Foundation of Image Processing-Inner Product and Dot Product

Defined in mathematics, dotproduct (scalarproduct)Scalar Product, Point product, and point Multiplication) Is a binary operation that accepts two vectors on the real number R and returns a real number scalar. It is the standard inner product of Euclidean space. Two vectors A = [a1, a2 ,..., An] and B = [b1, b2 ,..., The dot product of BN] is defined as a · B = a1b1 + A2B2 + ...... + Anbn uses matrix multiplication and regards the (column) vector as the N × 1 matrix. The dot product can also be written as a · B = a ^ t * B, here a ^ t indicates the transpose of matrix.

Inner Product, also known as scalar product and Dot Product)
It is a vector operation, but the result is a certain number, not a vector.
Set vector A = [a1, a2,... an], B = [B1, B2. .. bn]
The Inner Product of vectors A and B is expressed:
A · B =A1 × B1 + A2 × B2 + ...... + An × bn 
A · B = | A | × | B | × cos θ
| A | = (A1 ^ 2 + A2 ^ 2 +... + an ^ 2) ^ (1/2 );
| B | = (b1 ^ 2 + B2 ^ 2 +... + BN ^ 2) ^ (1/2 ).
Where, | A | and | B | are the modulus of vectors A and B respectively, which are the angle between θ vectors A and B (θ ε [0, π]).
If B is a unit vector, that is, when | B | = 1, A · B = | A | × cos θ indicates the projection length of vector A in the B direction.
The same is true when vector A is a unit vector.
When vector A is perpendicular to vector B, A · B = 0.

Exmple:

　　

Set ann = [AIJ] (1 <= I, j <= N), BNN = [bij] (1 <= I, j <= N );
The Inner Product of matrix A and B is c1n = [Σ (I = 1 to n summation) AIJ * bij] (where 1 <= I, j <= N ).
Note that c1n is a matrix of one row and n columns.
Examples of child matrices A and B are:
[1 2 3]
[4 5 6]
[7 8 9]
And
[9 8 7]
[6 5 4]
[3 2 1]
The inner product is:
[1*9 + 4*6 + 7*3 2*8 + 5*5 + 8*2 3*7 + 6*4 + 1*9] = [54 57 54]

Appendix:

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