Image structure tensor

Although the image structure tensor has recently been introduced in the medical imaging field, where clear structures exist and need to be identified, there is limited amoimt of work done in the field of seismic images. This may be explained by the fact that structures present in the Earth s subsurface are not of the same nature one can find in other images depicting specific objects that need to be identified and segmented. 

The image structure tensor has often been used especially for corner detection [8,16, 29]. The idea is based on the fact that the change of image intensity... 

The eigenvalues A of the image structure tensor can be used to detect lines, corners or constant grey value regions. The characteristic equation of matrix M is... 

Harris Corner Detector. In this subsection, we briefly describe the detection of corners as introduced by Harris and Stephens [16]. They used the image structure tensor M with the window w x, y) being a Gaussian. Instead of estimating the eigenvalues Ai and A2 of M separately, they introduced a cornerness measure or corner response R which is a function of the eigenvalues but it can be expressed directly in terms of the matrix elements. Thus, the corner response is defined as... 

Structure tensors have been shown to work well in segmenting and locating structures of specific shape. Several books published in the recent years present extensive literature reviews on structure tensors and their applications [13, 15, 19, 48]. In this chapter, we present in Section 2 a brief review of their application in image processing, focusing on their application to the analysis of seismic data. Section 3, describes the 2D first and second order structure tensors and their properties in more detail. Also, two well known corner detectors, based on the structure tensors, namely the Harris and the... 

Image sequences have been regarded as functions of space and time, thus they define hyper-surfaces. The curvature of the hyper-surfaces gives important information, so the structure tensor has been used in relation to motion parameter estimation [3, 4, 5] and optical flow estimation [20, 40]. 

Finally, Medioni and colleagues [25, 42, 43, 44] used the tensor voting technique in different applications of image analysis. Tensor voting is an approach of extracting salient structures by encoding data and corresponding uncertainties in the Hessian matrix. 

This new image tensor A is the Hessian matrix of the image function, and is defined in terms of the second partial derivatives of the image. This tensor measures the deviation from a straight line and it is also expected to have a zero determinant in a region with line-like structure. 

The structure tensor consisting of the first image derivatives was used for corner detection. It was shown to be useful in isolating chaotic regions. The chaotic area was also successfully detected by the use of a simple average of the local orientation of the image. 

Bums J, Job D, Bastin ME, Whalley H, Macgillivray T, et al. 2003. Structural disconnectivity in schizophrenia A diffusion tensor magnetic resonance imaging study. Br. J. Psychiatry 182 439-443. 

Diffusion tensor imaging (DTI) is a MRI method that allows examination of white matter integrity by measuring the directional diffusion of water. When there are no structural boundaries, water diffuses in all directions equally. When there are structural barriers, including myelin and cell membranes, diffusion tends... 

If the S, are groups, rings, etc., and the are homomorphisms, the limit acquires the same structure. (Elements in different St or Sj are added or multiplied by passing to some Sk containing images of them both.) The direct limit commutes in all ways with tensor product that is, if M, and N, are modules over S, and all maps are compatible,... 

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