Low-Pass Filter (Frequency)

Implement an ideal low-pass filter in the frequency domain to remove high-frequency components from an image spectrum. This process involves modifying the frequency representation of an image to only retain frequency components below a specified cutoff frequency.

The concept of filtering in the frequency domain is rooted in the Fourier Transform, which decomposes a signal into its constituent frequencies. In image processing, this allows for the separation of low-frequency components, representing overall brightness and shape, from high-frequency components, representing details and noise. By applying a low-pass filter, we can reduce noise and blur an image.

To achieve this, we follow these steps:

Obtain the frequency spectrum of the image.
Identify the cutoff frequency, below which frequency components are retained.
Set frequency components above the cutoff to zero.

F'(u, v) = \begin{cases} F(u, v) & \text{if } |u| \leq D_0 \text{ and } |v| \leq D_0 \\ 0 & \text{otherwise} \end{cases}

This technique is widely used in image denoising applications.

Example:

Input:

low_pass_filter([10, 5, 3, 1, 1, 3, 5], 2)

Output:

[10, 5, 3, 0, 0, 3, 5]

Reasoning:

The input array represents frequency components at indices $0$ to $6$ : $[10, 5, 3, 1, 1, 3, 5]$ , with a cutoff frequency of $2$ .
We keep components where $|\text{index} - \text{center}| \le 2$ ; here the center is index $3$ , so indices $1, 2, 3, 4, 5$ are within the cutoff and stay the same.
We zero out components outside that range (indices $0$ and $6$ ), giving $[10, 5, 3, 0, 0, 3, 5]$ .

Constraints:

cutoff is the maximum frequency index to keep (0-indexed)
Return filtered magnitude spectrum

Example:

Constraints:

Test Results