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On discrete cosine transform
Jianqin Zhou
TL;DR
Existing discrete transforms have established applications, but their generalized forms and orthogonality warrant further study. This paper introduces parameterized cosine and W transforms, including a new cosine type, and proves orthogonality in several cases.
Problem
The paper addresses the need to study broader discrete cosine and W transform forms while establishing when these transforms remain orthogonal.
Method
The paper generalizes discrete cosine and W transforms with additional parameters, proposes new cosine-related transforms, and uses orthogonality proofs.
Results
The proposed transforms are proved orthogonal, including generalized cosine cases, a new cosine transform, and generalized W-transform cases.
Takeaways & Limitations
The work expands the collection of discrete cosine, sine-related, and W transforms with mathematically established orthogonality.
Abstract
from arXiv · showhide
The discrete cosine transform (DCT), introduced by Ahmed, Natarajan and Rao, has been used in many applications of digital signal processing, data compression and information hiding. There are four types of the discrete cosine transform. In simulating the discrete cosine transform, we propose a generalized discrete cosine transform with three parameters, and prove its orthogonality for some new cases. A new type of discrete cosine transform is proposed and its orthogonality is proved. Finally, we propose a generalized discrete W transform with three parameters, and prove its orthogonality for some new cases.
I. INTRODUCTION
The paper develops generalized discrete cosine and W transforms with additional parameters, establishing orthogonality in new cases and proposing several new transform types. It is motivated by the DCT’s broad applications and its ability to approximate the ideal K-L transform.
- Motivation: The DCT is widely used in digital signal processing, image processing, data compression, and information hiding, and can approximate the ideal K-L transform.For Markov-process data models, the DCT performance is described as closely approximating the K-L transform.
- Generalized cosine transforms: A unified DCT-III-E form with parameters p, q, and r is generalized to identify new orthogonal transforms.The generalized form is introduced as a parameterized extension of DCT-III-E.
- Generalized cosine transforms: When (p, q, r) = (1, 1, 1), the new transform is orthogonal; generally, orthogonality holds when gcd(pq, N) = 1 and gcd(pr, 2) = 1.Here, p, q, and r are positive integers.
- Generalized cosine transforms: The paper generalizes DCT-II-E and DCT-IV-E and proposes new cosine, sine, and sine-cosine transforms with proved orthogonality.These extensions add multiple transform families beyond the unified DCT-III-E form.
- Generalized W transform: A three-parameter generalized discrete W transform is proposed, with orthogonality proved for some new cases.The discrete W transform has four useful types and previously had a unified form with two parameters.
II. GENERALIZED DISCRETE COSINE TRANSFORM
This section generalizes DCT-III-E into a three-parameter transform and proves its orthogonality under stated gcd conditions. It also derives related orthogonal transforms, including generalized DCT-IV-E and discrete sine transforms.
- DCT-III-E is generalized into a unified transform with positive-integer parameters p, q, and r.
- When gcd(pq, N) = 1 and gcd(pr, 2) = 1, the generalized transform is orthogonal.The proof represents the transform as a matrix and shows that distinct rows have zero inner product while each row has unit norm.
- The product of the transform matrix and its transpose is the identity, yielding the inverse transform.
- The transpose of the transform matrix defines another orthogonal transform under the same gcd conditions.
- The construction produces a generalized DCT-IV-E transform and can similarly generalize DST-II-E, DST-III-E, and DST-IV-E.
III. A NEW TYPE OF DISCRETE COSINE TRANSFORM
This section introduces a new discrete cosine transform and proves that it is orthogonal, with an inverse obtained from its transpose. It also presents new discrete sine and discrete sine-cosine transforms whose orthogonality follows similarly.
- New discrete cosine transform: A new form of the discrete cosine transform is defined for a real vector x(n), n = 0, 1, 2, · · ·, N −1.The transform is subsequently represented in matrix form using column vectors X(N) and x(N).
- New discrete cosine transform: The transform is orthogonal because C(N) is an orthogonal matrix.The paper establishes orthogonality by showing that row inner products vanish for distinct indices, including the final row.
- New discrete cosine transform: The inverse transform follows from C(N)C(N)^T being the identity matrix.The inverse-transform expression is omitted.
- Related transforms: New forms of the discrete sine transform and discrete sine-cosine transform are also presented.Their orthogonality follows from an analysis similar to that used for the new discrete cosine transform.
IV. GENERALIZED DISCRETE W TRANSFORM
This section extends the unified discrete W transform with a third parameter and proves orthogonality for the resulting transform and special cases. It also establishes the inverse transform and notes a corresponding generalization of DWT-IV.
- Three-parameter generalization: The unified discrete W transform is extended by introducing a third parameter γ.The new transform is presented as a three-parameter generalization of the unified form.
- Orthogonality: For positive integers p, q, and r satisfying gcd(pq, N) = 1, transform (6) is orthogonal.The stated condition gives a family of parameterized orthogonal transforms.
- Orthogonality proof: The proof shows that distinct rows of H(N) have zero inner product, establishing that H(N) is an orthogonal matrix.This verifies pairwise row orthogonality for 1 ≤ k1 < k2 ≤ N.
- Inverse transform: Each row of H(N) has unit inner product with itself, so H(N) multiplied by its transpose is the identity matrix and the inverse transform follows.The result directly yields the inverse of transform (7).
- Further generalization: The new transform permits arbitrary positive integer r, and DWT-IV can be generalized similarly.The generalized DWT-IV reduces to transform (2) when r is even.
- Special cases: Two special cases of transform (6) are also orthogonal.The paper identifies special cases involving parameter choices and conditions on N or gcd(q, N).