684 IEEE SIGNAL PROCESSING LETTERS, VOL. 14, NO. 10, OCTOBER 2007 A Fast 4 4 Forward Discrete Tchebichef Transform Algorithm Kiyoyuki Nakagaki and Ramakrishnan Mukundan Abstract—The discrete Tchebichef transform (DTT) is a transform method based on discrete orthogonal Tchebichef polynomials, which have applications recently found in image analysis and compression. This letter introduces a new fast 4 4 forward DTT algorithm. The new algorithm requires only 32 multiplications and 66 additions, while the best-known method using two properties of the DTT requires 64 multiplications and 96 additions. The proposed method could be used as the base case for recursive computation of transform coefficients. Exper- imental results showing performance improvement over existing techniques are presented. Index Terms—Algorithms, discrete cosine transforms, discrete Tchebichef transform, image coding, signal processing. I. INTRODUCTION I MAGE transform methods using orthogonal kernel func- tions are commonly used in image compression. One of the most widely used image transform methods is the discrete co- sine transform (DCT) [5], used in JPEG image compression standard [11]. Due to its popularity, there have been many fast algorithms proposed for the DCT. Many of them are based on the recursion on the size of input data [3], [10], [12], where the problem of computing the DCT coefficients of size is re- cursively reduced to the problems of computing the DCT coeffi- cients of size . Some fast algorithms are developed specif- ically for two-dimensional DCT [1], [2], [7]. In addition, [6] proposes a 4 4 DCT algorithm to serve the base case for the recursive methods. The discrete Tchebichef transform (DTT) is another trans- form method using Tchebichef polynomials [4], [9], which has as good energy compaction properties as the DCT and works better for a certain class of images [8]. Due to its high en- ergy compaction property, the DTT has been used in image pro- cessing applications such as image compression and image fea- ture extraction. The DTT has the additional advantage of requiring the evalu- ation of only algebraic expressions, whereas certain implemen- tations of DCT require lookup tables for computing trigono- metric functions. However, the algebraic form of the DTT does not permit a recursive reduction of polynomial order as in the case of the DCT. Therefore, not many fast algorithms for the Manuscript received October 27, 2006; revised February 16, 2007. The as- sociate editor coordinating the review of this manuscript and approving it for publication was Prof. Benoit Champagne. The authors are with the Department of Computer Science, University of Can- terbury, Christchurch, New Zealand (e-mail: kna23@student.canterbury.ac.nz; mukund@cosc.canterbury.ac.nz). Digital Object Identifier 10.1109/LSP.2007.898331 DTT have so far appeared in the literature. In this letter, a new fast 4 4 forward discrete Tchebichef transform is proposed. The proposed algorithm will be useful for the computation of base case for a recursive evaluation of transform coefficients in DTT-based image compression algorithms. The definition of the DTT is given in Section II. Two of its properties are listed in Section III. Section IV gives a descrip- tion of the proposed algorithm. Comparative analysis and the conclusion are given in Sections V and VI, respectively. II. DISCRETE TCHEBICHEF TRANSFORM Given a set of input values (image intensity values for image compression) of size , the forward discrete Tchebichef transform of order is defined as (1) where is the orthonormal version of Tchebichef polyno- mials given by the following recursive relation: (2) (3) (4) where Both DCT and DTT satisfy the properties of separability and even symmetry as outlined in Section III. III. PROPERTIES OF THE DTT A. Separability The definition of DTT can be written in separable form as (5) 1070-9908/$25.00 © 2007 IEEE