shape approximation based on sample points and decomposition data
Syntax: dataApprox = approximateDec(hcMatrix, dec, 'option', value, ...)
This function approximates the points in heMatrix (one sample point each
column) by an orthogonal projection to a linear subspace of the given
Hc-representation (dec). The approximated hE vectors are computed by:
heApprox = P * hE
where the projection matrix is computed from
P = baseMatrix * baseMatrix'.
The base matrix contains nTargetDim columns that span the desired
linear subspace in the embedding hE-space. Both results are returned in
dataApprox as the fields
.projectionMatrix - this is P
.mappingReducedToEmbedding - this is baseMatrix
.mappingEmbeddingToreduced - this is pinv(baseMatrix) or simply
the baseMatrix' since the base vectors are orthonormal
The method of the approximation is determined by:
approximateDec(.., 'method', method)
where method='PCA' uses principal component analysis (a quick first
result), 'nonlinear' uses a global nonlinear optimization (usually the
thing you wan't to do) and 'nonlinearLocal' finds only one local
optimum, starting from the PCA optimum.
For the nonlinear approach, you can specify options a string cell array
of measures that shalle be consideres ('measureList', {'volume',
'meanWidth'}). A corresponding vector assigns the weights
('measureWeightVector', [1 0]).0001 function dataApprox = approximateDec(heMatrix, dec, varargin) 0002 % shape approximation based on sample points and decomposition data 0003 % 0004 % Syntax: dataApprox = approximateDec(hcMatrix, dec, 'option', value, ...) 0005 % 0006 % This function approximates the points in heMatrix (one sample point each 0007 % column) by an orthogonal projection to a linear subspace of the given 0008 % Hc-representation (dec). The approximated hE vectors are computed by: 0009 % heApprox = P * hE 0010 % where the projection matrix is computed from 0011 % P = baseMatrix * baseMatrix'. 0012 % The base matrix contains nTargetDim columns that span the desired 0013 % linear subspace in the embedding hE-space. Both results are returned in 0014 % dataApprox as the fields 0015 % .projectionMatrix - this is P 0016 % .mappingReducedToEmbedding - this is baseMatrix 0017 % .mappingEmbeddingToreduced - this is pinv(baseMatrix) or simply 0018 % the baseMatrix' since the base vectors are orthonormal 0019 % 0020 % The method of the approximation is determined by: 0021 % approximateDec(.., 'method', method) 0022 % where method='PCA' uses principal component analysis (a quick first 0023 % result), 'nonlinear' uses a global nonlinear optimization (usually the 0024 % thing you wan't to do) and 'nonlinearLocal' finds only one local 0025 % optimum, starting from the PCA optimum. 0026 % 0027 % For the nonlinear approach, you can specify options a string cell array 0028 % of measures that shalle be consideres ('measureList', {'volume', 0029 % 'meanWidth'}). A corresponding vector assigns the weights 0030 % ('measureWeightVector', [1 0]). 0031 % 0032 0033 %! ToDo: rewrite documentation 0034 0035 0036 % 0037 % The output structure contains 0038 % .time - computation time for approximation and conversion 0039 % .optimalGrassmanVector - defines the optimal projection subspace based 0040 % on the PCA subspace - different optimal projections might be compared 0041 % according to this data 0042 %/ 0043 % 0044 % 0045 % see also: obtainDec, obtainCrystalDec, addMeasureDec 0046 0047 % The elk-library: convex geometry applied to crystallization modeling. 0048 % Copyright (C) 2013 Alexander Reinhold 0049 % 0050 % This program is free software: you can redistribute it and/or modify it 0051 % under the terms of the GNU General Public License as published by the 0052 % Free Software Foundation, either version 3 of the License, or (at your 0053 % option) any later version. 0054 % 0055 % This program is distributed in the hope that it will be useful, but 0056 % WITHOUT ANY WARRANTY; without even the implied warranty of 0057 % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU 0058 % General Public License for more details. 0059 % 0060 % You should have received a copy of the GNU General Public License along 0061 % with this program. If not, see <http://www.gnu.org/licenses/>. 0062 0063 %% input handling 0064 par = mapOptionStruct(varargin, ... 0065 'method', 'PCA', ... 0066 'targetDim', 1, ... 0067 'nResample', 0, ... 0068 'measureList', 'volume', ... 0069 'measureWeightVector', [], ... 0070 'errorType', 'absoluteSquared', ... 0071 'errorGroupVector', [], ... 0072 'projectionType', 'true', ... 0073 'manualScaling', 0.5, ... 0074 'manualBoundingCone', [], ... 0075 'numTrialPoints', 500, ... 0076 'numStageOnePoints', 10, ... 0077 'zerotol', elkZerotol); 0078 0079 %% Scaling / Resampling 0080 if par.nResample > 0 0081 disp('Reducing set of sample points..') 0082 resampleData = resampleHeMatrix(heMatrix, par.nResample); 0083 heMatrix = resampleData.heMatrixResampled; 0084 disp(' done.') 0085 end 0086 0087 par = validateInput(heMatrix, dec, par); 0088 0089 % add measure values for original hE-vectors 0090 % measure value matrix 0091 par.measureValueMatrix = computeMeasureDec(dec, 'hC', heMatrix, ... 0092 par.measureList, par.zerotol, 0, 1); 0093 0094 %% handle approximation 0095 tic 0096 switch lower(par.method) 0097 case 'pca' 0098 dataApprox = approxPca(heMatrix, par); 0099 case 'nonlinear' 0100 dataApprox = approxNonlinear(heMatrix, dec, par); 0101 case 'nonlinearlocal' 0102 % call fmincon instead of global search based on fmincon to provide 0103 % a local minimum (right thig to use this for testing) 0104 dataApprox = approxNonlinear(heMatrix, dec, par); 0105 otherwise 0106 error('elk:approximation', 'This should not happen.'); 0107 end 0108 0109 % dataApprox.mappingEmbeddingToReduced = orth(dataApprox.projectionMatrix)'; 0110 % dataApprox.mappingReducedToFull = pinv(dataApprox.mappingFullToReduced); 0111 dataApprox.heMatrix = heMatrix; 0112 if strcmpi(par.projectionType, 'true') 0113 dataApprox.heMatrixApprox = dataApprox.projectionMatrix*heMatrix; 0114 else 0115 dataApprox.heMatrixApprox = ... 0116 estimateHeMatrix(heMatrix, dataApprox.projectionMatrix, ... 0117 dataApprox.nonlinearData.sampleProjectionData); 0118 end 0119 dataApprox.parameterStruct = par; 0120 if exist('resampleData', 'var') 0121 dataApprox.resampleData = resampleData; 0122 end 0123 dataApprox.time = toc;