Linear functions of images. In this problem we consider several linear functions

Question

Linear functions of images. In this problem we consider several linear functions of monochrome image with N x N pixels. To keep the matrices small enough to work o by hand, we will consider the case with N = 3 (which would hardly qualify as an image We represent a 3 x 3 image as a 9-vector using the ordering of pixels shown below (This ordering is called column-major.) Each of the operations or transformations below defines a function y = f(x), where the 9-vector z represents the original image, and the 9-vector y represents the resulting or transformed image. For each of these operations give the 9 x 9 matrix A for which y = Ax. (a) Turn the original image z upside-down. (b) Rotate the original image z clockwise 90° (c) Translate the image up by 1 pixel and to the right by 1 pixel. In the translated image, assign the value y 0 to the pixels in the first column and the last row (d) Set each pixel value y, to be the average of the neighbors of pixel i in the original image. By neighbors, we mean the pixels immediately above and below, and imme diately to the left and right. The center pixel has 4 neighbors; corner pixels have 2 neighbors, and the remaining pixels have 3 neighbors.

Explanation / Answer

The Original 3x3 image, as 9-vector using ordering of pixel is

1 4    7

2      5   8

3      6      9

This above is image is original image (x).

Now, let us solve (a)

(a) Turn the original image x upside-down

Here, the original 9-vector image x will turn upside-down as following,

It is image y

So, y= f(x)= Ax

So, transformed image y is below...

9    6      3

8     5     2

7      4     1

This above transformed image y is upside-down.

Let us now solve (b)

(b) Rotate the original image x clockwise 90°

Here, the original 9-vector image x will will turn clockwise 90°

as follows.

It is y

So, y = f(x) = Ax

So, transformed image y is below....

8 6    3

8        5        2

7        4       1

This above transformed image y is rotated clockwise 90°.

Let us now solve (c)

(c) [ See description of (c) in question....... ]

Up by 1 pixel and right by 1 pixel...

Here, the original 9- Vector image x will turn upside 1 pixel and right by 1 pixel as follows...

It is image y

So, y= f(x)= Ax

So, transformed image y is below....

2    3    4

5    6    7

8    9    10

Comparing with original x this is 1 pixel up and one pixel right.

Let us see the pixel 5 in original x is at middle

but here in transformed y, it is 6, so it is one pixel up

Also, in transformed y the A12 is 3,

where as in original x it was 2, it is one pixel right also.

Also, in the transform

Now, let us solve, (d)

(d)...[ See detail in question]

Here, the original 9-vector image will transformed with average of neighbour as follows.

Let it will be 'Bij'

The original was 'Aij'

If A11= 1 then B11= (4+ 2)/2 = 3

If A12= 4 then B12= (7+1+5)/3 = 13/3

If A13= 7 then B13= (4+8)/2 = 6

Similarly, B21= 3, B22= 5, B23= 7,

B31= 4, B32=17/3, B33= 7

So transformed y is as follows...

3    13/3    6

3   5         7

4     17/3     7

This above transformed image y is required solution.

Now, let us solve for

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