An image gradient is a directional change in the intensity or color in an image. Image gradients may be used to extract information from images.

In graphics software

Graphics software

In computer graphics, graphics software or image editing software is a program or collection of programs that enable a person to manipulate visual images on a computer....

 for digital image editing, the term gradient is used for a gradual blend of color

Color

Color or colour is the visual perceptual property corresponding in humans to the categories called red, green, blue and others. Color derives from the spectrum of light interacting in the eye with the spectral sensitivities of the light receptors...

 which can be considered as an even gradation from low to high values, as used from white to black in the images to the right. Another name for this is color progression.

Mathematically, the gradient of a two-variable function (here the image intensity function) is at each image point a 2D vector with the components given by the derivative

Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

s in the horizontal and vertical directions. At each image point, the gradient vector points in the direction of largest possible intensity increase, and the length of the gradient vector corresponds to the rate of change in that direction.

Since the intensity function of a digital image is only known at discrete points, derivatives of this function cannot be defined unless we assume that there is an underlying continuous

Continuous function

In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

 intensity function which has been sampled at the image points. With some additional assumptions, the derivative of the continuous intensity function can be computed as a function on the sampled intensity function, i.e., the digital image. It turns out that the derivatives at any particular point are functions of the intensity values at virtually all image points. However, approximations of these derivative functions can be defined at lesser or larger degrees of accuracy.

The Sobel operator represents a rather inaccurate approximation of the image gradient, but is still of sufficient quality to be of practical use in many applications. More precisely, it uses intensity values only in a 3×3 region around each image point to approximate the corresponding image gradient, and it uses only integer values for the coefficients which weight the image intensities to produce the gradient approximation.

The gradient of the image is one of the fundamental building blocks in image processing

Image processing

In electrical engineering and computer science, image processing is any form of signal processing for which the input is an image, such as a photograph or video frame; the output of image processing may be either an image or, a set of characteristics or parameters related to the image...

. For example the Canny edge detector uses image gradient for edge detection

Edge detection

Edge detection is a fundamental tool in image processing and computer vision, particularly in the areas of feature detection and feature extraction, which aim at identifying points in a digital image at which the image brightness changes sharply or, more formally, has discontinuities...

.

Image gradients are often utilized in maps

MAPS

Maps is the plural of map, a visual representation of an area.As an acronym, MAPS may refer to:* Mail Abuse Prevention System, an organisation that provides anti-spam support...

 and other visual representations of data

Data

The term data refers to qualitative or quantitative attributes of a variable or set of variables. Data are typically the results of measurements and can be the basis of graphs, images, or observations of a set of variables. Data are often viewed as the lowest level of abstraction from which...

 in order to convey additional information. GIS

Geographic Information System

A geographic information system, geographical information science, or geospatial information studies is a system designed to capture, store, manipulate, analyze, manage, and present all types of geographically referenced data...

 tools use color progressions to indicate elevation

Elevation

The elevation of a geographic location is its height above a fixed reference point, most commonly a reference geoid, a mathematical model of the Earth's sea level as an equipotential gravitational surface ....

 and population density

Population density

Population density is a measurement of population per unit area or unit volume. It is frequently applied to living organisms, and particularly to humans...

, among others.

Computer vision

Image gradients can be used to extract information from images. Gradient images are created from the original image (generally by convolving with a filter, one of the simplest being the Sobel filter) for this purpose. Each pixel of a gradient image measures the change in intensity of that same point in the original image, in a given direction. To get the full range of direction, gradient images in the x and y directions are computed.

One of the most common uses is in edge detection. After gradient images have been computed, pixels with large gradient values become possible edge pixels. The pixels with the largest gradient values in the direction of the gradient become edge pixels, and edges may be traced in the direction perpendicular to the gradient direction. One example of an edge detection algorithm that uses gradients is the Canny edge detector.

Image gradients can also be used robust feature and texture matching. Different lighting or camera properties can cause two images of the same scene to have drastically different pixel values. This can cause matching algorithms to fail to match very similar or identical features. One way to solve this is to compute texture or feature signatures based on gradient images computed from the original images. These gradients are less susceptible to lighting and camera changes, so matching errors are reduced.

Math

The gradient of an image is given by the formula :
,

where :
is the gradient in the x direction
is the gradient in the y direction.

The gradient direction can be calculated by the formula :
.

An example of a gradient patch to use is

-1

1

However this will shift the image by half a pixel. The Sobel filter

-1

0

1

provides a good median

External links

• Noise robust gradient operators

• Mathematica GradientFilter function

n.