Fourier Transform
Uploaded by
Uma Maheswari BHC Information TechnologyFourier Transform
Uploaded by
Uma Maheswari BHC Information TechnologyFourier transform
The Fourier Transform is for analysing the frequency content of images.
It is used to transform an image from its spatial domain into its frequency domain.
The process of transforming an image from its spatial domain to its frequency domain involves applying
mathematical operations to represent the image in terms of its frequency components rather than its pixel
values.
In the spatial domain, an image is represented as an array of pixel values. Each pixel has a specific
position in the image, and its intensity value represents the color or brightness at that location.
In the frequency domain, the image is represented as a sum of sinusoidal components of different
frequencies. Each frequency component is associated with a specific spatial frequency in the original
image.
High-frequency components in the frequency domain correspond to rapid changes or details in the spatial
domain. Low-frequency components correspond to smoother variations and global features.
The distribution of energy in the frequency domain can give insights into the overall structure and
characteristics of the image.
This transformation allows for the separation of image information based on different frequencies, which
can be useful for various image processing tasks.
The two main types of Fourier Transforms used in digital image processing are the 2D Discrete Fourier
Transform (DFT) and its computationally efficient counterpart, the Fast Fourier Transform (FFT).
Applications:
Image Enhancement
Image Compression
Image Filtering
Image Reconstruction
Pattern Recognition
Image Analysis
Removal of Periodic Noise
Working of Fourier transform:
Convert Image to Grayscale: If the image is in color, convert it to grayscale.
Apply 2D Fourier Transform: Compute the 2D DFT or use the FFT algorithm to transform the image into
the frequency domain.
Analyze Frequency Components: Examine the frequency components of the image to identify features or
patterns.
Apply Processing Operations: Perform operations in the frequency domain, such as filtering or enhancing
certain frequencies.
Inverse Fourier Transform: Transform the image back to the spatial domain using the inverse Fourier
Transform.
2DFT:
The 2D Fourier Transform (2DFT) is a mathematical operation applied to a two-dimensional array, such
as a digital image, to analyze its frequency content in both the horizontal and vertical directions.
The mathematical expression for the 2D Fourier Transform of a function f(x,y) is given by:
F(u,v) is the 2D Fourier Transform of f(x,y). It represents a complex-valued function in the frequency
domain, where u and v are the spatial frequencies in the horizontal and vertical directions, respectively.
f(x,y) is the input function in the spatial domain. It represents the original image or signal in terms of its
spatial coordinates x and y. The function f(x,y) is the image in the spatial domain.
−j2π(ux+vy), is a complex number that depends on the spatial frequencies u and v, as well as the spatial
coordinates x and y. here, u and v determine the frequency components and x and y are the spatial
coordinates in the image.
Properties: Shift Theorem: Shifting an image in the spatial domain results in a phase shift in the
frequency domain. This property is particularly useful in image registration and alignment.
Scaling Theorem: Scaling an image in the spatial domain (resizing) affects the frequency domain.
Enlarging an image in the spatial domain results in a spread of frequencies in the frequency domain, and
vice versa.
Rotation Theorem: Rotating an image in the spatial domain corresponds to a rotation in the frequency
domain. This property is essential in image transformation tasks.
Convolution Theorem: The convolution operation in the spatial domain is equivalent to the element-wise
multiplication in the frequency domain. This property is exploited in filtering and image processing
operations.
Separability: For a 2D image, the Fourier Transform is separable, meaning it can be computed as the
product of two 1D transforms along the rows and columns of the image. This property is utilized for
efficient computation in image processing.
Frequency Component Analysis: The Fourier Transform allows for the analysis of an image's frequency
components. High-frequency components correspond to edges and fine details, while low-frequency
components represent smooth variations and global features.
Filtering in Frequency Domain: Applying filters in the frequency domain, such as high-pass or low-pass
filters, can enhance or suppress specific frequency components. This property is employed in image
enhancement and noise reduction.
Inverse Fourier Transform: The inverse Fourier Transform allows the reconstruction of an image from its
frequency components. This property is fundamental in restoring an image after frequency domain
manipulation.
Zero Padding: Zero padding an image in the spatial domain results in interpolation in the frequency
domain. This property is used to increase the frequency resolution during Fourier Transform
computation.