convolving with the first kernel and

be a point in an image formed by convolving with the second kernel. The gradient can then be defined as:

direction of the gradient:

Prewitt

where here denotes the 2-dimensional convolution operation.

Prewitt it is a separable filter (can be decomposed as the products of an averaging and a differentiation kernel), therefore can be written as:

Sobel

Sobel is a separable filter (can be decomposed as the products of an averaging and a differentiation kernel), therefore

Refs: 1

Edge Detection Using Second Gradient

differential approach of detecting zero-crossings

Second-order derivatives can be computed from the scale space representation according to:

corresponding to the following filter masks:

• Edges are zero crossing in Laplacian image
• Laplacian doesn't provide direction of edges

Difference of Gaussians (DoG)

Convolution Properties: Associativity

First derivative is linear operation, Gaussian smoothing is also linear operation so we can find the first derivative of Gaussian

Laplacian of Gaussian (LoG)

Canny Edge

Gabor Filter

Separable Filter

A 2-dimensional convolution operation is separated into two 1-dimensional filters.

This reduces the computational costs on an image with a

filter from down to

Examples:

1. Smoothing filter:

2. Weighted smoothing filter:

3. Sobel operator:

# Convolution

Filtering often involves replacing the value of a pixel in the input image F with the weighted sum of its neighbors

convolution of 5×5 sized image matrix x with the kernel h of size 3×3,

kernel flipping and mirroring:

sliding the kernel over the image:

Non Linear Filtering

Bilateral Filter

Median Filter

1. The function does actually compute correlation, not the convolution
2. That is, the kernel is not mirrored around the anchor point. If you need a real convolution, flip the kernel using flip and set the new anchor to (kernel.cols - anchor.x - 1, kernel.rows - anchor.y - 1).
3. The function uses the DFT-based algorithm in case of sufficiently large kernels (~11 x 11 or larger) and the direct algorithm for small kernels.

void filter2D(Mat src,
              Mat dst,
              int ddepth,
              Mat kernel,
              Point anchor,
              double delta,
              int borderType);

Refs: 1