de Rigo, D. (2012). Multi-dimensional weighted median: the module "wmedian" of the Mastrave modelling library. In: Semantic Array Programming with Mastrave - Introduction to Semantic Computational Modelling. http://mastrave.org/doc/mtv_m/wmedian

Multi-dimensional weighted median: the module "wmedian" of the Mastrave modelling library

Copyright and license notice of the function wmedian

Copyright © 2007,2008,2009,2010,2011 Daniele de Rigo

The file wmedian.m is part of Mastrave.

Mastrave is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.

Mastrave is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with Mastrave. If not, see http://www.gnu.org/licenses/.

Function declaration

answer = wmedian( values        ,
                  dim     = []  ,
                  weights = []  )

Description

Utility to extend the function median(.) providing weighted medians of the elements of the array values along a given dimension dim .

The weighted median of a column vector v with integer weights w is equivalent to the median of the vector [ v(1)◇w(1) ; v(2)◇w(2) ; ... ], where the operator ◇ denotes duplications (Yin et al., 1996), i.e.

v(1)◇w(1) = repmat( v(1), w(1), 1 )

The weighted median wm of v with nonnegative weights w2 is defined as

wm = arg min( w2' * abs( v - wm ) )

If one or more weights are Inf, the corresponding elements of values aligned along the dimension dim are weighted as if the Inf weights were al having the same weight and all remaining weights were zeros. This does not affect elements of values which are not aligned with Inf-weighted elements. If one or more weights are NaN, they are considered as zero weights. In case one or more all-zeros sequences of weights are aligned along the dimension dim of weights , the corresponding elements of values are weighted as if all weights were uniform.

References

Yin, L., Yang, R., Gabbouj, M., Neuvo, M. (1996): Weighted Median
Filters: A Tutorial. IEEE Transactions on Circuits and Systems II:
Analog and Digital Signal Processing, Vol. 43, No. 3, pp. 157-192,
March 1996.
DOI: 10.1109/82.486465.
Free access version:
http://www.cs.tut.fi/~moncef/publications/weighted-median-filters.pdf

Input arguments

 values            ::numeric::
                   Vector, matrix or multi-dimensional array of numbers.

 dim               ::scalar_index|empty::
                   Scalar positive integer representing the dimension along
                   which the weighted medians have to be computed.
                   If  dim  is an empty array  [] , the dimension is the 
                   first non-singleton dimension.  In case  values  is a
                   vector, this definition means that the default
                   dimension is the one along which the elements of the
                   vector  values  are aligned.
                   If omitted, the default value is  [].

 weights           ::nonnegative::
                   Vector, matrix or multi-dimensional array of nonnegative
                   numbers representing the weights to be associeted to the
                   corresponding elements of  values .
                   If  weights  and  values  do not have the same size, they
                   are expected to be instances of flats (linear manifolds)
                   suitable to be combined within a bsxfun(...) call.
                   If  weights  is an empty array  [], then it is considered
                   as an array of ones with the same size as  values .
                   If omitted, the default value is  [].

Example of usage

   % Basic usage

   % Vectors:
   v   = ceil( rand(1,7)* 100 )
   wm  = wmedian( v )
   assert( wm == median(v) )
   w   = bsxfun( @power, 1:7, [0:4].' );
   w   = [w(end:-1:2,end:-1:1); w]
   wm  = wmedian( v,2,w )

   % Verifying the definition of weighted median
   def = @(v,w,x)abs( bsxfun( @minus, v(:).', x(:) ) )*w(:)
   x   = [1:100].';
   hold off
   for i=1:size(w,1) 
      wi = w(i,:).';
      mi = abs( v - wm(i) ) * wi;
      semilogy( x, def(v,wi,x), wm(i), mi , 'or');
      text( wm(i), mi*.8, sprintf( 'w( %d, : )', i ) )
      hold on;
   end; hold off

   % Matrices:
   v   = ceil( rand(5,7)* 100 )
   wm  = wmedian( v )
   assert( wm == median(v) )
   

   % Passing a custom dimension
   wm  = wmedian( v , 2 )
   assert( wm == median(v,2) )

   % Dealing with multi-dimensional arrays
   v   = ceil( rand(5,7,3)* 100 )
   wm  = wmedian( v , 1 )
   assert( wm == median(v,1) )

   wm  = wmedian( v , 2 )
   assert( wm == median(v,2) )
   

Memory requirements:
   O( numel( bsxfun( @plus,  values ,  weights  ) ) )



See also:
   cumstd, cumvar, cumsumsq, groupfun



Keywords:
   weighted operators, reduction



Version: 0.5.5

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