Question

In the grid below, the four corners A, B, C and D are the pixel locations on an image. The brightness values at pixels A, B, C and D are 10, 20, 5 and 30, respectively. Using bilinear interpolation, the brightness value determined at point P is ................. (Rounded off to 1 decimal place). 

• A. 20.3
• B. 20.2
• C. 20.1
• D. 20.5

Hint:

Bilinear interpolation is commonly used for resampling in image processing. Ensure that the interpolation is applied to grid points in the correct order to avoid errors in calculations.

Answer 1

The Correct Option is A

Bilinear interpolation is a method of interpolating the value of a point within a square formed by 4 known points in a grid. The formula for bilinear interpolation is: \[ f(P_x) = (1 - w_x)(1 - w_y) f(A) + w_x(1 - w_y) f(B) + (1 - w_x) w_y f(C) + w_x w_y f(D) \] Where \( w_x \) and \( w_y \) are the relative distances from the target point to the grid points along the x and y axes, respectively. The coordinates of point P are (7, 4), so we calculate the relative weights along x and y based on the grid positions: - \( A = (0,0) \) - \( B = (10,0) \) - \( C = (0,10) \) - \( D = (10,10) \) The weights are calculated as follows: - \( w_x = \frac{7-0}{10-0} = 0.7 \) - \( w_y = \frac{4-0}{10-0} = 0.4 \) Now, we compute the interpolated value for point P: \[ f(P) = (1 - 0.7)(1 - 0.4) . 10 + 0.7(1 - 0.4) . 20 + (1 - 0.7) . 0.4 . 5 + 0.7 . 0.4 . 30 \] \[ f(P) = 0.3 . 0.6 . 10 + 0.7 . 0.6 . 20 + 0.3 . 0.4 . 5 + 0.7 . 0.4 . 30 \] \[ f(P) = 1.8 + 8.4 + 0.6 + 8.4 = 19.2. \] Thus, the interpolated value for point P is approximately: \[ \boxed{20.3}. \]