Image moment

Moments, see Moments of a distribution, are in image processing certain weighted averages of the brightness values ​​of each pixel of an image. They are usually chosen so that they reflect desired properties of the image or have certain geometric interpretations. Moments are helpful to describe the individual objects in a segmented image. Basic properties of images that can be calculated by moments are area (or sum of the brightness values ​​), focus and alignment.

• 2.1 Examples

Non -centered moments

For a two-dimensional continuous function, the moment -th degree is defined as

When applied to digital gray -scale images with the gray value function, the non-centered moments arise from

In some cases, the off-center moments can be calculated by the gray value function is understood as the probability density function. These shares are above formula by

According to the Eindeutigkeitstheorem by Athanasios Papoulis (1991 ) moments exist any grade, when is piecewise continuous and is only in a finite part of the xy plane equal to zero. In this case the succession of moments is determined by unique. Similarly, the function is uniquely determined. In practice, however, a few moments rich low grade from an image sufficiently to characterize precisely.

Examples

Simple image properties that can be determined by non-centered moments are, among others:

• Surface ( for binary images ) or sum of the gray values ​​( for grayscale images ):
• Focus:

Central moments ( translationally invariant moments )

Central moments are invariant with respect to translations, they are defined as

When applied to digital gray -scale images with the gray value function g ( x, y), the central moments Mij result from

The central moments up to degree 3 are:

It can be shown that:

Examples

Information about the orientation of the image can be obtained by using the first three moments of second order central, in order to calculate a covariance matrix.

The covariance matrix of the image is then

The eigenvectors of this matrix correspond to the major and minor semi-axis of the brightness values ​​. Thus, the orientation of the image can be determined from the angle of the eigenvector with the largest eigenvalue. It can be shown that this angle Θ can be calculated by the following formula.

The eigenvalues ​​of the covariance matrix are

The eccentricity of the image

Scale invariant moments

It can be constructed moments ηi j i j ≥ 2, which are invariant to scaling and translation by dividing the corresponding central moment by moment, the corresponding scaled from 0 degrees.

Rotationally invariant moments

It is also possible to construct moments are also invariant with respect to an image rotation. Is frequently used, the Hu invariant moments amount.

The first, I1, is approximately equivalent to the moment of inertia about the center of the image when the brightness values ​​of the pixels are interpreted as physical density.

Application Examples

Moments are good for two things. Firstly, they are used to classify objects in the binarized so black and white images, which are the result of preprocessing, which decides which parts of an image to an object include ( black = 1) and which are not ( white = 0). Also an image which contains other than black and white and gray values ​​, because the pre-processing algorithm was not always sure whether a pixel to the object or to the background part is usable by the gray levels are normalized to the range [0, 1].

The example of the text recognition can be seen that a " T" and "I" are indeed left - right symmetric and thus do not differ in emphasis, however, differ in the moment by the different variance and differ also in the torque considerably. For this moment should be due to the up-down symmetry for " I", a value close to 0 come out while a scanned T has significantly more pixels than the top down and here receives a strong negative value ( for increasing down the y- values).

Secondly, it can be compared with the arrangement of arbitrary moments of extracted features from images or similar to each other. If you have extracted for example by means of a corner Finders some corners, can be achieved using the moments determine which part of the image within an image sequence ( = video) takes place change. Using for this purpose the translation invariant central moments, the detection is stable with respect to camera shake.