Image Convolution  for meta-data

2021 James B. Wilson, Colorado State University
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      * * * * * * *                           - +           + -
    * * * * * * * * *                       - +               + -
    * * * * * * * * *                       - +               + -
    * * * * * * * * *                       - +               + -
    * * * * * * * * *                       - +               + -
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Convolution

Original Goals:

  • sharpen an image,
  • find edges,
  • find textures, etc.

In the past it was for "photo-shop" (trade mark?).

 

Today its image feature extraction to give to machine learning.

1 1 1		  
0 _ 0    -->  3
0 0 0
\begin{aligned} \begin{bmatrix} 1 & 1 & 1\\ 0 & 0 & 0\\ -1 & -1 & -1 \end{bmatrix} & \boxtimes \begin{bmatrix} a & b & c\\ d & e & f\\ g & h & i\\ \end{bmatrix} \to \begin{bmatrix} 1\cdot a & 1\cdot b & 1\cdot c\\ 0\cdot d & 0\cdot e & 0\cdot f\\ -1\cdot g & -1\cdot h & -1\cdot i\\ \end{bmatrix} \\ & \to a+b+c-g-h-i \end{aligned}

Image Convolution

0 0 0		  
0 _ 0    -->  -3
1 1 1
0 0 1		  
0 _ 0    -->  0
1 0 0
1 1 1		  
1 _ 0    -->  2
1 0 0
0 0 1		  
0 _ 1    -->  -2
1 1 1
1 0 1		  
1 _ 1    -->  0
1 0 1
1 1 1		  
1 _ 1    -->  0
1 1 1
1 0 0		  
0 _ 0    -->  1
0 0 0
0 0 0		  
0 _ 0    -->  -1
0 1 0
\begin{aligned} \begin{bmatrix} a & b & c & d\\ e & f & g & h\\ i & j & k & l\\ \end{bmatrix} & \to \left[ \begin{bmatrix} a & b & c \\ e & f & g \\ i & j & k \\ \end{bmatrix} , \begin{bmatrix} b & c & d\\ f & g & h\\ j & k & l\\ \end{bmatrix} \right]\\ & \to \left[ \begin{bmatrix} 1 & 1 & 1 \\ 0 & 0 & 0 \\ -1 & -1 &-1 \\ \end{bmatrix} \boxtimes \begin{bmatrix} a & b & c \\ e & f & g \\ i & j & k \\ \end{bmatrix} , \begin{bmatrix} 1 & 1 & 1 \\ 0 & 0 & 0 \\ -1 & -1 &-1 \\ \end{bmatrix} \boxtimes \begin{bmatrix} b & c & d\\ f & g & h\\ j & k & l\\ \end{bmatrix} \right]\\ \end{aligned}

Image Convolution

0 0 0 1        0 0 0     0 0 1 		  
0 0 0 1  -->   0 _ 0     0 _ 1  --> [ -3, -2 ]
1 1 1 1        1 1 1     1 1 1
1 1 0 0        1 1 0     1 0 0 		  
0 0 0 0  -->   0 _ 0     0 _ 0  --> [ 1, -1 ]
0 0 1 1        0 0 1     0 1 1
                                                + + + + + + +
        * * * * *                             + + + + + + + + +
      * * * * * * *                         + + + +       + + + +
    * * * * * * * * *                       + + +           + + +
    * * * * * * * * *
    * * * * * * * * *
    * * * * * * * * *
    * * * * * * * * *                       - - -           - - -
      * * * * * * *                         - - - -       - - - -
        * * * * *                             - - - - - - - - -
                                                - - - - - - -

Detect horizontal edges

Now working on the interior we can step through all the points and record if the answer is + or -.

 

Here are the results on a octagon.

                                                + +       - -
        * * * * *                             + + +       - - -
      * * * * * * *                         + + + +       - - - -
    * * * * * * * * *                       + + +           - - -
    * * * * * * * * *                       + +               - -
    * * * * * * * * *                       + +               - -
    * * * * * * * * *                       + +               - -
    * * * * * * * * *                       + + +           - - -
      * * * * * * *                         + + + +       - - - -
        * * * * *                             + + +       - - -
                                                + +       - -
\begin{bmatrix} 1 & 0 & -1\\ 1 & 0 & -1\\ 1 & 0 & -1 \end{bmatrix}

Convolutions detecting Verticality

Use a different mask, notice it detect something different, similar to the mask...

                                                  - - - - -
        * * * * *                               - + + + + + -
      * * * * * * *                           - +           + -
    * * * * * * * * *                       - +               + -
    * * * * * * * * *                       - +               + -
    * * * * * * * * *                       - +               + -
    * * * * * * * * *                       - +               + -
    * * * * * * * * *                       - +               + -
      * * * * * * *                           - +           + -
        * * * * *                               - + + + + + -
                                                  - - - - -
\begin{bmatrix} 0 & -1 & 0\\ -1 & 4 & -1\\ 0 & -1 & 0 \end{bmatrix}

All around edge detection?

Only near the edges do we find nonzeros.

Convolution Tensors

Inert Axis

Linear Axis

Image Meta Data

By James Wilson

Image Meta Data

Basic image convolution

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