Deconvolution/1D: Difference between revisions
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The deconv function for floating point data is built into Julia. |
The deconv function for floating point data is built into Julia. |
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Integer inputs may need to be converted and copied to floating point to use deconv(). |
Integer inputs may need to be converted and copied to floating point to use deconv(). |
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<lang Julia> |
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</lang> |
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{{output}} |
{{output}} |
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<pre> |
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which is the same as h = [-8, -9, -3, -1, -6, 7] |
which is the same as h = [-8, -9, -3, -1, -6, 7] |
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The deconvolution deconv(g,h) is [-3.0, -6.0, -1.0, 8.0, -6.0, 3.0, -1.0, -9.0, -9.0, 3.0, -2.0, 5.0, 2.0, -2.0, -7.0, -1.0], |
The deconvolution deconv(g, h) is [-3.0, -6.0, -1.0, 8.0, -6.0, 3.0, -1.0, -9.0, -9.0, 3.0, -2.0, 5.0, 2.0, -2.0, -7.0, -1.0], |
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which is the same as f = [-3, -6, -1, 8, -6, 3, -1, -9, -9, 3, -2, 5, 2, -2, -7, -1] |
which is the same as f = [-3, -6, -1, 8, -6, 3, -1, -9, -9, 3, -2, 5, 2, -2, -7, -1]</pre> |
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</pre> |
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=={{header|Kotlin}}== |
=={{header|Kotlin}}== |
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Revision as of 17:55, 28 January 2018
You are encouraged to solve this task according to the task description, using any language you may know.
The convolution of two functions and of an integer variable is defined as the function satisfying
for all integers . Assume can be non-zero only for ≤ ≤ , where is the "length" of , and similarly for and , so that the functions can be modeled as finite sequences by identifying with , etc. Then for example, values of and would determine the following value of by definition.
We can write this in matrix form as:
or
For this task, implement a function (or method, procedure, subroutine, etc.) deconv to perform deconvolution (i.e., the inverse of convolution) by constructing and solving such a system of equations represented by the above matrix for given and .
- The function should work for of arbitrary length (i.e., not hard coded or constant) and of any length up to that of . Note that will be given by .
- There may be more equations than unknowns. If convenient, use a function from a library that finds the best fitting solution to an overdetermined system of linear equations (as in the Multiple regression task). Otherwise, prune the set of equations as needed and solve as in the Reduced row echelon form task.
- Test your solution on the following data. Be sure to verify both that
deconvanddeconvand display the results in a human readable form.
h = [-8,-9,-3,-1,-6,7]
f = [-3,-6,-1,8,-6,3,-1,-9,-9,3,-2,5,2,-2,-7,-1]
g = [24,75,71,-34,3,22,-45,23,245,25,52,25,-67,-96,96,31,55,36,29,-43,-7]
BBC BASIC
As several others, this is a translation of the D solution. <lang bbcbasic> *FLOAT 64
DIM h(5), f(15), g(20)
h() = -8,-9,-3,-1,-6,7
f() = -3,-6,-1,8,-6,3,-1,-9,-9,3,-2,5,2,-2,-7,-1
g() = 24,75,71,-34,3,22,-45,23,245,25,52,25,-67,-96,96,31,55,36,29,-43,-7
PROCdeconv(g(), f(), x())
PRINT "deconv(g,f) = " FNprintarray(x())
x() -= h() : IF SUM(x()) <> 0 PRINT "Error!"
PROCdeconv(g(), h(), y())
PRINT "deconv(g,h) = " FNprintarray(y())
y() -= f() : IF SUM(y()) <> 0 PRINT "Error!"
END
DEF PROCdeconv(g(), f(), RETURN h())
LOCAL f%, g%, i%, l%, n%
f% = DIM(f(),1) + 1
g% = DIM(g(),1) + 1
DIM h(g% - f%)
FOR n% = 0 TO g% - f%
h(n%) = g(n%)
IF n% < f% THEN l% = 0 ELSE l% = n% - f% + 1
IF n% THEN
FOR i% = l% TO n% - 1
h(n%) -= h(i%) * f(n% - i%)
NEXT
ENDIF
h(n%) /= f(0)
NEXT n%
ENDPROC
DEF FNprintarray(a())
LOCAL i%, a$
FOR i% = 0 TO DIM(a(),1)
a$ += STR$(a(i%)) + ", "
NEXT
= LEFT$(LEFT$(a$))</lang>
- Output:
deconv(g,f) = -8, -9, -3, -1, -6, 7 deconv(g,h) = -3, -6, -1, 8, -6, 3, -1, -9, -9, 3, -2, 5, 2, -2, -7, -1
C
Using FFT: <lang C>#include <stdio.h>
- include <stdlib.h>
- include <math.h>
- include <complex.h>
double PI; typedef double complex cplx;
void _fft(cplx buf[], cplx out[], int n, int step) { if (step < n) { _fft(out, buf, n, step * 2); _fft(out + step, buf + step, n, step * 2);
for (int i = 0; i < n; i += 2 * step) { cplx t = cexp(-I * PI * i / n) * out[i + step]; buf[i / 2] = out[i] + t; buf[(i + n)/2] = out[i] - t; } } }
void fft(cplx buf[], int n) { cplx out[n]; for (int i = 0; i < n; i++) out[i] = buf[i]; _fft(buf, out, n, 1); }
/* pad array length to power of two */ cplx *pad_two(double g[], int len, int *ns) { int n = 1; if (*ns) n = *ns; else while (n < len) n *= 2;
cplx *buf = calloc(sizeof(cplx), n); for (int i = 0; i < len; i++) buf[i] = g[i]; *ns = n; return buf; }
void deconv(double g[], int lg, double f[], int lf, double out[]) { int ns = 0; cplx *g2 = pad_two(g, lg, &ns); cplx *f2 = pad_two(f, lf, &ns);
fft(g2, ns); fft(f2, ns);
cplx h[ns]; for (int i = 0; i < ns; i++) h[i] = g2[i] / f2[i]; fft(h, ns);
for (int i = 0; i >= lf - lg; i--) out[-i] = h[(i + ns) % ns]/32; free(g2); free(f2); }
int main() { PI = atan2(1,1) * 4; double g[] = {24,75,71,-34,3,22,-45,23,245,25,52,25,-67,-96,96,31,55,36,29,-43,-7}; double f[] = { -3,-6,-1,8,-6,3,-1,-9,-9,3,-2,5,2,-2,-7,-1 }; double h[] = { -8,-9,-3,-1,-6,7 };
int lg = sizeof(g)/sizeof(double); int lf = sizeof(f)/sizeof(double); int lh = sizeof(h)/sizeof(double);
double h2[lh]; double f2[lf];
printf("f[] data is : "); for (int i = 0; i < lf; i++) printf(" %g", f[i]); printf("\n");
printf("deconv(g, h): "); deconv(g, lg, h, lh, f2); for (int i = 0; i < lf; i++) printf(" %g", f2[i]); printf("\n");
printf("h[] data is : "); for (int i = 0; i < lh; i++) printf(" %g", h[i]); printf("\n");
printf("deconv(g, f): "); deconv(g, lg, f, lf, h2); for (int i = 0; i < lh; i++) printf(" %g", h2[i]); printf("\n"); }</lang>
- Output:
f[] data is : -3 -6 -1 8 -6 3 -1 -9 -9 3 -2 5 2 -2 -7 -1deconv(g, h): -3 -6 -1 8 -6 3 -1 -9 -9 3 -2 5 2 -2 -7 -1 h[] data is : -8 -9 -3 -1 -6 7
deconv(g, f): -8 -9 -3 -1 -6 7
Common Lisp
Uses the routine (lsqr A b) from Multiple regression and (mtp A) from Matrix transposition.
<lang lisp>;; Assemble the mxn matrix A from the 2D row vector x. (defun make-conv-matrix (x m n)
(let ((lx (cadr (array-dimensions x)))
(A (make-array `(,m ,n) :initial-element 0)))
(loop for j from 0 to (- n 1) do
(loop for i from 0 to (- m 1) do
(setf (aref A i j)
(cond ((or (< i j) (>= i (+ j lx)))
0)
((and (>= i j) (< i (+ j lx)))
(aref x 0 (- i j)))))))
A))
- Solve the overdetermined system A(f)*h=g by linear least squares.
(defun deconv (g f)
(let* ((lg (cadr (array-dimensions g)))
(lf (cadr (array-dimensions f)))
(lh (+ (- lg lf) 1))
(A (make-conv-matrix f lg lh)))
(lsqr A (mtp g))))</lang>
Example:
<lang lisp>(setf f #2A((-3 -6 -1 8 -6 3 -1 -9 -9 3 -2 5 2 -2 -7 -1))) (setf h #2A((-8 -9 -3 -1 -6 7))) (setf g #2A((24 75 71 -34 3 22 -45 23 245 25 52 25 -67 -96 96 31 55 36 29 -43 -7)))
(deconv g f)
- 2A((-8.0)
(-9.000000000000002) (-2.999999999999999) (-0.9999999999999997) (-6.0) (7.000000000000002))
(deconv g h)
- 2A((-2.999999999999999)
(-6.000000000000001) (-1.0000000000000002) (8.0) (-5.999999999999999) (3.0000000000000004) (-1.0000000000000004) (-9.000000000000002) (-9.0) (2.9999999999999996) (-1.9999999999999991) (5.0) (1.9999999999999996) (-2.0000000000000004) (-7.000000000000001) (-0.9999999999999994))</lang>
D
<lang d>T[] deconv(T)(in T[] g, in T[] f) pure nothrow {
int flen = f.length;
int glen = g.length;
auto result = new T[glen - flen + 1];
foreach (int n, ref e; result) {
e = g[n];
immutable lowerBound = (n >= flen) ? n - flen + 1 : 0;
foreach (i; lowerBound .. n)
e -= result[i] * f[n - i];
e /= f[0];
}
return result;
}
void main() {
import std.stdio;
immutable h = [-8,-9,-3,-1,-6,7];
immutable f = [-3,-6,-1,8,-6,3,-1,-9,-9,3,-2,5,2,-2,-7,-1];
immutable g = [24,75,71,-34,3,22,-45,23,245,25,52,25,-67,
-96,96,31,55,36,29,-43,-7];
writeln(deconv(g, f) == h, " ", deconv(g, f));
writeln(deconv(g, h) == f, " ", deconv(g, h));
}</lang>
- Output:
true [-8, -9, -3, -1, -6, 7] true [-3, -6, -1, 8, -6, 3, -1, -9, -9, 3, -2, 5, 2, -2, -7, -1]
Fortran
This solution uses the LAPACK95 library. <lang fortran> ! Build ! Windows: ifort /I "%IFORT_COMPILER11%\mkl\include\ia32" deconv1d.f90 "%IFORT_COMPILER11%\mkl\ia32\lib\*.lib" ! Linux:
program deconv
! Use gelsd from LAPACK95. use mkl95_lapack, only : gelsd
implicit none real(8), allocatable :: g(:), href(:), A(:,:), f(:) real(8), pointer :: h(:), r(:) integer :: N character(len=16) :: cbuff integer :: i intrinsic :: nint
! Allocate data arrays allocate(g(21),f(16)) g = [24,75,71,-34,3,22,-45,23,245,25,52,25,-67,-96,96,31,55,36,29,-43,-7] f = [-3,-6,-1,8,-6,3,-1,-9,-9,3,-2,5,2,-2,-7,-1]
! Calculate deconvolution h => deco(f, g)
! Check result against reference
N = size(h)
allocate(href(N))
href = [-8,-9,-3,-1,-6,7]
cbuff = ' '
write(cbuff,'(a,i0,a)') '(a,',N,'(i0,a),i0)'
if (any(abs(h-href) > 1.0d-4)) then
write(*,'(a)') 'deconv(f, g) - FAILED'
else
write(*,cbuff) 'deconv(f, g) = ',(nint(h(i)),', ',i=1,N-1),nint(h(N))
end if
! Calculate deconvolution r => deco(h, g)
cbuff = ' '
N = size(r)
write(cbuff,'(a,i0,a)') '(a,',N,'(i0,a),i0)'
if (any(abs(r-f) > 1.0d-4)) then
write(*,'(a)') 'deconv(h, g) - FAILED'
else
write(*,cbuff) 'deconv(h, g) = ',(nint(r(i)),', ',i=1,N-1),nint(r(N))
end if
contains
function deco(p, q) real(8), pointer :: deco(:) real(8), intent(in) :: p(:), q(:)
real(8), allocatable, target :: r(:) real(8), allocatable :: A(:,:) integer :: N
! Construct derived arrays
N = size(q) - size(p) + 1
allocate(A(size(q),N),r(size(q)))
A = 0.0d0
do i=1,N
A(i:i+size(p)-1,i) = p
end do
! Invoke the LAPACK routine to do the work
r = q
call gelsd(A, r)
deco => r(1:N) end function deco
end program deconv </lang> Results: <lang fortran> deconv(f, g) = -8, -9, -3, -1, -6, 7 deconv(h, g) = -3, -6, -1, 8, -6, 3, -1, -9, -9, 3, -2, 5, 2, -2, -7, -1 </lang>
Go
<lang go>package main
import "fmt"
func main() {
h := []float64{-8, -9, -3, -1, -6, 7}
f := []float64{-3, -6, -1, 8, -6, 3, -1, -9, -9, 3, -2, 5, 2, -2, -7, -1}
g := []float64{24, 75, 71, -34, 3, 22, -45, 23, 245, 25, 52, 25, -67, -96,
96, 31, 55, 36, 29, -43, -7}
fmt.Println(h)
fmt.Println(deconv(g, f))
fmt.Println(f)
fmt.Println(deconv(g, h))
}
func deconv(g, f []float64) []float64 {
h := make([]float64, len(g)-len(f)+1)
for n := range h {
h[n] = g[n]
var lower int
if n >= len(f) {
lower = n - len(f) + 1
}
for i := lower; i < n; i++ {
h[n] -= h[i] * f[n-i]
}
h[n] /= f[0]
}
return h
}</lang>
- Output:
[-8 -9 -3 -1 -6 7] [-8 -9 -3 -1 -6 7] [-3 -6 -1 8 -6 3 -1 -9 -9 3 -2 5 2 -2 -7 -1] [-3 -6 -1 8 -6 3 -1 -9 -9 3 -2 5 2 -2 -7 -1]
<lang go>package main
import (
"fmt" "math" "math/cmplx"
)
func main() {
h := []float64{-8, -9, -3, -1, -6, 7}
f := []float64{-3, -6, -1, 8, -6, 3, -1, -9, -9, 3, -2, 5, 2, -2, -7, -1}
g := []float64{24, 75, 71, -34, 3, 22, -45, 23, 245, 25, 52, 25, -67, -96,
96, 31, 55, 36, 29, -43, -7}
fmt.Printf("%.1f\n", h)
fmt.Printf("%.1f\n", deconv(g, f))
fmt.Printf("%.1f\n", f)
fmt.Printf("%.1f\n", deconv(g, h))
}
func deconv(g, f []float64) []float64 {
n := 1
for n < len(g) {
n *= 2
}
g2 := make([]complex128, n)
for i, x := range g {
g2[i] = complex(x, 0)
}
f2 := make([]complex128, n)
for i, x := range f {
f2[i] = complex(x, 0)
}
gt := fft(g2)
ft := fft(f2)
for i := range gt {
gt[i] /= ft[i]
}
ht := fft(gt)
it := 1 / float64(n)
out := make([]float64, len(g)-len(f)+1)
out[0] = real(ht[0]) * it
for i := 1; i < len(out); i++ {
out[i] = real(ht[n-i]) * it
}
return out
}
func fft(in []complex128) []complex128 {
out := make([]complex128, len(in)) ditfft2(in, out, len(in), 1) return out
}
func ditfft2(x, y []complex128, n, s int) {
if n == 1 {
y[0] = x[0]
return
}
ditfft2(x, y, n/2, 2*s)
ditfft2(x[s:], y[n/2:], n/2, 2*s)
for k := 0; k < n/2; k++ {
tf := cmplx.Rect(1, -2*math.Pi*float64(k)/float64(n)) * y[k+n/2]
y[k], y[k+n/2] = y[k]+tf, y[k]-tf
}
}</lang>
- Output:
Some results have errors out in the last decimal place or so. Only one decimal place shown here to let results fit in 80 columns.
[-8.0 -9.0 -3.0 -1.0 -6.0 7.0] [-8.0 -9.0 -3.0 -1.0 -6.0 7.0] [-3.0 -6.0 -1.0 8.0 -6.0 3.0 -1.0 -9.0 -9.0 3.0 -2.0 5.0 2.0 -2.0 -7.0 -1.0] [-3.0 -6.0 -1.0 8.0 -6.0 3.0 -1.0 -9.0 -9.0 3.0 -2.0 5.0 2.0 -2.0 -7.0 -1.0]
Haskell
<lang haskell>import Data.List
h, f, g :: [Double] h = [-8,-9,-3,-1,-6,7] f = [-3,-6,-1,8,-6,3,-1,-9,-9,3,-2,5,2,-2,-7,-1] g = [24,75,71,-34,3,22,-45,23,245,25,52,25,-67,-96,96,31,55,36,29,-43,-7]
scale x ys = map (x*) ys
deconv1d :: (Fractional a) => [a] -> [a] -> [a] deconv1d xs ys = takeWhile (/=0) $ deconv xs ys
where [] `deconv` _ = []
(0:xs) `deconv` (0:ys) = xs `deconv` ys
(x:xs) `deconv` (y:ys) =
q : zipWith (-) xs (scale q ys ++ repeat 0) `deconv` (y:ys)
where q = x / y</lang>
Check: <lang haskell>*Main> h == deconv1d g f True
- Main> f == deconv1d g h
True</lang>
J
This solution borrowed from Formal power series:
<lang J>Ai=: (i.@] =/ i.@[ -/ i.@>:@-)&# divide=: [ +/ .*~ [:%.&.x: ] +/ .* Ai</lang>
Sample data:
<lang J>h=: _8 _9 _3 _1 _6 7 f=: _3 _6 _1 8 _6 3 _1 _9 _9 3 _2 5 2 _2 _7 _1 g=: 24 75 71 _34 3 22 _45 23 245 25 52 25 _67 _96 96 31 55 36 29</lang>
Example use: <lang J> g divide f _8 _9 _3 _1 _6 7
g divide h
_3 _6 _1 8 _6 3 _1 _9 _9 3 _2 5 2 _2 _7 _1</lang>
That said, note that this particular implementation is slow since it uses extended precision intermediate results. It will run quite a bit faster for this example with no notable loss of precision if floating point is used. In other words:
<lang J>divide=: [ +/ .*~ [:%. ] +/ .* Ai</lang>
Java
<lang java>import java.util.Arrays;
public class Deconvolution1D {
public static int[] deconv(int[] g, int[] f) {
int[] h = new int[g.length - f.length + 1];
for (int n = 0; n < h.length; n++) {
h[n] = g[n];
int lower = Math.max(n - f.length + 1, 0);
for (int i = lower; i < n; i++)
h[n] -= h[i] * f[n - i];
h[n] /= f[0];
}
return h;
}
public static void main(String[] args) {
int[] h = { -8, -9, -3, -1, -6, 7 };
int[] f = { -3, -6, -1, 8, -6, 3, -1, -9, -9, 3, -2, 5, 2, -2, -7, -1 };
int[] g = { 24, 75, 71, -34, 3, 22, -45, 23, 245, 25, 52, 25, -67, -96,
96, 31, 55, 36, 29, -43, -7 };
StringBuilder sb = new StringBuilder();
sb.append("h = " + Arrays.toString(h) + "\n");
sb.append("deconv(g, f) = " + Arrays.toString(deconv(g, f)) + "\n");
sb.append("f = " + Arrays.toString(f) + "\n");
sb.append("deconv(g, h) = " + Arrays.toString(deconv(g, h)) + "\n");
System.out.println(sb.toString());
}
}</lang>
- Output:
h = [-8, -9, -3, -1, -6, 7] deconv(g, f) = [-8, -9, -3, -1, -6, 7] f = [-3, -6, -1, 8, -6, 3, -1, -9, -9, 3, -2, 5, 2, -2, -7, -1] deconv(g, h) = [-3, -6, -1, 8, -6, 3, -1, -9, -9, 3, -2, 5, 2, -2, -7, -1]
Julia
The deconv function for floating point data is built into Julia. Integer inputs may need to be converted and copied to floating point to use deconv().
<lang julia>h = [-8, -9, -3, -1, -6, 7] g = [24, 75, 71, -34, 3, 22, -45, 23, 245, 25, 52, 25, -67, -96, 96, 31, 55, 36, 29, -43, -7] f = [-3, -6, -1, 8, -6, 3, -1, -9, -9, 3, -2, 5, 2, -2, -7, -1]
hanswer = deconv(float.(g), float.(f)) println("The deconvolution deconv(g, f) is $hanswer, which is the same as h = $h\n")
fanswer = deconv(float.(g), float.(h)) println("The deconvolution deconv(g, h) is $fanswer, which is the same as f = $f\n")</lang>
- Output:
The deconvolution deconv(g, f) is [-8.0, -9.0, -3.0, -1.0, -6.0, 7.0], which is the same as h = [-8, -9, -3, -1, -6, 7] The deconvolution deconv(g, h) is [-3.0, -6.0, -1.0, 8.0, -6.0, 3.0, -1.0, -9.0, -9.0, 3.0, -2.0, 5.0, 2.0, -2.0, -7.0, -1.0], which is the same as f = [-3, -6, -1, 8, -6, 3, -1, -9, -9, 3, -2, 5, 2, -2, -7, -1]
Kotlin
<lang scala>// version 1.1.3
fun deconv(g: DoubleArray, f: DoubleArray): DoubleArray {
val fs = f.size
val h = DoubleArray(g.size - fs + 1)
for (n in h.indices) {
h[n] = g[n]
val lower = if (n >= fs) n - fs + 1 else 0
for (i in lower until n) h[n] -= h[i] * f[n -i]
h[n] /= f[0]
}
return h
}
fun main(args: Array<String>) {
val h = doubleArrayOf(-8.0, -9.0, -3.0, -1.0, -6.0, 7.0)
val f = doubleArrayOf(-3.0, -6.0, -1.0, 8.0, -6.0, 3.0, -1.0, -9.0,
-9.0, 3.0, -2.0, 5.0, 2.0, -2.0, -7.0, -1.0)
val g = doubleArrayOf(24.0, 75.0, 71.0, -34.0, 3.0, 22.0, -45.0,
23.0, 245.0, 25.0, 52.0, 25.0, -67.0, -96.0,
96.0, 31.0, 55.0, 36.0, 29.0, -43.0, -7.0)
println("${h.map { it.toInt() }}")
println("${deconv(g, f).map { it.toInt() }}")
println()
println("${f.map { it.toInt() }}")
println("${deconv(g, h).map { it.toInt() }}")
}</lang>
- Output:
[-8, -9, -3, -1, -6, 7] [-8, -9, -3, -1, -6, 7] [-3, -6, -1, 8, -6, 3, -1, -9, -9, 3, -2, 5, 2, -2, -7, -1] [-3, -6, -1, 8, -6, 3, -1, -9, -9, 3, -2, 5, 2, -2, -7, -1]
Lua
Using metatables: <lang lua>function deconvolve(f, g)
local h = setmetatable({}, {__index = function(self, n)
if n == 1 then self[1] = g[1] / f[1]
else
self[n] = g[n]
for i = 1, n - 1 do
self[n] = self[n] - self[i] * (f[n - i + 1] or 0)
end
self[n] = self[n] / f[1]
end
return self[n]
end})
local _ = h[#g - #f + 1]
return setmetatable(h, nil)
end</lang>
Tests: <lang lua> local f = {-3,-6,-1,8,-6,3,-1,-9,-9,3,-2,5,2,-2,-7,-1} local g = {24,75,71,-34,3,22,-45,23,245,25,52,25,-67,-96,96,31,55,36,29,-43,-7} local h = {-8,-9,-3,-1,-6,7} print(unpack(deconvolve(f, g))) --> -8 -9 -3 -1 -6 7 print(unpack(deconvolve(h, g))) --> -3 -6 -1 8 -6 3 -1 -9 -9 3 -2 5 2 -2 -7 -1</lang>
Mathematica / Wolfram Language
This function creates a sparse array for the A matrix and then solves it with a built-in function. It may fail for overdetermined systems, though. Fast approximate methods for deconvolution are also built into Mathematica. See Deconvolution/2D+ <lang Mathematica> deconv[f_List, g_List] :=
Module[{A =
SparseArray[
Table[Band[{n, 1}] -> fn, {n, 1, Length[f]}], {Length[g], Length[f] - 1}]},
Take[LinearSolve[A, g], Length[g] - Length[f] + 1]]
</lang> Usage:
f = {-3, -6, -1, 8, -6, 3, -1, -9, -9, 3, -2, 5, 2, -2, -7, -1};
g = {24, 75, 71, -34, 3, 22, -45, 23, 245, 25, 52, 25, -67, -96, 96, 31, 55, 36, 29, -43, -7};
deconv[f,g]
- Output:
{-8, -9, -3, -1, -6, 7}
MATLAB
The deconvolution function is built-in to MATLAB as the "deconv(a,b)" function, where "a" and "b" are vectors storing the convolved function values and the values of one of the deconvoluted vectors of "a". To test that this operates according to the task spec we can test the criteria above: <lang MATLAB>>> h = [-8,-9,-3,-1,-6,7]; >> g = [24,75,71,-34,3,22,-45,23,245,25,52,25,-67,-96,96,31,55,36,29,-43,-7]; >> f = [-3,-6,-1,8,-6,3,-1,-9,-9,3,-2,5,2,-2,-7,-1]; >> deconv(g,f)
ans =
-8.0000 -9.0000 -3.0000 -1.0000 -6.0000 7.0000
>> deconv(g,h)
ans =
-3 -6 -1 8 -6 3 -1 -9 -9 3 -2 5 2 -2 -7 -1</lang>
Therefore, "deconv(a,b)" behaves as expected.
Perl 6
Translation of Python, using a modified version of the Reduced Row Echelon Form subroutine rref() from here.
<lang perl6>sub deconvolve (@g, @f) {
my $h = 1 + @g - @f;
my @m;
@m[^@g;^$h] >>+=>> 0;
@m[^@g;$h] >>=<< @g;
for ^$h -> $j { for @f.kv -> $k, $v { @m[$j + $k][$j] = $v } }
return rref( @m )[^$h;$h];
}
sub convolve (@f, @h) {
my @g = 0 xx + @f + @h - 1; @g[^@f X+ ^@h] >>+=<< (@f X* @h); return @g;
}
- Reduced Row Echelon Form simultaneous equation solver.
- Can handle over-specified systems of equations.
- (n unknowns in n + m equations)
sub rref ($m is copy) {
return unless $m; my ($lead, $rows, $cols) = 0, +$m, +$m[0];
# Trim off over specified rows if they exist. # Not strictly necessary, but can save a lot of # redundant calculations. [remove until debugged]
- if $rows >= $cols {
- $m = trim_system($m);
- $rows = +$m;
- }
for ^$rows -> $r {
$lead < $cols or return $m;
my $i = $r;
until $m[$i][$lead] {
++$i == $rows or next;
$i = $r;
++$lead == $cols and return $m;
}
$m[$i, $r] = $m[$r, $i] if $r != $i;
my $lv = $m[$r][$lead];
$m[$r] >>/=>> $lv;
for ^$rows -> $n {
next if $n == $r;
$m[$n] >>-=>> $m[$r] >>*>> ($m[$n][$lead]//0);
}
++$lead;
}
return $m;
# Reduce a system of equations to n equations with n unknowns.
# Looks for an equation with a true value for each position.
# If it can't find one, assumes that it has already taken one
# and pushes in the first equation it sees. This assumtion
# will alway be successful except in some cases where an
# under-specified system has been supplied, in which case,
# it would not have been able to reduce the system anyway.
sub trim_system ($m is rw) {
my ($vars, @t) = +$m[0]-1, ();
for ^$vars -> $lead {
for ^$m -> $row {
@t.push( $m.splice( $row, 1 ) ) and last if $m[$row][$lead];
}
}
while (+@t < $vars) and +$m { @t.push( $m.splice( 0, 1 ) ) };
return @t;
}
}
my @h = (-8,-9,-3,-1,-6,7);
my @f = (-3,-6,-1,8,-6,3,-1,-9,-9,3,-2,5,2,-2,-7,-1);
my @g = (24,75,71,-34,3,22,-45,23,245,25,52,25,-67,-96,96,31,55,36,29,-43,-7);
.say for ~@g, ~convolve(@f, @h),;
.say for ~@h, ~deconvolve(@g, @f),;
.say for ~@f, ~deconvolve(@g, @h),;</lang>
- Output:
24 75 71 -34 3 22 -45 23 245 25 52 25 -67 -96 96 31 55 36 29 -43 -7 24 75 71 -34 3 22 -45 23 245 25 52 25 -67 -96 96 31 55 36 29 -43 -7 -8 -9 -3 -1 -6 7 -8 -9 -3 -1 -6 7 -3 -6 -1 8 -6 3 -1 -9 -9 3 -2 5 2 -2 -7 -1 -3 -6 -1 8 -6 3 -1 -9 -9 3 -2 5 2 -2 -7 -1
Phix
<lang Phix>function deconv(sequence g, sequence f) integer lf = length(f) sequence h = repeat(0,length(g)-lf+1)
for n = 1 to length(h) do
atom e = g[n]
for i=max(n-lf,0) to n-2 do
e -= h[i+1] * f[n-i]
end for
h[n] = e/f[1]
end for
return h
end function
constant h = {-8,-9,-3,-1,-6,7} constant f = {-3,-6,-1,8,-6,3,-1,-9,-9,3,-2,5,2,-2,-7,-1} constant g = {24,75,71,-34,3,22,-45,23,245,25,52,25,-67,
-96,96,31,55,36,29,-43,-7}
sequence r r = deconv(g, f) ?{r==h,r} r = deconv(g, h) ?{r==f,r}</lang>
- Output:
{1,{-8,-9,-3,-1,-6,7}}
{1,{-3,-6,-1,8,-6,3,-1,-9,-9,3,-2,5,2,-2,-7,-1}}
PicoLisp
<lang PicoLisp>(load "@lib/math.l")
(de deconv (G F)
(let A (pop 'F)
(make
(for (N . H) (head (- (length F)) G)
(for (I . M) (made)
(dec 'H
(*/ M (get F (- N I)) 1.0) ) )
(link (*/ H 1.0 A)) ) ) ) )</lang>
Test: <lang PicoLisp>(setq
F (-3. -6. -1. 8. -6. 3. -1. -9. -9. 3. -2. 5. 2. -2. -7. -1.) G (24. 75. 71. -34. 3. 22. -45. 23. 245. 25. 52. 25. -67. -96. 96. 31. 55. 36. 29. -43. -7.) H (-8. -9. -3. -1. -6. 7.) )
(test H (deconv G F)) (test F (deconv G H))</lang>
Python
Inspired by the TCL solution, and using the ToReducedRowEchelonForm function to reduce to row echelon form from here
<lang python>def ToReducedRowEchelonForm( M ):
if not M: return
lead = 0
rowCount = len(M)
columnCount = len(M[0])
for r in range(rowCount):
if lead >= columnCount:
return
i = r
while M[i][lead] == 0:
i += 1
if i == rowCount:
i = r
lead += 1
if columnCount == lead:
return
M[i],M[r] = M[r],M[i]
lv = M[r][lead]
M[r] = [ mrx / lv for mrx in M[r]]
for i in range(rowCount):
if i != r:
lv = M[i][lead]
M[i] = [ iv - lv*rv for rv,iv in zip(M[r],M[i])]
lead += 1
return M
def pmtx(mtx):
print ('\n'.join(.join(' %4s' % col for col in row) for row in mtx))
def convolve(f, h):
g = [0] * (len(f) + len(h) - 1)
for hindex, hval in enumerate(h):
for findex, fval in enumerate(f):
g[hindex + findex] += fval * hval
return g
def deconvolve(g, f):
lenh = len(g) - len(f) + 1
mtx = [[0 for x in range(lenh+1)] for y in g]
for hindex in range(lenh):
for findex, fval in enumerate(f):
gindex = hindex + findex
mtx[gindex][hindex] = fval
for gindex, gval in enumerate(g):
mtx[gindex][lenh] = gval
ToReducedRowEchelonForm( mtx )
return [mtx[i][lenh] for i in range(lenh)] # h
if __name__ == '__main__':
h = [-8,-9,-3,-1,-6,7] f = [-3,-6,-1,8,-6,3,-1,-9,-9,3,-2,5,2,-2,-7,-1] g = [24,75,71,-34,3,22,-45,23,245,25,52,25,-67,-96,96,31,55,36,29,-43,-7] assert convolve(f,h) == g assert deconvolve(g, f) == h</lang>
R
Here we won't solve the system but use the FFT instead. The method :
- extend vector arguments so that they are the same length, a power of 2 larger than the length of the solution,
- solution is ifft(fft(a)*fft(b)), truncated.
<lang R>conv <- function(a, b) { p <- length(a) q <- length(b) n <- p + q - 1 r <- nextn(n, f=2) y <- fft(fft(c(a, rep(0, r-p))) * fft(c(b, rep(0, r-q))), inverse=TRUE)/r y[1:n] }
deconv <- function(a, b) { p <- length(a) q <- length(b) n <- p - q + 1 r <- nextn(max(p, q), f=2) y <- fft(fft(c(a, rep(0, r-p))) / fft(c(b, rep(0, r-q))), inverse=TRUE)/r return(y[1:n]) } </lang>
To check :
<lang R> h <- c(-8,-9,-3,-1,-6,7) f <- c(-3,-6,-1,8,-6,3,-1,-9,-9,3,-2,5,2,-2,-7,-1) g <- c(24,75,71,-34,3,22,-45,23,245,25,52,25,-67,-96,96,31,55,36,29,-43,-7)
max(abs(conv(f,h) - g)) max(abs(deconv(g,f) - h)) max(abs(deconv(g,h) - f)) </lang>
This solution often introduces complex numbers, with null or tiny imaginary part. If it hurts in applications, type Re(conv(f,h)) and Re(deconv(g,h)) instead, to return only the real part. It's not hard-coded in the functions, since they may be used for complex arguments as well.
R has also a function convolve,
<lang R>
conv(a, b) == convolve(a, rev(b), type="open")
</lang>
Racket
<lang racket>
- lang racket
(require math/matrix) (define T matrix-transpose)
(define (convolution-matrix f m n)
(define l (matrix-num-rows f))
(for*/matrix m n ([i (in-range 0 m)] [j (in-range 0 n)])
(cond [(or (< i j) (>= i (+ j l))) 0]
[(matrix-ref f (- i j) 0)])))
(define (least-square X y)
(matrix-solve (matrix* (T X) X) (matrix* (T X) y)))
(define (deconvolve g f)
(define lg (matrix-num-rows g)) (define lf (matrix-num-rows f)) (define lh (+ (- lg lf) 1)) (least-square (convolution-matrix f lg lh) g))
</lang> Test: <lang racket> (define f (col-matrix [-3 -6 -1 8 -6 3 -1 -9 -9 3 -2 5 2 -2 -7 -1])) (define h (col-matrix [-8 -9 -3 -1 -6 7])) (define g (col-matrix [24 75 71 -34 3 22 -45 23 245 25 52 25 -67 -96 96 31 55 36 29 -43 -7]))
(deconvolve g f) (deconvolve g h) </lang>
- Output:
<lang racket>
- <array '#(6 1) #[-8 -9 -3 -1 -6 7]>
- <array '#(16 1) #[-3 -6 -1 8 -6 3 -1 -9 -9 3 -2 5 2 -2 -7 -1]>
</lang>
Tcl
This builds the a command, 1D, with two subcommands (convolve and deconvolve) for performing convolution and deconvolution of these kinds of arrays. The deconvolution code is based on a reduction to reduced row echelon form.
<lang tcl>package require Tcl 8.5
namespace eval 1D {
namespace ensemble create; # Will be same name as namespace
namespace export convolve deconvolve
# Access core language math utility commands
namespace path {::tcl::mathfunc ::tcl::mathop}
# Utility for converting a matrix to Reduced Row Echelon Form # From http://rosettacode.org/wiki/Reduced_row_echelon_form#Tcl proc toRREF {m} {
set lead 0 set rows [llength $m] set cols [llength [lindex $m 0]] for {set r 0} {$r < $rows} {incr r} { if {$cols <= $lead} { break } set i $r while {[lindex $m $i $lead] == 0} { incr i if {$rows == $i} { set i $r incr lead if {$cols == $lead} { # Tcl can't break out of nested loops return $m } } } # swap rows i and r foreach j [list $i $r] row [list [lindex $m $r] [lindex $m $i]] { lset m $j $row } # divide row r by m(r,lead) set val [lindex $m $r $lead] for {set j 0} {$j < $cols} {incr j} { lset m $r $j [/ [double [lindex $m $r $j]] $val] }
for {set i 0} {$i < $rows} {incr i} { if {$i != $r} { # subtract m(i,lead) multiplied by row r from row i set val [lindex $m $i $lead] for {set j 0} {$j < $cols} {incr j} { lset m $i $j \ [- [lindex $m $i $j] [* $val [lindex $m $r $j]]] } } } incr lead } return $m
}
# How to apply a 1D convolution of two "functions"
proc convolve {f h} {
set g [lrepeat [+ [llength $f] [llength $h] -1] 0] set fi -1 foreach fv $f { incr fi set hi -1 foreach hv $h { set gi [+ $fi [incr hi]] lset g $gi [+ [lindex $g $gi] [* $fv $hv]] } } return $g
}
# How to apply a 1D deconvolution of two "functions"
proc deconvolve {g f} {
# Compute the length of the result vector set hlen [- [llength $g] [llength $f] -1]
# Build a matrix of equations to solve set matrix {} set i -1 foreach gv $g { lappend matrix [list {*}[lrepeat $hlen 0] $gv] set j [incr i] foreach fv $f { if {$j < 0} { break } elseif {$j < $hlen} { lset matrix $i $j $fv } incr j -1 } }
# Convert to RREF, solving the system of simultaneous equations set reduced [toRREF $matrix]
# Extract the deconvolution from the last column of the reduced matrix for {set i 0} {$i<$hlen} {incr i} { lappend result [lindex $reduced $i end] } return $result
}
}</lang> To use the above code, a simple demonstration driver (which solves the specific task): <lang tcl># Simple pretty-printer proc pp {name nlist} {
set sep ""
puts -nonewline "$name = \["
foreach n $nlist {
puts -nonewline [format %s%g $sep $n] set sep ,
} puts "\]"
}
set h {-8 -9 -3 -1 -6 7} set f {-3 -6 -1 8 -6 3 -1 -9 -9 3 -2 5 2 -2 -7 -1} set g {24 75 71 -34 3 22 -45 23 245 25 52 25 -67 -96 96 31 55 36 29 -43 -7}
pp "deconv(g,f) = h" [1D deconvolve $g $f] pp "deconv(g,h) = f" [1D deconvolve $g $h] pp " conv(f,h) = g" [1D convolve $f $h]</lang>
- Output:
deconv(g,f) = h = [-8,-9,-3,-1,-6,7] deconv(g,h) = f = [-3,-6,-1,8,-6,3,-1,-9,-9,3,-2,5,2,-2,-7,-1] conv(f,h) = g = [24,75,71,-34,3,22,-45,23,245,25,52,25,-67,-96,96,31,55,36,29,-43,-7]
Ursala
The user defined function band constructs the required
matrix as a list of lists given the pair of sequences to be
deconvolved, and the lapack..dgelsd function solves the system. Some other library functions used are zipt (zipping two unequal length
lists by truncating the longer one) zipp0 (zipping unequal length lists by padding the
shorter with zeros) and pad0 (making a list of lists all
the same length by appending zeros to the short ones).
<lang Ursala>#import std
- import nat
band = pad0+ ~&rSS+ zipt^*D(~&r,^lrrSPT/~<K33tx zipt^/~&r ~&lSNyCK33+ zipp0)^/~&rx ~&B->NlNSPC ~&bt
deconv = lapack..dgelsd^\~&l ~&||0.!**+ band </lang> test program: <lang Ursala>h = <-8.,-9.,-3.,-1.,-6.,7.> f = <-3.,-6.,-1.,8.,-6.,3.,-1.,-9.,-9.,3.,-2.,5.,2.,-2.,-7.,-1.> g = <24.,75.,71.,-34.,3.,22.,-45.,23.,245.,25.,52.,25.,-67.,-96.,96.,31.,55.,36.,29.,-43.,-7.>
- cast %eLm
test =
<
'h': deconv(g,f), 'f': deconv(g,h)>
</lang>
- Output:
<
'h': <
-8.000000e+00,
-9.000000e+00,
-3.000000e+00,
-1.000000e+00,
-6.000000e+00,
7.000000e+00>,
'f': <
-3.000000e+00,
-6.000000e+00,
-1.000000e+00,
8.000000e+00,
-6.000000e+00,
3.000000e+00,
-1.000000e+00,
-9.000000e+00,
-9.000000e+00,
3.000000e+00,
-2.000000e+00,
5.000000e+00,
2.000000e+00,
-2.000000e+00,
-7.000000e+00,
-1.000000e+00>>
zkl
Using GNU Scientific Library: <lang zkl>var [const] GSL=Import("zklGSL"); // libGSL (GNU Scientific Library) fcn dconv1D(f,g){
fsz,hsz:=f.len(), g.len() - fsz +1;
A:=GSL.Matrix(g.len(),hsz);
foreach n,fn in ([0..].zip(f)){ foreach rc in (hsz){ A[rc+n,rc]=fn } }
h:=A.AxEQb(g);
h
}</lang> <lang zkl>f:=GSL.VectorFromData(-3,-6,-1,8,-6,3,-1,-9,-9,3,-2,5,2,-2,-7,-1); g:=GSL.VectorFromData(24,75,71,-34,3,22,-45,23,245,25,52,25,-67,-96,96,31,55,36,29,-43,-7); h:=dconv1D(f,g); h.format().println();
f:=dconv1D(h,g); f.format().println();</lang>
- Output:
-8.00,-9.00,-3.00,-1.00,-6.00,7.00 -3.00,-6.00,-1.00,8.00,-6.00,3.00,-1.00,-9.00,-9.00,3.00,-2.00,5.00,2.00,-2.00,-7.00,-1.00
Or, using lists:
<lang zkl>fcn deconv(g,f){
flen, glen, delta:=f.len(), g.len(), glen - flen + 1;
result:=List.createLong(delta); // allocate list with space for items
foreach n in (delta){
e:=g[n];
lowerBound:=(if (n>=flen) n - flen + 1 else 0);
foreach i in ([lowerBound .. n-1]){ e-=result[i]*f[n - i]; }
result.append(e/f[0]);
}
result;
}</lang> <lang zkl>h:=T(-8,-9,-3,-1,-6,7); f:=T(-3,-6,-1,8,-6,3,-1,-9,-9,3,-2,5,2,-2,-7,-1); g:=T(24,75,71,-34,3,22,-45,23,245,25,52,25,-67,
-96,96,31,55,36,29,-43,-7);
println(deconv(g, f) == h, " ", deconv(g, f)); println(deconv(g, h) == f, " ", deconv(g, h));</lang>
- Output:
True L(-8,-9,-3,-1,-6,7) True L(-3,-6,-1,8,-6,3,-1,-9,-9,3,-2,5,2,-2,-7,-1)