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Here is a crude Matlab code. Can you test i, and if OK, i'll send a one-liner (if I can).

nRow = 8;
nCol = 8;
HalfSizeCentralKernel = 1;
x = zeros(nRow,nCol);
x(nRow/2-HalfSizeCentralKernel:nRow/2+HalfSizeCentralKernel,nCol/2-HalfSizeCentralKernel:nCol/2+HalfSizeCentralKernel)=rand(2*HalfSizeCentralKernel+1);
figure(1);
imagesc(x);

% 2D version
s1=[1,0,-1]';
s2=[1 2 1];
y = s1*s2;
diff1=conv2(x,y,'same')-conv2(conv2(x,s1,'same'),s2,'same');
disp(['Max error 1: ',num2str(max(abs(diff1(:))))]);

% 1D version
xRowFlat1 = x(:);
xRowFlat1FiltCol = conv(xRowFlat1,s1,'same');
xRowFlat2 = (reshape(xRowFlat1FiltCol,nRow,nCol))';
xRowFlat2 = xRowFlat2(:);
xRowFlat2FiltRow = conv(xRowFlat2,s2,'same');
xRowFlatFilt2Row = reshape(xRowFlat2FiltRow,nRow,nCol)';

figure(2);
subplot(1,2,1)
imagesc([conv2(x,y,'same')])
subplot(1,2,2)
imagesc(xRowFlatFilt2Row)