# Download Algebre, solutions developpees des exercices, 1ere partie, by Mac Lane, Birkhoff (WEIL, HOCQUEMILLER) PDF

By Mac Lane, Birkhoff (WEIL, HOCQUEMILLER)

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The environment of medical imaging greatly reduces the number of options, however, owing partly to the properties of the projection matrix and partly to the demands set by clinical practice. The system matrix A of Eq. (5) has the following features: - large. The number of entries of A is not seldom of the order of 10 1 &. In most applications, M is larger than N (overdetermined system). - sparse. The number of non-zero entries of A is small, typically less than 1%. - singular. The rank of A will generally be less than the number of columns N.

194), we get n-1 L j=O F(j) ow+kj n nIl ( l/n onIl f(l)OW-lj) w+kj j=O 1=0 n I n I/n o n-l n-l L L j=O 1=0 f(l)ow(k-l)j n f(k) , (201) which is identical with Eq. (195). Theorem of periodicity: F(j+n) = F(j), f(k+n) = f(k). This property is easy to prove by substitution. It is useful for certain numerical computations, nevertheless it is important to remember that the so-called periodicity of f(k) and F(j) is purely formal; in fact f(k) and F(j) are sets of n terms, defined only for k and j integers from 0 to n-l.

To obtain this set we suppose that the supports of all relevant pictures lie inside a square, say Ixl ~ 1, Iyl ~ 1 if x and yare Cartesian coordinates. This square is divided into N(=n 2 ) identical subsquares referred to as picture elements or pixels (the 3D analogue is called voxels) , identified by an index j, j = 1(1)N. The pixel basis then consists of basis functions b j of the form inside pixel j, b,(X,y)={l J 0 j 1 (1)N (7) elsewhere, while the coefficients f, in Eq. (1) are defined as the average values of J ¢(x,y) in the corresponding pixel.

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