### Instruction: Suppose an image has pixel values of 50, 100, 150, and 200 arranged in a single row. Apply the Fourier transform to this image and find the amplitude of the Fourier coefficients for the frequency corresponding to the pixel values.### Response: To apply the Fourier transform to this image, we will use the Discrete Fourier Transform (DFT) formula, given that we have a discrete set of pixel values. The DFT formula is:

X(k) = Σ [x(n) * e^(-j * 2 * π * k * n / N)]

where X(k) is the k-th Fourier coefficient, x(n) is the n-th pixel value, N is the total number of pixels, and j is the imaginary unit (j^2 = -1).

In this case, we have 4 pixel values: 50, 100, 150, and 200. Thus, N = 4. We will calculate the Fourier coefficients for k = 0, 1, 2, and 3.

X(0) = Σ [x(n) * e^(-j * 2 * π * 0 * n / 4)]
     = 50 + 100 + 150 + 200
     = 500

X(1) = Σ [x(n) * e^(-j * 2 * π * 1 * n / 4)]
     = 50 * e^(0) - 100 * e^(-j * π / 2) - 150 * e^(-j * π) + 200 * e^(-j * 3 * π / 2)
     = 50 - 100j - 150 + 200j
     = -100 + 100j

X(2) = Σ [x(n) * e^(-j * 2 * π * 2 * n / 4)]
     = 50 * e^(0) + 100 * e^(-j * π) + 150 * e^(-j * 2 * π) + 200 * e^(-j * 3 * π)
     = 50 - 100 + 150 - 200
     = -100

X(3) = Σ [x(n) * e^(-j * 2 * π * 3 * n / 4)]
     = 50 * e^(0) - 100 * e^(-j * 3 * π / 2) - 150 * e^(-j * 3 * π) + 200 * e^(-j * 9 * π / 2)
     = 50 + 100j - 150 - 200j
     = -100 - 100j

Now, we will find the amplitude of the Fourier coefficients:

A(k) = |X(k)|

A(0) = |500| = 500
A(1) = |-100 + 100j| = sqrt((-100)^2 + (100)^2) = sqrt(20000) = 100 * sqrt(2)
A(2) = |-100| = 100
A(3) = |-100 - 100j| = sqrt((-100)^2 + (-100)^2) = sqrt(20000) = 100 * sqrt(2)

So, the amplitudes of the Fourier coefficients for the given pixel values are 500, 100 * sqrt(2), 100, and 100 * sqrt(2).