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The goal of edge detection is to mark the points in a digital image at which the luminous intensity changes sharply. Sharp changes in image properties usually reflect important events and changes in properties of the world. These include (i) discontinuities in depth, (ii) discontinuities in surface orientation, (iii) changes in material properties and (iv) variations in scene illumination. Edge detection is a research field within image processing and computer vision, in particular within the area of feature extraction. A digital image is a representation of a two-dimensional image as a finite set of digital values, called picture elements or pixels. ...
Luminous intensity is a measure of the energy emitted by a light source in a particular direction. ...
UPIICSA IPN - Binary image Image processing is any form of information processing for which the input is an image, such as photographs or frames of video; the output is not necessarily an image, but can be for instance a set of features of the image. ...
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Feature extraction is an area of image processing which involves using algorithms to detect and isolate various desired portions of a digitized image or video stream. ...
Edge detection of an image reduces significantly the amount of data and filters out information that may be regarded as less relevant, preserving the important structural properties of an image. There are many methods for edge detection, but most of them can be grouped into two categories, search-based and zero-crossing based. The search-based methods detect edges by looking for maxima and minima in the first derivative of the image, usually local directional maxima of the gradient magnitude. The zero-crossing based methods search for zero crossings in the second derivative of the image in order to find edges, usually the zero-crossings of the Laplacian or the zero-crossings of a non-linear differential expression.
Edge properties
Edges may be viewpoint dependent - these are edges that may change as the viewpoint changes, and typically reflect the geometry of the scene, objects occluding one another and so on, or may be viewpoint independent - these generally reflect properties of the viewed objects such as surface markings and surface shape. In two dimensions, and higher, the concept of perspective projection has to be considered. The word projection can mean more than one thing. ...
A typical edge might be (for instance) the border between a block of red color and a block of yellow; in contrast a line can be a small number of pixels of a different color on an otherwise unchanging background. There will be one edge on each side of the line. Edges play quite an important role in many applications of image processing. During recent years, however, substantial (and successful) research has also been made on computer vision methods that do not explicitly rely on edge detection as a pre-processing step. Three lines — the red and blue lines have same slope, while the red and green ones have same y-intercept. ...
Simple edge model Edges in natural images are usually not ideal step edges. Instead they are normally affected by one or several of the following effects: Although the following model cannot be regarded as perfect, the error function  is commonly used for modelling the effects of edge blur in practical applications. Thus, a one-dimensional image f which has exactly one edge placed at 0 may be modelled as follows: An example of very shallow depth of field in a macro photograph. ...
It has been suggested that this article or section be merged into Umbra. ...
Shading refers to depicting depth in 3D models by varying levels of darkness. ...
Specularity is the quality used in many 3D Rendering programs to set the size and the brightness of a textures reflection to light. ...
Diagram of diffuse reflection Diffuse reflection is the reflection of light from an uneven or granular surface such that an incident ray is seemingly reflected at a number of angles. ...
Plot of the error function In mathematics, the error function (also called the Gauss error function) is a non-elementary function which occurs in probability, statistics and partial differential equations. ...
Then, left of the edge the intensity is  , and right of the edge it is  ; σ is called the blur scale of the edge. Please note that f can be written as a convolution f = gσ * u where gσ is the gaussian kernel with standard deviation σ, and u is a step function defined as follows: In mathematics and, in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. ...
A Gaussian function (named after Carl Friedrich Gauss) is a function of the form: for some real constants a > 0, b, and c. ...
In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. ...
A function on the reals is a step function if it can be written as a finite linear combination of semi-open intervals. ...
Detecting an edge Taking an edge to be a change in intensity taking place over a number of pixels, edge detection algorithms generally compute a derivative of this intensity change. To simplify matters, we can consider the detection of an edge in one dimension. In this instance, our data can be a single line of pixel intensities. For instance, we can intuitively say that there should be an edge between the 4th and 5th pixels in the following 1-dimensional data: To firmly state a specific threshold on how large the intensity change between two neighbouring pixels must be for us to say that there should be an edge between these pixels is, however, not always an easy problem. Indeed, this is one of the reasons why edge detection may be a non-trivial problem unless the objects in the scene are particularly simple and the illumination conditions can be well controlled.
Computing the 1st derivative Many edge-detection operators are based upon the 1st derivative of the intensity - this gives us the intensity gradient of the original data. Using this information we can search an image for peaks in the intensity gradient. This article assumes an understanding of algebra, analytic geometry, and the limit. ...
In graphics programs for digital image editing such as Photoshop (a bitmap graphics editor) and Adobe Illustrator (a vector graphics editor), the term gradient is used for a gradual blend of colour which can be considered as an even gradation from low to high values, as used from white to...
If I(x) represents the intensity of pixel x, and I′(x) represents the first derivative (intensity gradient) at pixel x, we therefore find that:
 For higher performance image processing, the 1st derivative can therefore be calculated (in 1D) by convolving the original data with a mask: In mathematics and, in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. ...
Computing the 2nd derivative Some other edge-detection operators are based upon the 2nd derivative of the intensity. This is essentially the rate of change in intensity gradient. In the ideal continuous case, detection of zero-crossings in the second derivative captures local maxima in the gradient. Peak detection in the second derivative, on the other hand, is a method for line detection, provided that the image operators are expressed at a proper scale. As noted above, a line is a double edge, hence we will see an intensity gradient on one side of the line, followed immediately by the opposite gradient on the opposite site. Therefore we can expect to see a very high change in intensity gradient where a line is present in the image. To find lines, we can alternatively search for zero-crossings in the second derivative of the image gradient. In mathematics, the derivative of a function is one of the two central concepts of calculus. ...
If I(x) represents the intensity at point x, and I"(x) is the second derivative at point x:
 Again most algorithms use a convolution mask to quickly process the image data: In mathematics and, in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. ...
Thresholding Once we have calculated our derivative, the next stage is to apply a threshold, to determine where the result suggest an edge to be present. The lower the threshold, the more lines will be detected, and the results become increasingly susceptible to noise, and also to picking out irrelevant features from the image. Conversely a high threshold may miss subtle lines, or segmented lines. Image noise is unwanted and manifested in the pixels of an image. ...
A commonly used compromise is thresholding with hysteresis. This method uses multiple thresholds to find edges. We begin by using the upper threshold to find the start of a line. Once we have a start point, we trace the edge's path through the image pixel by pixel, marking an edge whenever we are above the lower threshold. We stop marking our edge only when the value falls below our lower threshold. This approach makes the assumption that edges are likely to be in continuous lines, and allows us to follow a faint section of an edge we have previously seen, without meaning that every noisy pixel in the image is marked down as an edge. Hysteresis is a property of systems (usually physical systems) that do not instantly follow the forces applied to them, but react slowly, or do not return completely to their original state: that is, systems whose states depend on their immediate history. ...
Edge detection operators Currently, the Canny operator (or variations of this operator) is most commonly used edge detection method. A large number of edge detection operators have been published but so far none has shown significant advantages over the Canny-type operators in general situations. In his original work, Canny studied the problem of designing an optimal pre-smoothing filter for edge detection, and then showed that this filter could be well approximated by a first-order Gaussian derivative kernel. Canny also introduced the notion of non-maximum suppression, which means that edges are defined as points where the gradient magnitude assumes a maximum in the gradient direction. In Computer vision one of the earliest edge detection algorithms, the Roberts Cross operator works by computing the sum of the squares of the differences between diagonally adjacent pixels. ...
Prewitt is a method of edge detection in computer graphics which calculates the maximum response of a set of convolution kernels to find the local edge orientation for each pixel. ...
In computer vision, the Sobel operator is a simple edge detection algorithm using the 1st derivative of the intensity information. ...
The Canny edge detection operator was developed by John F. Canny in 1986 and uses a multiple stage algorithm to detect a wide range of edges. ...
In computer vision, the Marr-Hildreth algorithm is a method of detecting edges in digital images. ...
In alternating current, the zero crossing is the instantaneous point at which there is no voltage present. ...
The Canny edge detection operator was developed by John F. Canny in 1986 and uses a multiple stage algorithm to detect a wide range of edges. ...
In statistics and image processing, to smooth a data set is to create a function that attempts to capture important patterns in the data, while leaving out noise. ...
Gaussian curves parametrised by expected value and variance (see normal distribution) A Gaussian function (named after Carl Friedrich Gauss) is a function of the form: for some real constants a > 0, b, and c. ...
On a discrete grid, the non-maximum suppression stage can be implemented by estimating the gradient direction using first-order derivatives, then rounding off the gradient direction to multiples of 45 degrees, and finally comparing the values of the gradient magnitude in the estimated gradient direction. A more refined approach to obtain edges with sub-pixel accuracy is by using the following differential approach of detecting zero-crossings of the second-order directional derivative in the gradient direction (Lindeberg 1998)
 that satisfy a sign-condition on the third-order directional derivative in the same direction (for more details, please see the relations between edge detection and ridge detection in the article on ridge detection) In a 2-D function, a (bright) ridge is a connected set of points that are maximal in at least one dimension. ...
 where Lx, Ly ... Lyyy denote partial derivatives computed from a scale-space representation L obtained by smoothing the original image with a Gaussian kernel. In this way, the edges will be automatically obtained as continuous curves with subpixel accuracy. Hysteresis thresholding can also be applied to these differential and subpixel edge segments. Scale space theory is a framework for multi-scale signal representation developed by the computer vision and image processing communities. ...
References - Canny, J., A Computational Approach To Edge Detection, IEEE Trans. Pattern Analysis and Machine Intelligence, 8:679-714, 1986.
- Lindeberg, T., "Edge detection and ridge detection with automatic scale selection", International Journal of Computer Vision, 30, 2, pp 117--154, 1998.
- Ziou, D. and Tabbone, S.: Edge Detection Techniques An Overview, International Journal of Pattern Recognition and Image Analysis, 8(4):537--559, 1998 (Contains an extensive set of references.)
See also
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