The idea of a vector space can be extended to include objects that you would not initially consider to be ordinary vectors. Matrix spaces. Consider the set M_(2x3)(R)\mathbb{M}_{2x3}(\mathbb{R}) of 2 by 3 matrices with real entries. This set is closed under addition, since the sum of a pair of 2 by 3 matrices is again a 2 by 3 matrix, and when such a matrix is multiplied by a real scalar, the resulting matrix is in the set also. Since M_(2x3)(R)\mathbb{M}_{2x3}(\mathbb{R}), with the usual algebraic operations, is closed under addition and scalar multiplication, it is a real Euclidean vector space. The objects in the vector space, the vectors, are now matrices. This proof is trivial and it is left as an exercise for the reader.
Since M_(2x3)(R)\mathbb{M}_{2x3}(\mathbb{R}) is a vector space, what is its dimension? First, note that any 2 by 3 matrix is a unique linear combination of the following six matrices:
Therefore, they span M_(2x3)(R)\mathbb{M}_{2x3}(\mathbb{R}).
Furthermore, these elements of the vector space are linearly independent: none of these matrices is a linear combination of the others. (ie, the only way k_(1)E_(1)+k_(2)E_(2)+k_(3)E_(3)+k_(4)E_(4)+k_(5)E_(5)+k_(6)E_(6)k_1 E_1 + k_2 E_2 + k_3 E_3 + k_4 E_4 + k_5 E_5 + k_6 E_6 will give the 2 by 3 zero matrix is if each scalar coefficient, k_(i)k_i , in this combination is zero) These six elemments therefore form a basis for M_(2x3)(R)\mathbb{M}_{2x3}(\mathbb{R}), :.dim(M_(2x3)(R))=6\therefore\dim(\mathbb{M}_{2x3}(\mathbb{R}))=6
If the entries in a given 2 by 3 matrix are written out in a single row (or column), the result is a vector in R^(6)\mathbb{R}^6. For example,
The rule here is simple: Given a 2 by 3 matrix, form a 6-vector by writing the entries in the first row of the matrix followed by the entries in the second row. Then, to every matrix in M_(2x3)(R)\mathbb{M}_{2x3}(\mathbb{R}) there corresponds a unique vector in R^(6)\mathbb{R}^6, and vice versa. This one-to-one correspondence:
{:[M_(2x3)(R)longrightarrow^(varphi)R^(6)],[],[[[a,b,c],[d,e,f]]{:[longrightarrow^(varphi)],[longleftarrow^(varphi^(-1))]:}(a","b","c","d","e","f)]:}\begin{array}{c}
\mathbb{M}_{2x3}(\mathbb{R}) \stackrel{\varphi}{\longrightarrow}\mathbb{R}^6 \\
\\
\left[\begin{array}{lll}
a & b & c \\
d & e & f
\end{array}\right]
\begin{array}{c}
\stackrel{\varphi}{\longrightarrow}\\
{\stackrel{\varphi^{-1}}{\longleftarrow}}
\end{array}(a, b, c, d, e, f)
\end{array}
is compatible with the vector space operations of addition and scalar multiplication. This means that
varphi(k xx A)=k xx varphi(A)\varphi(k \times A)=k\times \varphi(A)
The conclusion is that the spaces M_(2x3)(R)\mathbb{M}_{2x3}(\mathbb{R}) and R^(6)\mathbb{R}^6 are structurally identical, that is, isomorphic, a fact which is denoted M_(2x3)(R)~=R^(6)\mathbb{M}_{2x3}(\mathbb{R}) \cong \mathbb{R}^6.
One consequence of this structural identity is that each basis of the vector space E_(i)E_i given above for M_(2x3)(R)\mathbb{M}_{2x3}(\mathbb{R}) corresponds to the standard basis vector e_(i)e_i
for R^(6)\mathbb{R}^6.
The only real difference between the spaces R^(6)\mathbb{R}^6 and M_(2x3)(R)\mathbb{M}_{2x3}(\mathbb{R}) is in the notation: The six entries denoting an element in R^(6)\mathbb{R}^6 are written as a single row (or
column), while the six entries denoting an element in M_(2x3)(R)\mathbb{M}_{2x3}(\mathbb{R}) are written in two rows of three entries each.
This example can be generalized further.
Group Equivariant Convolutional Networks in Medical Image Analysis
Group Equivariant Convolutional Networks in Medical Image Analysis
This is a brief review of G-CNNs' applications in medical image analysis, including fundamental knowledge of group equivariant convolutional networks, and applications in medical images' classification and segmentation.