Image of matrix

Bremer, C., Bredow, S., Mahmood, U., Weissleder, R. and Tung, C. H. (2001). Optical imaging of matrix metalloproteinase-2 activity in tumors feasibility study in a mouse model. Radiology 221, 523-529. 

M. Schafers, B. Riemann, K. Kopka, H.J. Breyholz, S. Wagner, K.P. Schafers, M.P. Law, O. Schober, B. Levkau, Scintigraphic imaging of matrix metalloproteinase activity in the arterial wall in vivo. Circulation 109 (2004) 2554-2559. 

To quantitatively control the amount and the quality of the matrix deposited, a clear glass side was coated under each sublimating condition in parallel. UV absorbance of the matrix layer on the side and the microscopic images of matrix morphology were collected to monitor the thickness and uniformity of matrix deposition. Empirically it has been found that a matrix layer with UV absorbance between 1.0 and 2.0 was, in general, suitable for ME-SALDI. 

Manual thresholding by detecting peaks of matrix diagonal (4) to segment X-Rays images. 

The parameters of this matrix are the image / and the vector d written by [dx, dy] in cartesian coordinates or [ r, 0] in polar coordinates. The number of co-occurrence on the image / of pairs of pixels separated by vector d. The latter pairs have i and j intensities respectively, i.e. 

Finally, each coefficient were standardized by the division of the sum of all coefficients(2). This definition allows also to regard as the co-occurrence matrix as a function of probability distribution, it can be represented by an image of KxK dimensions. 

Approach to restoring of stresses SD in the three-dimensional event requires for each pixel determinations of matrix with six independent elements. Type of matrixes depends on chosen coordinate systems. It is arised a question, how to present such result for operator that he shall be able to value stresses and their SD. One of the possible ways is a calculation and a presenting in the form of image of SD of stresses tensor invariants. For three-dimensional SDS relative increase of time of spreading of US waves, polarized in directions of main axises of stresses tensor ... 

One more application area is composite materials where one wants to investigate the 3D structure and/or reaction to external influences. Fig.3a shows a shadow image of a block of composite material. It consists of an epoxy matrix with glass fibers. The reconstructed cross-sections, shown in Fig.3b, clearly show the fiber displacement inside the matrix. The sample can be loaded in situ to investigate the reaction of matrix and fibers to external strain. Also absorption and transmission by liquids can be visualized directly in three-dimensions. This method has been applied to the study of oil absorption in plastic granules and water collection inside artificial plant grounds. 

Figure 2 Bright-field TEM image of polyether sulphone inclusions (dark objects see arrows) in a polystyrene matrix.

Figure 4.3 shows the fluorescence SNOM images of single PMMA-Pe chains embedded in the unlabeled PMMA matrix. In the SNOM image, each PMMA-Pe chain was observed as an isolated fluorescent spot. The molecular weight of each chain can be estimated from the integrated fluorescence intensity [19], and Figure 4.3... 

The matrix X defines a pattern P" of n points, e.g. x, in which are projected perpendicularly upon the axis v. The result, however, is a point s in the dual space S". This can be understood as follows. The matrix X is of dimension nxp and the vector V has dimensions p. The dimension of the product s is thus equal to n. This means that s can be represented as a point in S". The net result of the operation is that the axis v in 5 is imaged by the matrix X as a point s in the dual space 5". For every axis v in 5 we will obtain an image s formed by X in the dual space. In this context, we use the word image when we refer to an operation by which a point or axis is transported into another space. The word projection is reserved for operations which map points or axes in the same space [11]. The imaging of v in S into s in S" is represented geometrically in Fig. 29.9a. Note that the patterns of points P" and P are represented schematically by elliptic envelopes. 

From the above we conclude that the product of a matrix with a vector can be interpreted geometrically as an operation by which a pattern of points is projected upon an axis. This projection produces an image of the axis at a point in the dual space. The concept can be extended to the product of a matrix with another matrix. In this case we can conceive of the latter as a set of axes, each of which produces image points in the dual space. In the special case when this matrix has only two columns, the product can be regarded as a projection of a pattern of points upon the plane formed by the two axes. As a result one obtains two image points (one for each axis that defines the plane of projection) in the dual space. 

---