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从黎曼空间投影到切空间#xff0c;其中P为黎曼均值#xff0c;也是切空间的参考中心点#xff0c;Pi是要投影到切空间的点。 从切空间投影回来#xff0c;其中Si为切空间中的向量。 function Tcov CovToTan(cov,Mcov)Cm12 Mcov^(-1/2);X_new logm(Cm…
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从黎曼空间投影到切空间其中P为黎曼均值也是切空间的参考中心点Pi是要投影到切空间的点。 从切空间投影回来其中Si为切空间中的向量。 function Tcov CovToTan(cov,Mcov)Cm12 Mcov^(-1/2);X_new logm(Cm12 * cov * Cm12);C12 Mcov^(1/2);Tcov Mupper(C12 * X_new * C12);
endfunction Cov TanToCov(vec,Mcov)X Munupper(vec);Cm12 Mcov^(-1/2);X Cm12 * X * Cm12;C12 Mcov^(1/2);Cov C12 * expm(X) * C12;
endfunction T Mupper(X)% Upper triangular part vectorization with diagonal preservation.% This function keeps the upper triangular part of the matrix and% vectorizes it while multiplying non-diagonal elements by sqrt(2).% Get the size of X[M, N] size(X);% Check if matrices are squareif M ~ Nerror(Matrices must be square);end% Initialize T with zerosT zeros(M, M, like, X);% Calculate the multiplier for non-diagonal elementsmultiplier sqrt(2);% Fill T with the upper triangular part, preserving the diagonalfor i 1:Mfor j i:Mif i jT(i, j) X(i, j); % Diagonal element remains the sameelseT(i, j) X(i, j) * multiplier; % Non-diagonal elements multiplied by sqrt(2)endendend% Flatten the upper triangular part of T to a vectorT T(triu(true(size(T))) 1);T T;
endfunction X Munupper(T, n)% Reverse the operation to reconstruct the matrix from its upper triangular part.% Calculate the size of the square matrix based on the length of the input vector Tn round((sqrt(1 8 * length(T)) - 1) / 2);% Check if T is a valid upper triangular vectorm n * (n 1) / 2;if numel(T) ~ merror(Invalid input. Input vector size does not match the expected size for upper triangular vectors.);end% Initialize the symmetric matrix X with zerosX zeros(n, n, like, T);% Calculate the indices for the upper triangular part[I, J] find(triu(ones(n)));% Reverse the vectorization and apply the appropriate scaling to non-diagonal elementsfor k 1:numel(I)i I(k);j J(k);if i jX(i, j) T(k); % Diagonal elements remain the sameelseX(i, j) T(k) / sqrt(2); % Reverse scaling for non-diagonal elementsX(j, i) X(i, j); % Symmetric matrixendend
end
