In an 8-bit greyscale image, the 2 x 2 image region below: 64, 182, 142, 41 is to be rescaled to occupy an area of 8 x 8 pixels using bilinear interpolation. In this new area, what... In an 8-bit greyscale image, the 2 x 2 image region below: 64, 182, 142, 41 is to be rescaled to occupy an area of 8 x 8 pixels using bilinear interpolation. In this new area, what's the value of the pixel at position (7,6)? Round your answer to be in ℕ, rounding 0.5 upwards. Read more

Steps to Solve

1. Identify the original pixel values We have a 2x2 pixel region with pixel values as follows:

[ \begin{bmatrix} 64 & 182 \ 142 & 41 \end{bmatrix} ]

2. Determine the coordinates in the original image The coordinate (7,6) in an 8x8 pixel area corresponds to a position in the original 2x2 area. We can find the scaling factor:

• The original width and height are both 2, and the new width and height are both 8.
• Each pixel in the 2x2 region corresponds to $4$ pixels in the 8x8 region, hence the scale factor is $M = \frac{8}{2} = 4$.

3. Calculate the corresponding position in the original image To find the corresponding (i, j) position in the original area, we use:

[ i = \left\lfloor \frac{7}{4} \right\rfloor = 1, \quad j = \left\lfloor \frac{6}{4} \right\rfloor = 1 ]

So, the corresponding region is (1,1).

4. Use bilinear interpolation formula Based on the bilinear interpolation formula:

[ L_{i,j} = L_{1,1} \left(1 - \frac{M - (i-1)}{MN}\right) + L_{1,2} \left( \frac{M - (i-1)}{MN}\right) + L_{2,1} \left(1 - \frac{N - (j-1)}{MN}\right) + L_{2,2} \left(\frac{N - (j-1)}{MN}\right) ]

Where:

• $L_{1,1} = 64$, $L_{1,2} = 182$, $L_{2,1} = 142$, $L_{2,2} = 41$

5. Substitute values into the formula Now we can substitute the values:

• For (M = 4), (N = 4) (since we scaled from 2x2 to 8x8) and substituting values:

[ L_{7,6} = 64 \left( 1 - \frac{4 - (1-1)}{8} \right) + 182 \left( \frac{4 - (1-1)}{8} \right) + 142 \left( 1 - \frac{4 - (1-1)}{8} \right) + 41 \left( \frac{4 - (1-1)}{8} \right) ]

6. Calculate each term Calculate the individual terms:

• First term: $64(1 - \frac{4 - 0}{8}) = 64(1 - 0.5) = 32$
• Second term: $182(\frac{4 - 0}{8}) = 182(0.5) = 91$
• Third term: $142(1 - \frac{4 - 0}{8}) = 142(0.5) = 71$
• Fourth term: $41(\frac{4 - 0}{8}) = 41(0.5) = 20.5$

7. Sum up the values Now, sum them up:

[ L_{7,6} = 32 + 91 + 71 + 20.5 = 214.5 ]

8. Round the final result Round up $214.5$ to the nearest integer to get $215$.