method1

function [fittedR,fittedP] = method1(IT,dlMAT)
sol = cell(size(dlMAT,1),size(dlMAT,2));
for ind1 = 1:size(dlMAT,1)
  for ind2 = 1:size(dlMAT,2)
    sol{ind1,ind2} = polyfit(log(IT(:)),log(squeeze(dlMAT(ind1,ind2,:))),1);
  end
end

fittedR = cellfun(@(x)exp(x(2)),sol); %// Estimate of rMAT
fittedP = cellfun(@(x)x(1),sol);      %// Estimate of pMAT

method2

function [fittedR,fittedP] = method2(IT,dlMAT)
sol = cell(size(dlMAT,1),size(dlMAT,2));
for ind1 = 1:size(dlMAT,1)
  for ind2 = 1:size(dlMAT,2)
    sol{ind1,ind2} = flipud([ones(numel(IT),1) log(IT(:))]\log(squeeze(dlMAT(ind1,ind2,:)))).';
  end
end

fittedR = cellfun(@(x)exp(x(2)),sol); %// Estimate of rMAT
fittedP = cellfun(@(x)x(1),sol);      %// Estimate of pMAT

method3

function [fittedR,fittedP] = method3(IT,dlMAT)
N = 1; %// degree of polynomial
VM = bsxfun(@power, log(IT(:)), 0:N); %// Vandermonde matrix
result = fliplr((VM\log(reshape(dlMAT,[],size(dlMAT,3)).')).');

fittedR = exp(real(reshape(result(:,2),size(dlMAT,1),size(dlMAT,2))));
fittedP = real(reshape(result(:,1),size(dlMAT,1),size(dlMAT,2)));

regressionBenchmark

function regressionBenchmark(numEl)
clc
%// Define some nominal values:
R = 5;
IT = 600:600:3000;
P = 0.97;
%// Impose some believable spatial variations:
pMAT = 0.01*randn(numEl)+P;
rMAT = 0.1*randn(numEl)+R;
%// Generate "fake" measurement data using the relation "DL = R*IT.^P"
dlMAT = bsxfun(@times,rMAT,bsxfun(@power,permute(IT,[3,1,2]),pMAT));
%% Method1: loops + polyval
tic; [fR,fP] = method1(IT,dlMAT); toc;
fprintf(1,'Regression performance:\nR: %d\nP: %d\n',norm(fR-rMAT,1),norm(fP-pMAT,1));
%% Method2: loops + Vandermonde
tic; [fR,fP] = method2(IT,dlMAT); toc;
fprintf(1,'Regression performance:\nR: %d\nP: %d\n',norm(fR-rMAT,1),norm(fP-pMAT,1));
%% Method3: vectorized Vandermonde
tic; [fR,fP] = method3(IT,dlMAT); toc;
fprintf(1,'Regression performance:\nR: %d\nP: %d\n',norm(fR-rMAT,1),norm(fP-pMAT,1));