The most efficient pairwise distance computation

Enhannced `pdist2`! Vectorized code that achieve 10x~100x efficienficy for nD-array (i.g., i-by-j-by-k). Support many distance metrics.

In human motion analysis, a commond need is the computation of the distance between defferent point sets.

The builtin function `pdist2` could be helpful, but it is inefficient for multiple timeframes data.

Currently avaliable codes in FEX are also insufficient as they can only compute (squared) Euclidean distance and they still cannot handle nD-array.

Therefore, I created this handy function `markerdistance` to enhance `pdist2`.

My code is vectorized to achieve 10x~100x efficienficy for nD-array (i.g., i-by-j-by-k).

Additionally, `markerdistance` supports many different distance metrics, including 'euclidean', 'cityblock', 'chebychev', 'minkowski', 'cosine', 'correlation', 'mahalanobis', 'chisquare', and 'Earth's mover' distance.

Note1.

This function heavly depends on the calculation of vector-wise norm.

If you don't have the builtin `vecnorm` that was introduced in R2017b, use local subfunction `ndnorm` as an alternative (all you need to do is simply replace `vecnorm` with `ndnorm`).

Note2.

Computation of Mahalanobis distance requires `pagetranspose`, `pagemtimes`, and 'pageinv'.

Note3.

I don't really know much about the measurements above as I am not working at these research fields.

All I did is vectorize the codes for nD-array. Don't blame on me if the results are not what you expected.

The following code is for demo.

metric = 'euclidean'; % euclidean, cityblock, chebychev, cosine, correlation, mahalanobis
Xn = randi([1, 10], 1);
Yn = randi([1, 10], 1);
d = 3;
slice = 123456;
X = rand(Xn, d, slice);
Y = rand(Yn, d, slice);
tic;
D1 = markerdistance(X, Y, metric);
t1 = toc

tic;
D2 = zeros(Xn, Yn, slice);
for pp2 = 1:slice
    D2(:, :, pp2) = pdist2(X(:, :, pp2), Y(:, :, pp2),  metric);
end
t2 = toc

efficiency = t2/t1
reserr = max(abs(D1(:)-D2(:)), [], 1)

if strncmpi(metric, 'e', 1)
    % `sqdist` could be found in FEX
    tic;
    D3 = zeros(Xn, Yn, slice);
    for pp3 = 1:slice
        D3(:, :, pp3) = sqdist(X(:, :, pp3).', Y(:, :, pp3).');
    end
    D3 = sqrt(D3);
    t3 = toc

    efficiency2 = t3/t1
    reserr2 = max(abs(D1(:)-D3(:)), [], 1)
    % Note. For large-scale data (e.g., Xn or Yn is large), `sqdist` might be better.
end

No more for-loop!

Enjoy!