A divergence operator to segment urban texture

Abstract

A divergence operator to measure urban texture from a multi-spectral image is proposed. A multi-spectral image is mapped to an n-dimensional vector field, n being the number of bands of the image. The obtainment of the vector field is carried out by considering a pixel of the image as an n-dimensional vector in a euclidian space. The ensemble of these vectors forms the vector field associated to the multi-spectral image. The flux variations of the vector field are related to changes in texture in the image. Null or small variations are related to a smooth texture, while medium and coarse textures are related to a certain flux change, i.e., the coarser the texture the greater the flux variation. A divergence operator measures. the flux variation and hence, texture. In order to save computer memory, speed up the divergence operator calculation, and lessen the content of noise of the image, the principal components decomposition is applied to the bands of the image. The best three principal components of a Landsat TM image are used to construct the vector field. The IR band is not taken into account since its pixel size is different. The partial derivatives involved in the divergence operator are written as weighted finite differences. To estimate such derivatives, cubes of three, five and seven voxels per side are considered. The cube is systematically displaced to cover the entire domain of the vector field. In each position of the cube, the divergence value is calculated using the weighted finite difference approximation. This value is written as a pixel in an output image file according to the cartesian coordinates defined by the location of the cube. This image file depicts the texture variations of the multi-spectral image. The relation: flux variation /spl rArr/ coarseness of texture is discussed. One example, of the Mexico City urban texture, is presented and analyzed in full detail.