(* kernel.sml * * COPYRIGHT (c) 2010 The Diderot Project (http://diderot.cs.uchicago.edu) * All rights reserved. * * QUESTION: should we *) structure Kernel : sig type coefficient = Rational.rat (* polynomial represented as list of coefficients, where ith element is * coefficient for x^i. *) type polynomial = coefficient list type kernel = { support : int, (* number of samples to left/right *) segs : polynomial list, (* piece-wise polynomial that defines the kernel. Since *) (* a kernel is symmetric, we only store the positive *) (* segments, with segs[i] covering the domain [i,i+1) *) } val differentiate : polynomial -> polynomial val evaluate : polynomial * int -> Rational.rat end = struct structure R = Rational type coefficient = R.rat val zero = R.fromInt 0 val one = R.fromInt 1 (* polynomial represented as list of coefficients, where ith element is * coefficient for x^i. *) type polynomial = coefficient list type kernel = { support : int, (* number of samples to left/right *) segs : polynomial list, (* piece-wise polynomial that defines the *) (* kernel; there are 2*support segments *) } fun differentiate [] = raise Fail "invalid polynomial" | differentiate [_] = [zero] | differentiate (_::coeffs) = let fun lp (_, []) = [] | lp (i, c::r) = R.mul(F.fromInt i, c) :: lp(i+1, r) in lp (1, coeffs) end (* evaluate a polynomial at an integer coordinate (used to test continuity) *) fun evaluate (poly, x) = let val x = F.fromInt x fun eval (sum, [], xn) = sum | eval (sum, c::r, xn) = eval(F.add(sum, R.mul(c, xn)), r, R.mul(x, xn)) in eval (zero, poly, one) end (* some standard kernels *) local val op / = R./ fun r i = R.fromInt i in val tent : kernel = { (* linear interpolation *) support = 1, segs = [[r 1, r ~1]] } val ctmr : kernel = { (* Catmull-Rom interpolation *) support = 2, segs = [ [r 1, r 0, ~5/2, 3/2], [r 2, r ~4, 5/2, ~1/2] ] } val bspl3 : kernel = { (* cubic bspline reconstruction, doesn't interpolate *) support = 2, segs = [ [ 2/3, r 0, r ~1, 1/2 ], [ 4/3, r ~2, r 1, ~1/6 ]' ] } val bspl5 : kernel = { (* quintic bspline reconstruction, doesn't interpolate *) support = 3, segs = [ [ 11/20, r 0, ~1/2, r 0, 1/4, ~1/12 ], [ 17/40, 5/8, ~7/4, 5/4, ~3/8, 1/24 ] [ 81/40, ~27/8, 9/4, ~3/4, 1/8, ~1/120 ] ] } end end
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The output has ended thus: 5/8, ~7/4, 5/4, ~3/8, 1/24 ] [ 81/40, ~27/8, 9/4, ~3/4, 1/8, ~1/120 ] ] } end end