The following are the questions:

1. Implement down-sampling with anti-aliasing using the Gaussian LPF (hint: reduce the house grayscale image four times, first by applying a Gaussian filter and then by filtering every other row and column. Compare the output images with and without pre-processing with LPF before down-sampling). 
2. Use the FFT to up-sample an image: first double the size of the lena grayscale image by padding zero rows/columns at every alternate positions, then use the FFT followed by an LPF and then by the IFFT to get the output image. Why does it work?
3. Try to apply the Fourier transform and image reconstruction with a color (RGB) image. (Hint: apply the FFT for each channel separately).
4. Show (mathematically and with a 2D kernel example) that the Fourier transform of a Gaussian kernel is another Gaussian kernel.

5. Use the lena image and the asymmetric ripple kernel to generate images with correlation and convolution. Show that output images are different. Now, flip the kernel twice (upside-down and left-right) and apply the correlation with the flipped kernel—is the output image the same as the one obtained using the original kernel with convolution?