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Exploring Digital Image Processing: A Study on Frequency Domain Filtering

Techniques

Research · April 2025

DOI: 10.13140/RG.2.2.25561.84320

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Anam Khizar Khanzada Noshin Sabuwala

Veermata Jijabai Technological Institute Veermata Jijabai Technological Institute
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 Exploring Digital Image Processing: A Study on
Frequency Domain Filtering Techniques
Anam Khanzadah Noshin Sabuwala
Computer Engineering and Information Technology Computer Engineering and Information Technology
Veermata Jijabai Technological Institute Veermata Jijabai Technological Institute
Mumbai, India Mumbai, India
Anam.khanzada26@gmail.com Nasabuwala@ce.vjti.ac.in

Abstract—Frequency domain filtering is a fundamental II. THEORETICAL BACKGROUND

technique in digital image processing, enabling efficient noise
reduction, edge enhancement, and feature extraction. Unlike A. Fourier Transform and Frequency Domain
spatial domain filtering, which directly manipulates pixel Representation
values, frequency domain filtering transforms an image using
the Fourier Transform, allowing modifications in the frequency The Fourier Transform (FT) is a mathematical technique that
spectrum before the inverse transformation. This research converts an image from the spatial domain to the frequency
explores the theoretical foundations of frequency domain domain. The 2D Discrete Fourier Transform (DFT) of an
filtering, including low-pass, high-pass, band-pass, and notch
filters, and their applications in image enhancement and image of size is given by:
restoration. We analyze the performance of various filters,
such as Gaussian, Butterworth, and Ideal filters, in different
image processing scenarios. Experimental results demonstrate
the advantages of frequency domain techniques in preserving ..Equation.2.1 In
image details while effectively removing noise and artifacts. equation 2.1, f(x,y) is the spatial domain 2D-function, which
The study highlights the computational efficiency and flexibility represents the spatial domain image, while f(u,v) represents
of frequency-based filtering in real-world applications, its frequency domain representation [1], [2], [5], [8].
including medical imaging, remote sensing, and digital If we consider u = v = 0, it means the slowest varying
photography.
frequency component.
Index Terms—Digital Image Processing, Frequency Domain,
Filtering, Fourier Transform, Image Enhancement.
..Equation .2.2
In equation 2.2 indicates zero frequency term of f(u,v). Low
I. INTRODUCTION frequencies indicate slow variation of intensity ,while as we
Digital image processing involves the manipulation of images move further away from the origin of transform, high
to enhance their quality or extract meaningful information. frequency component indicates that fast variation of intensity.
Frequency domain filtering is a powerful technique that The Inverse DFT (IDFT) reconstructs the spatial image from
modifies the frequency components of an image rather than its frequency components:
the pixel values directly. This method offers several
advantages, including efficient noise removal, edge
sharpening, and artifact reduction. ..Equation .2.3
For x = 0, 1, 2.M-1 and y = 0, 1, 2.N-1
In this paper, we investigate the performance of frequency
domain filters, such as low-pass, high-pass, and band-pass
B. Filtering in the Frequency Domain.
filters. We implement and compare Ideal, Butterworth, and
Frequency domain filtering involves the following steps:
Gaussian filters to evaluate their effectiveness in image
enhancement and noise reduction. 1. Fourier Transform: Convert the image into the
frequency domain using DFT.
 2. Apply the Filter: Multiply the frequency domain
image with the filter function.
3. Inverse Fourier Transform: Convert the filtered
image back to the spatial domain using IDFT

Fig 2.1 Frequency Domain filtering operations

III. IMAGE ENHANCEMENT USING FREQUENCY DOMAIN

TECHNIQUE Fig 3.1 Types of Frequency Domain Filters

Frequency domain filters play a key role in enhancing images 3.2 Low-Pass Filters (LPF)- Image Smoothing
by reducing noise, smoothing textures, and emphasizing
edges. Unlike spatial filtering, which works directly on pixels, Low-pass filters remove high-frequency noise while
frequency domain methods first apply a Fourier Transform to retaining smooth transitions in an image. This results in a
manipulate frequency components, offering precise control. blurring effect, making LPFs useful for de-noising and reducing
abrupt intensity variations.
3.1 Image Enhancement
3.2.1 Ideal Low-Pass Filter (ILPF)
Image enhancement in the frequency domain is achieved by
selectively amplifying or attenuating specific frequency An Ideal Low-Pass Filter (ILPF) applies an abrupt cutoff,
components. This process improves image quality for various allowing all frequency components within a radius Do to pass
applications such as medical imaging, remote sensing, and while completely eliminating those beyond it. It is
digital photography. mathematically defined as:

Common noise types affecting images include:

• Gaussian Noise: Sensor-related random intensity

variations
• Salt-and-Pepper Noise: Random black-and-white
pixel spikes Where D(u,v) is the Euclidean distance from the frequency
• Poisson Noise: Quantum sensor fluctuations center.

Filtering techniques are categorized into: 3.2.2 Butterworth Low-Pass Filter (BLPF)

• Low-Pass Filtering (LPF): Smooths images by Unlike ILPF, the Butterworth filter introduces a gradual
suppressing high-frequency noise while preserving transition rather than an abrupt cutoff, reducing unwanted
low-frequency details. artifacts such as ringing effects. It is represented as:

• High-Pass Filtering (HPF): Enhances sharp transitions

and edges by removing low-frequency components.

• Band-Pass and Notch Filters: Selectively retain or where n is the order of the filter, controlling the sharpness
remove specific frequency ranges, useful for of the transition.
removing periodic noise.
 3.2.3 Gaussian Low-Pass Filter (GLPF) IV. EXPERIMENTAL SETUP

The Gaussian filter provides a smooth, exponentially A. Dataset and Tools

decaying response, minimizing edge artifacts. Its function is:
• Dataset: Grayscale image.

• Tools: Python with NumPy, OpenCV, Matplotlib, and

skimage.metrics for filtering and evaluation
This filter is widely used in medical imaging due to its smooth
effect without introducing sharp discontinuities. B. Filtering Process

3.3 High-Pass Filters (HPF)- Image Sharpening 1. Image Loading & DFT:

High-pass filters emphasize rapid intensity changes, Convert the image to the frequency domain using
preserving edges and fine details while attenuating uniform Discrete Fourier Transform (DFT).
regions.
2. Filtering:
3.3.1 Ideal High-Pass Filter (IHPF)
Apply LPF, HPF, Band-Pass, and Notch filters using
The Ideal High-Pass Filter is defined as: Ideal, Butterworth, and Gaussian techniques..

3. Reconstruction:

Use Inverse DFT (IDFT) to revert filtered images back

This completely removes low-frequency components, to the spatial domain.
resulting in a sharp image but often introducing ringing
artifacts. 4. Performance Evaluation:

3.3.2 Butterworth High-Pass Filter (BHPF) Measured the filtering performance using:

Butterworth HPF provides a more controlled transition: • PSNR (Peak Signal-to-Noise Ratio): Evaluated image
quality by comparing the filtered image to the
original.

Higher-order Butterworth filters behave similarly to ideal • MSE (Mean Squared Error): Measured the average
squared difference between the original and filtered
filters but with reduced artifacts.
images.
3.3.3 Gaussian High-Pass Filter (GHPF)
• SSIM (Structural Similarity Index): Assessed the
The Gaussian High-Pass Filter is given by: structural similarity between the original and filtered
images.

While HPFs degrade PSNR/SSIM by design (as they

amplify high-frequency noise), they are effective for
edge enhancement.
It smoothly preserves edges while minimizing noise
amplification.

3.4 Band-Pass and Notch Filters V. RESULT AND ANALYSIS

• Band-Pass Filters: Retain only a specific frequency 5.1 Image Filtering and Performance Evaluation
range while eliminating frequencies outside the
range. These are useful for texture analysis. The filtering operations were performed on a grayscale
image using frequency domain techniques. The filters
• Notch Filters: Target and remove periodic noise, such applied include:
as interference patterns in scanned documents.
• Low-Pass Filters (LPF): Smooth the image by
removing high-frequency components.

• High-Pass Filters (HPF): Enhance edges by removing

low-frequency components.

• Band-Pass Filters: Retain frequencies within a specific

range.

• Notch Filters: Remove specific noise frequencies

while preserving the overall image structure.

The filtered images were reconstructed using IDFT

(Inverse Discrete Fourier Transform) and evaluated using
PSNR, MSE, and SSIM metrics.

Fig. 5.4 Ideal LPF image(Cutoff = 50)

Fig 5.1: Original grayscale image Fig. 5.6 Butterworth LPF image (Cutoff = 50)

Fig. 5.9 Ideal HPF image (Cutoff = 30)

Fig. 5.2 Fourier Transform of an Image
 Fig. 5.10 Ideal HPF image (Cutoff = 50) Fig. 5.16 Notch filtered image (30,30)

5.2 Performance Metrics and Analysis

The table below shows the PSNR, MSE, and SSIM values
for each filter type:

Fig. 5.12 Butterworth HPF image (Cutoff = 50)

Fig. 5.14 Bandpass image (20-60)

Analysis of Results o Both band-pass filters resulted in poor image
quality, with:
1. Low-Pass Filters (LPF)
▪ PSNR < 9 dB
• Purpose: Smooth images by removing high-frequency
noise. ▪ MSE > 7700

• Performance: ▪ SSIM ≤ 0.03

o The Gaussian LPF with a cutoff of 50 showed o This indicates significant image degradation
the best performance, achieving: and low structural similarity.

▪ PSNR = 29.00 dB o Band-pass filtering did not effectively

preserve image details.
▪ MSE = 75.01
4. Notch Filters
▪ SSIM = 0.96
• Purpose: Remove specific frequency noise while
o This indicates minimal noise and excellent preserving image structure.
structural similarity, making it highly
effective for denoising and smoothing. • Performance:

o The Butterworth LPF also performed well, o The Notch filter delivered exceptional
with a slightly lower SSIM compared to results, achieving:
Gaussian. Ideal LPF caused more blurring
and had lower SSIM. ▪ PSNR = 47.80 dB with MSE = 0.99
(center at (30,30))
2. High-Pass Filters (HPF)
▪ PSNR = 49.78 dB with MSE = 0.63
• Purpose: Enhance edges by removing low-frequency (center at (50,50))
components.
▪ SSIM = 1.00 in both cases,
• Performance: indicating perfect structural
similarity.
o All HPFs produced low PSNR and SSIM
values, indicating significant noise and o This demonstrates that the Notch filter
distortion. effectively removed noise while preserving
image quality.
o The Gaussian HPF with a cutoff of 50 had the
best performance among HPFs but still
showed:
VI. APPLICATIONS
▪ PSNR = 8.63 dB
• Medical Imaging: Noise removal and contrast
▪ MSE = 8161.70 enhancement in MRI and CT scans.

▪ SSIM = 0.02 • Remote Sensing: Removing atmospheric noise in

satellite images.
o Ideal HPF had the worst performance, with
significant artifacts and low structural • Photography: Artifact reduction and sharpening in
similarity. digital photos.

3. Band-Pass Filters

• Purpose: Retain frequencies within a specific range.

• Performance:
 Journal of Computer Sciences and Engineering, Vol. 7,
Special Issue 2, January 2019.
VII. CONCLUSION [4] Sarker, Orvila, Sheuly Akter, and Afrina Mishu. "Review on
the Performance of Different Types of Filters in the
In this research paper, we explored various frequency Presence of Various Noises." Engineering International,
domain filtering techniques for image enhancement, Vol. 4, No. 2, December 2016.R. Nicole, “Title of paper
including Low-Pass Filters (LPF), High-Pass Filters (HPF), Band- with only first word capitalized,” J. Name Stand. Abbrev.,
Pass Filters (BPF), and Notch Filters, and analyzed their in press.
performance using PSNR, MSE, and SSIM metrics. The results [5] "A Frequency-Domain Approach with Learnable Filters for
demonstrate that LPFs effectively reduce noise and smooth Image Classification." Applied Soft Computing, 2024.
images by attenuating high-frequency components, with Available:
https://www.sciencedirect.com/science/article/abs/pii/S
Gaussian LPF delivering the best smoothing performance.
1568494624002175
Conversely, HPFs sharpen the image by preserving edges and
[6] GeeksforGeeks. "Frequency Domain Filters and its Types."
enhancing fine details, but at the cost of increased noise, Available at:
resulting in lower PSNR and SSIM values. The Band-Pass Filters https://www.geeksforgeeks.org/frequency-domain-
combine smoothing and sharpening effects but show relatively filters-and-its-types/
lower performance in terms of image quality. Notably, Notch
Filters excel at removing periodic noise, achieving near-perfect
PSNR and SSIM values, making them highly effective for noise
suppression. The cutoff frequency plays a significant role in
determining the filtering outcome, where higher cutoff
frequencies in LPFs retain more details, while lower cutoff
frequencies in HPFs emphasize edges but may introduce
artifacts. Overall, Gaussian filters offer the most balanced
performance in smoothing, while Butterworth filters provide
flexibility due to their adjustable filter order. Notch filters
prove particularly effective in noise removal, making them
valuable for restoring degraded images.

ACKNOWLEDGMENT
I would like to express my sincere gratitude to my professor,
Noshin Sabuwala, for her continuous guidance and support
throughout this research. Her valuable insights and expertise
have been instrumental in helping me understand the intricate
concepts of frequency domain filtering techniques. I am also
thankful for the access to resources and the constructive
feedback that significantly contributed to the successful
completion of this research analysis.

REFERENCES
[1] Dewangan, Swati, and Anup Kumar Sharma. "Image
Smoothing and Sharpening using Frequency Domain
Filtering Technique." International Journal of Emerging
Technologies in Engineering Research (IJETER), Vol. 5,
Issue 4, April 2017.
[2] Shaikh, Md. Shahnawaz, Ankita Choudhry, and Rakhi
Wadhwani. "Analysis of Digital Image Filters in Frequency
Domain." International Journal of Computer Applications
(0975 – 8887), Vol. 140, No. 6, April 2016.
[3] Kanagalakshmi, K., and Shenbagavadivu, S. "Frequency
Domain Enhancement Filters: A Survey." International

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