Wavelet transforms in one dimension are a powerful mathematical tool used for analyzing and processing signals in both the time and frequency domains. Unlike traditional Fourier transforms, which represent a signal using infinite sinusoidal functions, wavelet transforms decompose a signal into localized wavelet functions. These wavelets are capable of capturing both global trends and fine-scale … Read more
Multiresolution expansions form a powerful mathematical framework for analyzing, approximating, and representing functions or signals at varying levels of detail. Rooted in the theory of wavelets and functional analysis, they provide a systematic way to decompose a signal into components that capture features at different scales. This multiscale representation is particularly valuable in applications where … Read more
The Haar Transform is a fundamental tool in signal processing and image analysis, known for its simplicity and efficiency. It is used to break down data into different levels of detail, making it especially useful in multiresolution analysis and compression algorithms. The Haar transform is part of a broader family of wavelet transforms that help … Read more
Subband coding is a signal processing technique primarily used in digital audio, speech, and image compression. The main idea behind subband coding is to split a signal into different frequency components, called subbands, and then process each subband independently. By compressing these subbands separately, we can achieve more efficient data representation and compression. This method … Read more
Image pyramids are hierarchical structures used in image processing to represent images at multiple resolutions. They are especially useful for applications like image compression, feature detection, image blending, and multiresolution analysis. The most common types of image pyramids are Gaussian pyramids and Laplacian pyramids. Gaussian Pyramid A Gaussian pyramid is constructed by repeatedly applying a … Read more
Color image compression is the process of reducing the size of a color image file while maintaining acceptable visual quality. Since color images contain more data than grayscale images, compression plays a critical role in reducing storage and transmission costs. Various mathematical techniques are used to eliminate redundancy in color images, enabling efficient storage and … Read more
Noise in color images refers to unwanted random variations in color and brightness values that distort the quality of the image. It can arise from multiple sources during image acquisition, processing, transmission, or storage. The presence of noise can significantly reduce the clarity and visual appeal of images, making it essential to understand how it … Read more
Image segmentation is a fundamental task in image processing and computer vision, where the goal is to partition an image into multiple segments or regions. Each segment typically corresponds to different objects or parts of the scene. Color-based segmentation is one of the most popular approaches to segment an image, especially when color information is … Read more
In image processing, smoothing and sharpening are fundamental techniques used to either reduce noise (smoothing) or enhance edges and fine details (sharpening). These processes are particularly useful when dealing with both grayscale and color images. In this article, we will focus on the mathematical concepts behind color image smoothing, as outlined in the image provided, … Read more
Color transformations involve processing the components of a color image within a single color model, rather than converting between different models. In these transformations, each pixel of the color image is represented as a triplet (for RGB or HSI) or quartet (for CMYK) of values corresponding to the chosen color space. Formulation of Color Transformations … Read more
