de Rigo, D. (2012). Multi-dimensional weighted median: the module "wmedian" of the Mastrave modelling library. Mastrave project technical report. http://mastrave.org/doc/mtv_m/wmedian
Multi-dimensional weighted median: the module "wmedian" of the Mastrave modelling library
Abstract: Weighted median (WM) filtering is a well known technique for dealing with noisy images and a variety of
WM-based algorithms have been proposed as effective ways for reducing uncertainties or reconstructing degraded
signals by means of available information with heterogeneous reliability. Here a generalized module for applying
weighted median filtering to multi-dimensional arrays of information with associated multi-dimensional arrays of
corresponding weights is presented. Weights may be associated to single elements or to groups of elements along
given dimensions of the multi-dimensional arrays. The filtered information derives from a reduction operator
applied along a custom dimension.
Copyright © 2007,2008,2009,2010,2011 Daniele de Rigo
The file wmedian.m is part of Mastrave.
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answer = wmedian( values , dim =  , weights =  )
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.
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,
Free access version:
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 .
% 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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