Edges

An edge in an image is a significant local change or discontinuity in the image intensity.

Edges

Edges come from discontinuity in:

Edges

Edges

Edges

Derivatives

The derivative is the rate of change of a function.

• 1D first order derivative: difference in consecutive pixels: \[\frac{\delta f}{\delta x} \approx f(x + 1) - f(x)\]

Derivatives

The derivative is the rate of change of a function.

• 1D second order derivative: acceleration of pixel intensity change: \[\frac{\delta^{2}f}{\delta {x}^2} \approx f(x + 1) + f(x - 1) - 2f(x)\]

Derivatives

Required properties of first derivatives:

Derivatives

Required properties of second derivatives:

Derivatives

Derivatives

Image Derivatives

For images, we must consider the derivative in both directions:

\[\frac{\delta f}{\delta x} \approx f(x + 1, y) - f(x, y)\]

\[\frac{\delta f}{\delta y} \approx f(x, y + 1) - f(x, y)\]

Image Derivatives

Image Derivatives

An image gradient is formed of two components:

\[\nabla f = \left[ \frac{\delta f}{\delta x}, \frac{\delta f}{\delta y} \right]\]

Image Derivatives

Image gradient is a vector:

\[\nabla f = \left[ \frac{\delta f}{\delta x}, \frac{\delta f}{\delta y} \right]\]

Image Derivatives

A vector has magnitude…

\[|\nabla f| = \sqrt{\left( \frac{\delta f}{\delta x} \right)^{2} + \left( \frac{\delta f}{\delta y} \right)^{2}}\]

Magnitude is the strength of the edge.

Image Derivatives

A vector has direction…

\[ \theta = \tan^{-1} \left( \frac{\delta f}{\delta y} / \frac{\delta f}{\delta x} \right) \]

Direction of an edge is perpendicular to the gradient direction.

Image Derivatives

• The gradient points in the direction of most rapid change in intensity.
• Perpendicular to the edge direction.

Image Derivatives

Image Derivatives

First order derivatives:

Second Order Derivatives

Second order derivatives:

Second Order Derivatives

\[\nabla^{2} f = \frac{\delta^{2} f}{\delta x^{2}} + \frac{\delta^{2} f}{\delta y^{2}}\]

Derivative in this form is known as the Laplacian.

Laplacian

So, the Laplacian is calculated as:

\[\nabla^{2} f = f(x + 1) + f(x - 1) + f(y + 1) + f(y - 1) - 4f(x, y)\]

Laplacian

\[ \begin{bmatrix} 0 & 1 & 0 \\ 1 & -4 & 1 \\ 0 & 1 & 0 \end{bmatrix} \]

The Laplacian can also be calculated by convolving the image with this filter.

Laplacian

Laplacian

Line Detection

What if we only wanted to detect lines that point in a certain direction?

\[ \begin{bmatrix} -1 & 2 & -1 \\ -1 & 2 & -1 \\ -1 & 2 & -1 \end{bmatrix} \]

Line Detection

Line Detection

What about detecting edges in other directions?

Line Detection

What about detecting edges in other directions?

Line Detection

Previous filter gives strong response along a line.

Line Detection

Look either side of candidate pixel…

Line Detection

Two popular first-order operators are Prewitt and Sobel.

Both provide approximations of derivatives.

Line Detection

Line Detection

Line Detection

Line Detection

For each pixel, find the maximum value from all of the filter responses, and then threshold.

Edge Detection

Four limitations with basic gradient-based edge detection:

Canny Edge Detector

Requirements for a good edge detector:

Canny Edge Detector

Canny provides an elegant solution to edge detection.

Canny Edge Detector

Canny Edge Detection is a four step process:

Canny Edge Detector

Step 1: Convolve image with Gaussians of particular scales.

• Smoothing helps ensure robustness to noise.
• The size of the Gaussian kernel affects the performance of the detector.

Canny Edge Detector

Step 2: Compute gradient magnitude and direction:

• Using Sobel operators.

Quantise the angle of the gradient:

• Discrete nature of image limits the possible angle.
• Angle can only be {0, 45, 90, 135} degrees.

Canny Edge Detector

Step 3: Perform non-maximal suppression.

• An edge-thinning technique.
• Searches for maximum value along direction of gradient and sets all others to zero.
• Result is a one pixel wide curve.

Canny Edge Detector

Step 4: Threshold edges with hysteresis.

• Hysteresis is the dependence of the state of a system on its history.

Canny Edge Detector

Step 4: Threshold edges with hysteresis.

Use two thresholds: \(T_{min}\) and \(T_{max}\).

\[ \begin{aligned} E(x, y) = \begin{cases} 1~ & E(x, y) \geq T_{max} \\ 0~ & E(x, y) < T_{min} \end{cases} \end{aligned} \]

Canny Edge Detector

Step 4: Threshold edges with hysteresis.

Canny Edge Detector

Canny Edge Detector