Image moment

In , computer vision and related fields, an image moment is a certain particular weighted average (moment) of the image pixels' intensities, or a function of such moments, usually chosen to have some attractive property or interpretation. Image moments are useful to describe objects after . Simple properties of the image which are found via image moments include area (or total intensity), its centroid, and information about its orientation. For a 2D continuous function f(x,y) the moment (sometimes called "raw moment") of order (p + q) is defined as for p,q = 0,1,2,... Adapting this to scalar (greyscale) image with pixel intensities I(x,y), raw image moments Mij are calculated by In some cases, this may be calculated by considering the image as a probability density function, i.e., by dividing the above by A uniqueness theorem (Hu [1962]) states that if f(x,y) is piecewise continuous and has nonzero values only in a finite part of the xy plane, moments of all orders exist, and the moment sequence (Mpq) is uniquely determined by f(x,y). Conversely, (Mpq) uniquely determines f(x,y). In practice, the image is summarized with functions of a few lower order moments. Simple image properties derived via raw moments include: Area (for binary images) or sum of grey level (for greytone images): Centroid: Central moments are defined as where and are the components of the centroid. If ƒ(x, y) is a digital image, then the previous equation becomes The central moments of order up to 3 are: It can be shown that: Central moments are translational invariant. Information about image orientation can be derived by first using the second order central moments to construct a covariance matrix. The covariance matrix of the image is now The eigenvectors of this matrix correspond to the major and minor axes of the image intensity, so the orientation can thus be extracted from the angle of the eigenvector associated with the largest eigenvalue towards the axis closest to this eigenvector.