Matrix newton interpolation and progressive 3D imaging: PC-based computation

For polynomials P(x) = Anxn + An−1xn−1 + ⋯ + A1x + A0 in a real scalar x, but with coefficients Aj that are rectangular matrices, a generalization of Newtonʹs divided difference interpolatory scheme is developed. Instances of P(x) at nodes xi 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.