Edge and contours are important for human visual system that a few lines are often sufficient to describe an object or a scene.
What is an Edge
Edges and contours perform a dominant role in human vision. They were described as image positions where the local intensity changes along a particular orientation. The stronger the local intensity change, the higher is the clue for an edge at that position.
Original image (a), Edge image (b).
Gradient-Based Edge Detection
Sample image and first derivative in one dimension for assuming that the image contains a single bright region at the center surrounded by a dark background image (a).
The horizontal intensity profile f(x) along the center image line would look like the one-dimensional function (b), and first derivative f'(x) (c).
Derivative Filters
Partial derivatives of a two-dimensional function: synthetic image function I (a).
The approximate first derivatives in the horizontal direction ∂I/∂u can be easily implemented by a linear filter (b). It reacts most strongly to rapid changes along the horizontal direction.
and the vertical direction ∂I/∂v (c). It also reacts most strongly to horizontal edges.
In (b) and (c), the lowest (negative) values are shown black, the maximum (positive) values are white, and zero values are gray.
The magnitude of the resulting gradient |∇I| (d).
Edge Operators
The strength of edge points and the local direction of the edge are an information that are contained in the gradient function and can be easily computed from the directional
components.
Prewitt and Sobel Operators
The operators are a gradient based method based on the first order derivatives.
Prewitt operator uses the filters…
Sobel operator uses the filters…
Edge strength and orientation
First, we denote the scaled filter results D(u,v). Then we defined the local edge strength E(u, v) as the gradient magnitude and the local edge orientation angle Φ(u, v) for each image position (u, v).
For example:
Edge strength and orientation obtained with a Sobel operator. Original images (a), the edge strength E(u, v) (b), and the local edge orientation Φ(u, v) (c).
Roberts Operator
It performs two small filters of size 2 × 2 for estimating the directional gradient along the image diagonals.
Compass Operators
A classic example is the extended Sobel operator, which employs the following eight filters with orientations spaced at 45◦
Other Edge Operators
Edge Detection Based on Second Derivatives
The idea is that edge points are located where the second derivative crosses through zero and the first derivative has a high magnitude.
Principle of edge detection with the second derivative. An original function (a), first derivative (b), and second derivative (c).
For example: Laplacian-of-Gaussian (LoG) operator which combines gaussian smoothing and computing the second derivatives into a single linear filter.
Edges at Different Scales
This typically amounts to detecting edges at various scale levels first and then deciding which edge at which scale level is dominant at each image position.
Canny Operator
Canny “filter” is a gradient method that uses the zero crossings of second derivatives for
precise edge localization.
The method tries to reach three main goals:
- minimize the number of false edge points
- achieve good localization of edges
- deliver only a single mark on each edge
Resulting edge maps for different settings of the smoothing (scale) parameter σ.
These figures below illustrate the different comparison of each edge operator technique.
Edge Sharpening
Edge Sharpening with the Laplace Filter
Edge sharpening with the second derivative. The sharpened intensity function ˇ f(x) = f(x)−w · f”(x) are shown.
Laplace operator perform sharpening of a two-dimensional function can be accomplished with the second derivatives in the horizontal and vertical directions.
Results of Laplace filter: Original image (a), second partial derivative ∂2I/∂2u in the horizontal direction (b), second partial derivative ∂2I/∂2v in the vertical direction (c), and Laplace filter ∇2I(u, v) (d).
Intensities in (b–d) are scaled such that maximally negative and positive values are shown as black and white, respectively, and zero values are gray.
Edge sharpening with the Laplace filter: original image with a horizontal profile taken from the marked line (a, b), result of Laplace filter HL (c, d), and sharpened image (e, f).
Unsharp Masking (USM)
USM filters with varying smoothing radii σ.
Original image (a) and the intensity profile along the marked image line (b); results of USM filtering with Gaussian smoothing radius σ = 2.5 (c, d) and σ = 10.0 (e, f)
Reference
Principles of digital image processing, Fundamental techniques, Wilhelm Burger Mark J. Burg
Image Processing 
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