Find minimum distance columnwise

Hi,

I have two matrices TRX and TRY [M x N] with cartesian coordinates. The coordinates are coordinates of an object which is moving by time (increasing N = timesteps) and is able to split or merge (M). I wanna calculate the shortest track through this points in direction N (from timestep to timestep). But notice, I wanna find the minimum track which means the points are depending on each other columnwise.

Example

TRX=[1 10 15 20; NaN 5 10 NaN; NaN NaN 12 NaN];

TRY=[5 15 20 25; NaN 7 19 NaN; NaN NaN 23 NaN];

I'm searching for the shortest distance from TRX(1,1) to TRX(1,end) and (TRY ofc). It should be taken into account that the shortest way between timestep 1 (first column) and timestep 2 (second column) is from TRX(1,1) to TRX(2,2) (and TRY). Thus the next way can only start from TRX(2,2)|TRY(2,2) and not be e.g. from TRX(1,2)Y(1,2) to TRX(:,3)TRY(:,3)

Is there any option to get this with a fast code? It is a kind of "find minimum over all distance by taking partial distances and connected neighbors into account"? I first tried nchoosk to get the hole bunch of variations but that won't work if the matrix increases in M and/or N direction due to to many possible combinations.

Now I calculate distances with

for i=1:size(TRX,2)-1
    
    Xts_1=TRX(:,i+1);
    Xts_0=TRX(:,i);
    diffX(:,1)=reshape(bsxfun(@minus, permute(Xts_0,[1,3,2]), permute(Xts_1,[3,1,2])),[],1);
    Yts_1=TRY(:,i+1);
    Yts_0=TRY(:,i);
    diffY(:,1)=reshape(bsxfun(@minus, permute(Yts_0,[1,3,2]), permute(Yts_1,[3,1,2])),[],1);
    trackdist=sqrt(diffX.^2+diffY.^2);
    
    dist(1:length(find(~isnan(trackdist)==1)),i)=trackdist(~isnan(trackdist));

    clear Xts_1 Xts_0 diffX diffY Yts_1 Yts_0 
end

Now I got the distances dist between each points at any connected time steps (N), but from now I struggle how to finde the best track on mentioned conditions.

At the end I wanna return the TRX and TRY coordinates for a "best track" including the indices for this coordinates.

Looking forward for any help, hints or tricks!

Thank and best regards

Manuel