How is the 2D filter function used in this code?

Question

Daigo 2022-1-13

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1628475-how-is-the-2d-filter-function-used-in-this-code

评论： Daigo 2022-1-14

In one of the cody answers (solution 712713), I found a following code:

function B = hanlelize(A)
    F = flip(eye(111)); 
    B1 = filter2(F,A); 
    B2 = filter2(F,A|1); 
    B = B1./B2; 
end

This function convert an input matrix A to a Hankel matrix B by replacing each skew-diagonal of A with its mean. For example, if the input is

A =
   7    10     2
   5     1     2
   3     2     7

then, the matrix B1, B2, B will be

B1 =
  10    21     6
  21     6     4
   6     4     7
    
B2 =
   2     3     3
   3     3     2
   3     2     1
B =
   5     7     2
   7     2     2
   2     2     7

The code looks elegant but I'm not so familiar with the 2D filter function used in this code.

How is the 2D filter working in the computation of B1? Why does the 2D filter by flipped identity matrix result in B1?
How is the 2D filter working in the computation of B2? What does 'A|1' mean? I checked the documentation of filter2 but I couldn't find such an expression.

I'm looking forward to your help.

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Answer 1

Steve Eddins 2022-1-13

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1628475-how-is-the-2d-filter-function-used-in-this-code#answer_874180

编辑：Steve Eddins 2022-1-13

在 MATLAB Online 中打开

For each element in A, filter2 essentially "slides" the filter F so that its center lines up with the element in A, and then it does a point-wise multiplication between the elements in A and the corresponding elements in F, and then sums all those products. With a filter that is just ones along the anti-diagonal, each element of the output matrix is found by summing up the input matrix elements along the corresponding anti-diagonal. Eh, it's a bit hard to describe in words. Let's try it with a small A and small F.

A = magic(5)
A = 5×5
    17    24     1     8    15
    23     5     7    14    16
     4     6    13    20    22
    10    12    19    21     3
    11    18    25     2     9
F = flip(eye(3))
F = 3×3
     0     0     1
     0     1     0
     1     0     0
B1 = filter2(F,A)
B1 = 5×5
    17    47     6    15    29
    47    10    21    42    36
     9    23    39    55    43
    16    36    57    68     5
    23    37    46     5     9

Where does B1(4,2) come from?

B1(4,2)
ans = 36

It is the sum of A(4,2), the element to its upper right, A(3,3), and the element to its lower left, A(5,1).

A(4,2) + A(3,3) + A(5,1)
ans = 36

In the expression A|1, the vertical bar is the element-wise OR operator. When you OR any value with 1, you always get 1, so this expression appears to be just a tricky way to compute true(size(A)). In Cody, solutions with a smaller code size (based on some metric) are scored higher, so perhaps A|1 had a slightly smaller code size metric than true(size(A)).