Question

Task 1 - Explain Convolution Theorem: Conduct a research on the Convolution Theorem,

emphasizing its relationship between Fourier Transform (frequency domain filtering) and
Convolution (spatial domain filtering). Write a report in which you provide an explanation using
mathematical expressions to illustrate how convolution in the spatial domain relates to Fourier
Transform in the frequency domain.
Task 2 - Practical Application (Spatial to Frequency Domain Transformation): Given a spatial
domain filter h(x,y) defined as: [(-1,-1,-1),(-1,8,-1),(-1,-1,-1)]
Compute the equivalent frequency domain filter H(u,v) using the Fourier Transform. Use a
programming language to accomplish this task. You can use image processing tools / libraries.
Provide your codes in your report.
Hint: Note that spatial filter h(x,y) has non-zero elements in 3x3 grid but can be considered to have
all zeros outside. Therefore, consider expanding it by padding zeros to obtain a larger filter. Then,
try to compute Fourier Transform.
Task 3 - Visualization and Image Filtering: Visualize the spatial domain filter h(x,y) and
frequency domain filter H(u,v) as an image. Interpret the results and explain how those filters in
different domain have equivalent job. Provide all visuals in your report.

Answer 1

The Convolution Theorem explains the relationship between convolution in the spatial domain and Fourier Transform in the frequency domain.

It states that the Fourier Transform of the convolution of two functions in the spatial domain is equal to the product of their individual Fourier Transforms in the frequency domain.

To compute the equivalent frequency domain filter H(u,v) from a spatial domain filter, the filter needs to be padded with zeros and then the Fourier Transform is applied.

The spatial and frequency domain filters have equivalent functions in filtering and modifying images or signals.

Convolution Theorem Explanation

The Convolution Theorem is a mathematical principle that relates the process of convolution in the spatial domain to the Fourier Transform in the frequency domain.

It states that the Fourier Transform of the convolution of two functions in the spatial domain is equal to the product of their individual Fourier Transforms in the frequency domain.

To illustrate this relationship, consider two functions, f(x) and g(x), in the spatial domain.

The convolution of these functions, h(x), can be calculated using the following formula:

h(x) = ∫f(t)g(x-t)dt

On the other hand, the Fourier Transform of h(x), H(u), can be calculated as the product of the individual Fourier Transforms of f(x) and g(x), F(u) and G(u), respectively:

H(u) = F(u)G(u)

Practical Application: Spatial to Frequency Domain Transformation

To compute the equivalent frequency domain filter H(u,v) using the given spatial domain filter h(x,y), you need to perform the following steps:

1. Pad zeros around the spatial filter h(x,y) to create a larger filter. For example, you can pad zeros to obtain a 3x3 grid filter.
2. Calculate the two-dimensional Fourier Transform of the padded filter using a programming language and image processing tools/libraries.
3. The resulting Fourier Transform H(u,v) represents the equivalent frequency domain filter.

Visualization and Image Filtering

Visualize the spatial domain filter h(x,y) and frequency domain filter H(u,v) as images using appropriate tools and libraries. Interpret the results by comparing the spatial and frequency domain representations of the filters.

Explain how these filters, despite being in different domains, have equivalent functions in filtering and modifying images or signals.