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Matrix Newton Interpolation and Progressive 3D Imaging: PC-Based Computation

E. Defez
Organization: Universidad Politécnica de Valencia, Valencia, Spain
A. Law
Organization: University of Waterloo, Waterloo, Ontario, Canada
J. Villanueva-Oller
Organization: Universidad Politécnica de Valencia, Valencia, Spain
R. Villanueva
Organization: Universidad Politecnica de Valencia
Department: Instituto de Matematica Multidisciplinar
Journal / Anthology

Mathematical and Computer Modelling
Year: 2002
Volume: 35
Page range: 303-322

For polynomials P(x) = A_n x^n + A_n-1 x^n-1 + ... + A_1 x + A_0 in a real scalar x, but with coefficients A_j that are rectangular matrices, a generalization of Newton's divided difference interpolatory scheme is developed. Instances of P(x) at nodes x_i may be interpreted as slices of a digital 3D object. Mathematica code for this machinery is given and its effectiveness illustrated for progressively-transmitted renderings. Analysis, with supporting Mathematica code, is extended to a piecewise matrix polynomial situation, to produce practicable software for a PC-based computational system. Two experiments about 3D progressive imaging, employing a 6 Mbyte data base consisting of 93 CT slices of a human head, are discussed along with PC-based performance evaluation. How a 3D object is decomposed into 2D subsets in preparation for progressive transmission, as well as their selected ordering for transmission, are seen to affect quality of the emerging reconstructions. Extension to 4D objects is also discussed briefly, to provide introduction to, for example, application of matrix polynomial machinery within the field of functional magnetic resonance imaging.

*Applied Mathematics > Numerical Methods
*Applied Mathematics > Visualization

progressive transmission of images, matrix Newton interpolation, matrix polynomial reconstruction, PC-based progressive rendering