Frequency-Domain-Filtering Frequency-Domain-Filtering
- 1. Frequency Domain Filtering for Image Processing Explore frequency domain filtering. Transform images into frequency components and filter. Useful for noise removal, edge detection, and image enhancement. NIKHIL DHIMAN 22010103034
- 2. Frequency Domain Filtering Steps: Transform Use the Fourier Transform to convert the image from the spatial domain to the frequency domain, decomposing it into its constituent frequencies. Filter Apply a filter to the frequency components, attenuating or amplifying certain frequencies to achieve a desired effect, like noise reduction or edge enhancement. Common filters include low-pass, high-pass, and band-pass filters. Inverse Transform Use the Inverse Fourier Transform to convert the image back to the spatial domain, reconstructing the filtered image from its modified frequency components.
- 3. From Pixels to Waves: Understanding the Frequency Domain Pixels Images are composed of pixels, arranged in a grid. Each pixel has a specific color value, typically represented as RGB (Red, Green, Blue) or grayscale intensity. Frequency In the context of images, frequency represents the rate of change in pixel values across the image. It describes how quickly the intensity or color changes from one pixel to the next. Waves High frequency corresponds to rapid changes in pixel values, representing edges, textures, and fine details. Low frequency indicates gradual changes, representing smooth regions and overall image structure.
- 4. ABOUT FOURIER: Fourier was a mathematician in 1822. He give Fourier series and Fourier transform to convert a signal into frequency domain. Fourier series: Fourier series simply states that, periodic signals can be represented into sum of sines and cosines when multiplied with a certain weight. It further states that periodic signals can be broken down into further signals with the following properties. •The signals are sines and cosines •The signals are harmonics of each other
- 5. Fourier series: The Fourier series can be denoted by this formula. The inverse can be calculated by this formula. Fourier transform: The Fourier transform simply states that that the non periodic signals whose area under the curve is finite can also be represented into integrals of the sines and cosines after being multiplied by a certain weight. The Fourier transform has many wide applications that include, image compression (e.g JPEG compression), filtering and image analysis.
- 6. Difference between Fourier series and transform: Although both Fourier series and Fourier transform are given by Fourier , but the difference between them is Fourier series is applied on periodic signals and Fourier transform is applied for non periodic signals Which one is applied on images ? Now the question is that which one is applied on the images , the Fourier series or the Fourier transform. Well, the answer to this question lies in the fact that what images are. Images are non – periodic. And since the images are non periodic, so Fourier transform is used to convert them into frequency domain. The formula for 2d-discrete Fourier transform is given below.
- 7. The Fourier Transform: Decomposition Breaks down an image into its fundamental sine and cosine components, revealing the underlying frequencies that constitute the image. Frequency Spectrum Represents the amplitude and phase of each frequency component, providing a comprehensive view of the image's frequency content. Mathematical Basis Based on complex numbers and Euler's formula, enabling the representation and manipulation of frequencies using mathematical principles.
- 8. The Inverse Fourier Transform Reconstruction This step synthesizes the filtered frequency components to reconstruct the image. It effectively reverses the Fourier Transform process, converting the modified frequencies back into spatial data. 1 Spatial Domain The transformation results in an image represented in the spatial domain, where each point corresponds to a pixel. This is the domain in which we can directly visualize and interpret the image. 2 Frequency Domain The Inverse Fourier Transform takes the frequency domain representation, which contains information about the amplitude and phase of different frequency components, as its input. 3 Transforms frequency components back into pixel values, recreating the original image from its frequency representation. This process relies on complex mathematical operations to accurately convert the modified frequencies back into a viewable image.
- 9. Filtering : Low-Pass, High-Pass, and Band-Pass Filters Low-Pass Allows low frequencies, blurs image. These filters are useful for reducing noise and smoothing images by attenuating high-frequency components, which often correspond to fine details and sharp edges. High-Pass Allows high frequencies, sharpens image. By enhancing high- frequency components, these filters can accentuate edges and fine details, making the image appear sharper and more defined. They are often used for feature extraction and edge detection. Band-Pass Allows specific frequency range. These filters isolate a particular range of frequencies, attenuating both higher and lower frequencies outside the specified band. They can be used to remove specific types of noise or to enhance certain features that are characterized by a particular frequency range.
- 10. Example : Sharpening Images with High-Pass Filtering Original Image A slightly blurred image of a cityscape, showing common imperfections and lack of clear details. This serves as the starting point for the sharpening process, where fine details are not easily discernible due to the blurring effect. High-Pass Filter Applies a high-pass filter to enhance edges and fine details by attenuating low-frequency components. This process accentuates the transitions in pixel intensity, effectively highlighting the boundaries and textures within the image. The filter amplifies the high-frequency components, which are responsible for the sharp details. Sharpened Image The resulting image with enhanced details and sharper edges, achieved through the application of the high- pass filter. Noticeably clearer and more defined, the sharpened image reveals previously hidden textures and details, providing a more visually appealing and informative representation of the scene.
- 12. Advantages and Limitations Compared to Spatial Domain Aspect Frequency Domain Spatial Domain Complexity More complex, requires understanding of Fourier transforms and frequency representations Simpler, involves direct manipulation of pixel values and local neighborhoods Computation Slower, needs transforms to and from the frequency domain, which can be computationally intensive Faster, direct manipulation allows for quicker processing, especially for small filters Types of Filters Powerful for frequency-based filters, such as low-pass, high-pass, and band-pass filters, offering precise control over frequency components Limited to spatial relationships, primarily useful for blurring, sharpening, and edge detection based on pixel proximity