3 Fourier Transforms and Interpolation In one of your programming projects, you will be asked to interpolate an image using a Fourier Transform. The general procedure involves (1) Fourier Transforming a signal (2) "padding" the signal with zero values, which makes the array longer while not adding any contribution from higher frequencies and (3) inverting the transform. Alternatively, we can interpolate our discrete data using the normalized sinc function', defined as: sin() sinc(x) = Y and use it as a kernel for upsampling the image. If the output image is the same size at the input image, we would expect that the image is unchanged by "interpolation. Does the normalized sine function satisfy this property? We have discussed in class the relationship between Fourier Transforms and filtering. Using a Fourier Transform for upsampling results in the same output as interpolating using the normalized sinc function, a result related As to the Whittaker-Shannon Interpolation Formula. The sinc function is the Fourier Transform of a rectangular function, which we have seen in the form of a low-pass filter in frequency space. Using this fact, explain (in words) why it makes sense that the two of these would be equivalent? Hint: What would happen if you took a high-resolution image, applied a low-pass filter, and then downsampled the image? 3 Fourier Transforms and Interpolation In one of your programming projects, you will be asked to interpolate an image using a Fourier Transform. The general procedure involves (1) Fourier Transforming a signal (2) "padding" the signal with zero values, which makes the array longer while not adding any contribution from higher frequencies and (3) inverting the transform. Alternatively, we can interpolate our discrete data using the normalized sinc function', defined as: sin() sinc(x) = Y and use it as a kernel for upsampling the image. If the output image is the same size at the input image, we would expect that the image is unchanged by "interpolation. Does the normalized sine function satisfy this property? We have discussed in class the relationship between Fourier Transforms and filtering. Using a Fourier Transform for upsampling results in the same output as interpolating using the normalized sinc function, a result related As to the Whittaker-Shannon Interpolation Formula. The sinc function is the Fourier Transform of a rectangular function, which we have seen in the form of a low-pass filter in frequency space. Using this fact, explain (in words) why it makes sense that the two of these would be equivalent? Hint: What would happen if you took a high-resolution image, applied a low-pass filter, and then downsampled the image