Image Processing: Frequency Domain Filtering

Smoothing in the frequency domain is achieved by removing high-frequency components.

A filter that only allows low-frequency components to pass through and blocks high frequencies.

A filter that only allows high-frequency components and blocks low frequencies.

Filtering processes in spatial and frequency domains are related.

Relates filtering in spatial and frequency domains.

The Discrete Fourier Transform of a two-dimensional image.

Discrete Fourier Transform. A powerful tool for analyzing the frequency components of a signal.

Frequency components visualisation of an image.

The range of frequencies in a signal is within [-M/2, M/2]. Goes from negative half the value of M to positive half of the value of M.

Frequencies in DFT range from -M/2 to M/2.

Filters that modify an image's frequency components to achieve effects like smoothing or sharpening.

A filter that reduces high-frequency components in an image, resulting in a smoother appearance.

A mathematical representation of how a filter modifies frequencies in the frequency domain.

Adding zeros to the edges of an image's data before applying the DFT, increasing its resolution and improving the accuracy of frequency analysis.

The DFT covers two consecutive half periods of a signal. For an M-point DFT, the frequency range goes from -M/2 to M/2.

To filter an image in the frequency domain, you:

1. Compute the DFT of the image (F(u,v)).
2. Multiply the DFT by a filter function (H(u,v)).
3. Compute the inverse DFT to obtain the filtered image.

Removing high frequencies in the frequency domain results in smoothing the image. This removes noise and sharp edges, creating a smoother and less detailed image.

Multiplying the DFT of an image by a filter function H(u,v) selectively amplifies or attenuates specific frequency components, effectively applying the filter to the image in the frequency domain.

The FFT is a powerful algorithm that efficiently calculates the Discrete Fourier Transform (DFT). It dramatically reduces the computational time for frequency domain analysis, making it practical for large images.

The FFT enables efficient frequency domain filtering, which can significantly improve image quality by removing noise, sharpening edges, or enhancing specific features.

Removing high frequencies in the frequency domain leads to smoothing. It makes the image appear less detailed, reducing noise and sharp edges.

Similar filtering tasks can be achieved in both the spatial and frequency domains. Spatial domain filtering operates directly on pixels, while frequency domain filtering works by manipulating frequency components.

Frequency domain filtering can be significantly faster than spatial domain filtering, especially for large images.

Adding zeros to the edges of an image's data before applying the DFT (Discrete Fourier Transform). This improves the accuracy of frequency analysis and increases the resolution of the image.

Hhfe(u,v) represents a highpass filter function in the frequency domain. It's a combination of a constant term 'a' and a scaled version of another highpass filter function Hhp(u,v) by 'b'.

'a' controls the baseline level of the filter, while 'b' determines the scaling factor for Hhp(u,v). 'a' is non-negative and 'b' is larger than 'a'.

The Laplacian operator in the frequency domain can be represented by a filter that multiplies the frequency components of an image by -(u^2 + v^2). This filter amplifies high frequencies, resulting in edge enhancement.

A Laplacian-filtered image is obtained by performing the inverse Fourier Transform of the product of the filter H(u,v) and the image's Fourier Transform F(u,v).

The Laplacian filter in the spatial domain can be applied using a kernel that emphasizes the difference between a pixel and its neighbors.

Highpass filtering removes low-frequency components (smooth areas) from an image, emphasizing high-frequency components (edges, details, noise) resulting in a sharper appearance.

The emphasis result is the enhanced version of the original image after highpass filtering. It highlights edges and details by emphasizing high-frequency components.

Histogram equalization is a technique used to improve the contrast of an image by distributing intensities more uniformly.

Zero-padding is the process of adding zeros to the edges of an image before applying the DFT. This increases the resolution of the frequency domain and improves the accuracy of frequency analysis.

A filter that allows all frequencies below a cutoff frequency to pass through and blocks all frequencies above it. It creates a sharp cutoff, leading to ringing artifacts in the output image.

A type of low-pass filter that has a smoother transition between the passband and stopband compared to an Ideal Low Pass Filter, resulting in less ringing artifacts in the output image.

The order of a Butterworth filter determines the steepness of the transition between the passband and stopband. A higher order results in a steeper transition but also increases processing time.

The cutoff frequency determines the highest frequency that is allowed to pass through. A higher cutoff frequency allows more high-frequency details to remain in the output image.

A type of low-pass filter that utilizes a Gaussian function to smoothly attenuate high-frequency components, leading to a very gradual transition and minimal ringing.

Undesirable oscillations or ripples around edges and sharp transitions in an image, often caused by sharp cutoff filters.

Demonstrations of applying different low-pass filters to images to perform tasks like removing noise, blurring, and connecting broken text.

BLPF has a smoother transition than ILPF but can still exhibit some ringing. GLPF has a more gradual transition than BLPF, leading to less ringing and more smoothing.

A higher cutoff frequency allows more high-frequency details to pass through, resulting in a less smoothed image. Conversely, a lower cutoff frequency removes more details and produces a smoother, blurred output.

Low-pass filters are useful for tasks like noise reduction, image blurring, smoothing, and emphasizing low-frequency content in images.

The choice of low-pass filter depends on the desired level of smoothing and ringing. ILPF produces strong ringing, BLPF provides a compromise, and GLPF achieves the most gradual transition with minimal ringing.

A technique for selectively modifying the frequency components of an image to achieve effects like blurring, smoothing, and removing noise.

Many image processing techniques rely on filtering in the frequency domain. Understanding the fundamentals of frequency domain analysis and filter design is crucial for effectively applying these techniques.