In a more subtle fashion, the boundary conditions are responsible for the "energy compactification" properties that make DCTs useful for image and audio compression, because the boundaries affect the rate of convergence of any Fourier-like series.

In particular, it is well known that any discontinuities in a function reduce the rate of convergence of the Fourier series, so that more sinusoids are needed to represent the function with a given accuracy.

DCT-IV has gained popularity for its applications in fast implementation of real-valued polyphase filtering banks, A variant, the modified discrete cosine transform, or MDCT (based on the DCT-IV), is used in the MP3, AAC, Vorbis and WMA audio compression formats.

Like any Fourier-related transform, discrete cosine transforms (DCTs) express a function or a signal in terms of a sum of sinusoids with different frequencies and amplitudes.

Like the discrete Fourier transform (DFT), a DCT operates on a function at a finite number of discrete data points.

The obvious distinction between a DCT and a DFT is that the former uses only cosine functions, while the latter uses both cosines and sines (in the form of complex exponentials).

Multidimensional DCTs (MD DCTs) are developed to extend the concept of DCT on MD Signals. A new variety of fast algorithms are also developed to reduce the computational complexity of implementing DCT. DCTs are also widely employed in solving partial differential equations by spectral methods, where the different variants of the DCT correspond to slightly different even/odd boundary conditions at the two ends of the array.

The discrete cosine transform (DCT) was first conceived by Nasir Ahmed while working at the University of Texas, and he proposed the concept to the National Science Foundation in 1972. DCTs are also closely related to Chebyshev polynomials, and fast DCT algorithms (below) are used in Chebyshev approximation of arbitrary functions by series of Chebyshev polynomials, for example in Clenshaw–Curtis quadrature.

(A similar problem arises for the DST, in which the odd left boundary condition implies a discontinuity for any function that does not happen to be zero at that boundary.) In contrast, a DCT where both boundaries are even always yields a continuous extension at the boundaries (although the slope is generally discontinuous).

This is why DCTs, and in particular DCTs of types I, II, V, and VI (the types that have two even boundaries) generally perform better for signal compression than DFTs and DSTs.

## Comments Discrete Cosine Transform Thesis

## Exploring Discrete Cosine Transform for Multi-resolution Analysis

EXPLORING DISCRETE COSINE TRANSFORM FOR MULTI-RESOLUTION ANALYSIS by SAFDAR ALI SYED ABEDI A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of…

## IMAGE COMPRESSION USING DISCRETE COSINE TRANSFORM AND WAVELET BASED.

This is to certify that the thesis entitled “Image compression using discrete cosine transform and wavelet transform and performance comparison’’ Submitted by Anshuman, Roll No 10306001, Gaurav Jaiswal, Roll No 10306004 & Ankit Rai, Roll…

## DCT based image compression - Thesis Projects

Discrete Cosine Transform DCT based Image Compression using MATLAB Project Description In the JPEG image compression algorithm, the input image is divided into 8-by-8 or 16-by-16 blocks, and the two-dimensional DCT is computed for each block.…

## The Discrete Cosine Transform DCT - edu

The Discrete Cosine Transform Like other transforms, the Discrete Cosine Transform DCT attempts to decorrelate the image data. After decorrelation each transform coefficient can be encoded independently without losing compression efficiency. This section describes the DCT and some of its important properties. 2.1. The One-Dimensional DCT…

## IMAGE COMPRESSION USING DISCRETE COSINE TRANSFORM & DISCRETE WAVELET.

This is to certify that the thesis entitled, “IMAGE COMPRESSION USING DISCRETE COSINE TRANSFORM AND DISCRETE WAVELET TRANSFORM” submitted by Bhawna Gautam in partial fulfillment of the requirements for the award of Bachelor of Technology Degree in Computer Science and Engineering at the National Institute of…

## Jpeg Image Compression Using Discrete Cosine Transform - arXiv

Transform coding Transform coding forms an integral part of compression techniques. the reversible linear transform in transform coding aims at mapping the image into a set of coefficients and the resulting coefficients are then quantized and coded. the first attempts is the discrete cosine transform DCT domain. 38…

## Digital Image Watermarking using Optimized DWT-DCT - Free-Thesis

This repository is for the free code of digital image watermarking. This code is for the invisible digital image watermarking using combination of three methods Discrete Wavelet Transform DWT, Discrete Cosine transform DCT and Bacterial Foraging optimization BFO.…

## THREE-DIMENSIONAL TEXTURE CLASSIFICATION USING THE DISCRETE COSINE.

This thesis proposes a novel approach to classifying texture in three dimensional volumetric images using a bank of 3-D ﬁlters derived from the discrete cosine transform DCT. This method is tested on collections of synthetic and natural 3-D textures. Classiﬁcation accuracy is compared to that obtained with 2-D DCT and 3-D Gaussian…