Fourier Transform in Remote Sensing

The Fourier Transform (FT) is a mathematical method used in remote sensing to break an image into its spatial frequency components.
Think of it as changing the view of an image—from shapes and objects (spatial domain) to patterns and textures (frequency domain).

Why Fourier Transform

Remote sensing images contain patterns such as:

• smooth water bodies

• rough mountains

• sharp boundaries

• regular textures (agriculture fields)

The Fourier transform helps us:

• Identify landscape changes

• Study surface texture (smooth, rough, periodic)

• Remove noise

• Sharpen or smooth images

• Detect repeated patterns (crop rows, sand ripples)

1. Spatial Domain

This is the original image in terms of rows and columns (x, y).
Here, pixel values represent brightness (DN values).

2. Frequency Domain

After applying Fourier Transform, the image is represented in terms of spatial frequencies:

• Low frequencies → slow changes (smooth areas, water, sky)

• High frequencies → rapid changes (edges, boundaries, sharp details)

• Directional frequencies → patterns along specific angles

How Fourier Transform Works 

Step 1: Decomposition (Breaking the Image into Waves)

The FT decomposes the image into many sinusoidal waves:

• each wave has a frequency (how fast brightness changes)

• an amplitude (strength)

• a direction

Simple meaning:
The image is represented as a combination of many small repeating patterns.

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Step 2: Moving into the Frequency Domain (Transforming Coordinates)

After decomposition, the image is converted into the frequency domain.

Here:

• The center of the transformed image = low frequencies

• The outer part = high frequencies

Instead of pixel position, we now look at "how often brightness changes".

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Step 3: Magnitude and Phase Components

The Fourier output is complex (has real and imaginary parts), represented as:

a) Magnitude Image

Shows how strong each frequency is.
Contains most of the shape and intensity information.

b) Phase Image

Shows where the patterns occur.
Contains spatial arrangement information.

Both are important for reconstructing the image.

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Step 4: Filtering (Improving Image Quality)

By modifying frequencies, we can filter the image:

High-pass filter

• keeps high frequencies

• removes low frequencies

• sharpens edges and boundaries

Low-pass filter

• keeps low frequencies

• removes high frequencies

• smooths the image and reduces noise

Band-pass filter

• keeps only selected frequencies

• highlights periodic textures (fields, ripples)

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Step 5: Inverse Fourier Transform (Returning to Spatial Domain)

After filtering, an Inverse Fourier Transform (IFT) converts the frequency-domain data back to the spatial domain.

This gives the final processed image:

• sharper

• smoother

• noise-reduced

• or highlighting specific patterns

depending on the filtering done.

Example

Example 1: Removing striping noise (common in satellite images)

• Striping = repeated pattern → high-frequency component

• Remove that specific frequency band → apply IFT

• Final image becomes clean

Example 2: Enhancing edges

• Keep high frequencies → edge sharpening

Example 3: Texture analysis

• Agricultural fields show periodic textures

• Frequency patterns reveal direction and spacing

The Fourier Transform converts an image into wave patterns.
Low waves = smooth areas.
High waves = sharp edges.
We can enhance or remove patterns in the frequency domain and then convert the image back.
This helps in sharpening, smoothing, noise removal, and texture analysis in remote sensing.