One dimensional sieve applied to images
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'siv4.mex' implemenation applies the m-sieve to a vector or column wise to a matrix
A Matlab function siv4_test.m illustrates how siv4.mex can be used to analyse columns of 1D data.
Consider a signal, <math>X</math>X=getData('PULSES3WIDE') >blue X=0 5 5 0 0 1 1 4 3 3 2 2 1 2 2 2 1 0 0 0 1 1 0 3 2 0 0 0 6 0 0 |
The data has minima and maxima of different scales (lengths). In one dimension we measure pulse length using a ruler, measuring tape or whatever - but not frequency or Gaussian scale. |
Filter
Lowpass siv4.mex
data{1}=siv4_alt('PULSES3WIDE',[2;5;10]); data{1} ans = y: {[34x1 double] [34x1 double] [34x1 double]} % outputs for the 3 specified scales scan: [34 34] % instructing single column processing X: [34x1 double] % input data options: [3x4 double] % options (see elsewhere) outputs: 'lll' % outputs all lowpass type: 'int' % input data may be double but only contains integers name: 'PULSES3WIDE'
Now think about scale-space.
scaleA=1; Y1=SIVND_m(X,scaleA,'o');
scaleB=2; Y2=SIVND_m(X,scaleB,'o');
red=double(X)-double(Y1); green=double(Y1)-double(Y2);
Repeat over scales 0 to 15
Increasing the scale (towards the front) removes extrema of increasing length. The algorithm cannot create new maxima (it is an 'o' sieve) it is, therefore, scale-space preserving. |
YY=ones([length(X),1+maxscale]);
for scale=0:maxscale
Y2=SIVND_m(Y1,scale,'o',1,'l',4);
YY(:,scale+1)=Y2';
Y1=Y2; % each stage of the filter (sieve) is idempotent
end
Label the granules
g=SIVND_m(X,maxscale,'o',1,'g',4); g = Number: 10 area: [1 1 1 2 2 2 3 3 5 12] value: [6 1 1 2 5 1 1 1 1 1] level: [6 4 3 2 5 1 3 2 2 1] deltaArea: [5 2 1 7 3 12 2 2 7 19] last_area: [6 3 2 9 5 14 5 5 12 31] root: [29 8 24 24 2 21 8 14 8 8] PictureElement: {1x10 cell}
g.PictureElement
Columns 1 through 9
[29] [8] [24] [2x1 double] [2x1 double] [2x1 double] [3x1 double] [3x1 double] [5x1 double] [12x1 double]
Tracing the granules through scale-space identifies candidate MSER's
We have candidate 1D MSER's
Which is the most stable?
This is a pragmatic judgement. Parameters might include
- how stable over scale (length)
- amplitude (value or level)
- a vector of amplitude over scale
- proximity to others
So far maxima. What about minima and more?
The filter (sieve) that finds maxima is a connect-set opening ('o' sieve). A 'c' sieve finds the connected-set closing, or minima. To work with minima we could:
- invert the signal, process it, and invert it back.
- OR, in this case, we could substitute a min for a max within SIVND_m.
YY=ones([length(X),1+maxscale]); for scale=0:maxscale Y2=SIVND_m(X,scale,'c',1,'l',4); YY(:,scale+1)=Y2'; Y1=Y2; % each stage of the filter (sieve) is idempotent end g=SIVND_m(X,maxscale,'c',1,'g',4);
This implementation also maintains lists of both maxima and minima throughout because there can be value in using the combined operators M, N, m
switch type case {'o'} % opening, merge all maximal runs of less than scale with their nearest value data=ND_connected_set_merging(data,scale,type,verbose); case {'c'} % closing, merge all minima runs of less than scale with their nearest value data=ND_connected_set_merging(data,scale,type,verbose); case {'C'} % closing, invert-open-invert data.workArray=uint8(-double(data.workArray)+256); data.value=uint8(-double(data.value)+256); data=ND_connected_set_merging(data,scale,'o',verbose); data.workArray=uint8(-double(data.workArray)+256); data.value=uint8(-double(data.value)+256); case 'M' % Open close data=ND_connected_set_merging(data,scale,'o',verbose); data=ND_connected_set_merging(data,scale,'c',verbose); case 'N' % Close open data=ND_connected_set_merging(data,scale,'c',verbose); data=ND_connected_set_merging(data,scale,'o',verbose); case 'm' % recursive median data=ND_connected_set_rmedian(data,scale,'m',verbose); otherwise error('type not recognised it should be (m, o, c, C, M or N)'); end