Deconvolution/1D: Difference between revisions
m (→{{header|R}}: typo (more R-like, though not mandatory)) |
m (→{{header|R}}: yet another typo : n <= p, so the "max" is simplified; also use local variables) |
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p <- length(a) |
p <- length(a) |
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q <- length(b) |
q <- length(b) |
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n <- |
n <- p - q + 1 |
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r <- upper(max(p, q |
r <- upper(max(p, q)) |
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y <- fft(fft(extend(a, r)) / fft(extend(b, r)), inverse=TRUE)/r |
y <- fft(fft(extend(a, r)) / fft(extend(b, r)), inverse=TRUE)/r |
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return(y[1:n]) |
return(y[1:n]) |
Revision as of 15:55, 27 February 2010
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.
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 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
deconv
anddeconv
and 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]
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
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> upper <- function(n) { r <- 1 while(r<n) r <- r + r r }
extend <- function(a, n) c(a, rep(0, n-length(a)))
conv <- function(a, b) { n <- length(a) + length(b) - 1 r <- upper(n) y <- fft(fft(extend(a, r)) * fft(extend(b, r)), inverse=TRUE)/r y[1:n] }
deconv <- function(a, b) { p <- length(a) q <- length(b) n <- p - q + 1 r <- upper(max(p, q)) y <- fft(fft(extend(a, r)) / fft(extend(b, r)), 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 introduce complex numbers in the solution, 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 use for complex arguments as well.
R has also a function convolve,
<lang R>
conv(a, b) == convolve(a, rev(b), type="open")
</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>>