IEEE SIGNAL PROCESSING LETTERS, VOL. 14, NO. 5, MAY 2007
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Direct Computation of Type-II Discrete Hartley Transform Huazhong Shu, Member, IEEE, Yuan Wang, Lotfi Senhadji, Senior Member, IEEE, and Limin Luo, Senior Member, IEEE
Abstract—We present in this letter an efficient direct method for the computation of a length- type-II generalized discrete Hartley 2 GDHT transform (GDHT) when given two adjacent lengthcoefficients. The computational complexity of the proposed method 8. is lower than that of the traditional approach for length The arithmetic operations can be saved from 16% to 24% for varying from 16 to 64. Furthermore, the new approach can be easily implemented. Fig. 1. Flow graph of traditional approach.
Index Terms—Compressed-domain processing, discrete Hartley transform, fast algorithm.
I. INTRODUCTION HE generalized discrete Hartley transform (GDHT) has been used in many digital signal and image processing applications. One advantage of the GDHT over the generalized discrete Fourier transform (GDFT) is that it can be used to avoid complex operations when the input sequence is real [1], [2]. It was shown [3] that the type-I, -II, -III, and -IV GDHT and the discrete transform (DWT) have a similar definition with a difference in constant scaling factors. Many fast methods for efficient computation of GDHT were reported in the literature. Hu [4] proposed a fast algorithm for computing the GDHT, and Bi and Chen [5] derived a split-radix algorithm for the computation of GDFT and GDHT. Both methods were focused on the sequences with length being a power of two. Recently, Bi [6] developed fast algorithms for calculating the GDHT of composite sequence lengths. Since the GDHT is widely used in signal or image compression techniques, the following question arises: How can we directly manipulate such compressed data stored in the GDHT domain? In other words, how can we construct a long GDHT sequence from several short GDHT sequences? This problem may be encountered when the signals are stored in the GDHT domain. In such a case, the signals need to be processed before being displayed, transmitted, etc. Some of the frequently used processing operations are scaling, filtering, translation, and
T
Manuscript received July 4, 2006; revised September 14, 2006. This work was supported in part by the National Basic Research Program of China under Grant 2003CB716102 and in part by the Program for New Century Excellent Talents in University under Grant No. NCET-04-0477. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Ljubisa Stankovic. H. Shu and L. Luo are with the Laboratory of Image Science and Technology and the Centre de Recherche en Information Biomédicale Sino-français (CRIBs), Southeast University, 210096 Nanjing, China (e-mail: shu.list@seu. edu.cn,
[email protected]). Y. Wang and L. Senhadji are with the INSERM U642, Laboratoire Traitement du Signal et de l’Image, and the Centre de Recherche en Information Médicale Sino-français (CRIBs) Université de Rennes 1, 35042 Rennes, France (e-mail:
[email protected];
[email protected]). Digital Object Identifier 10.1109/LSP.2006.888362
rotation. When the scaling operation is concerned, it is necessary to construct a long GDHT sequence from several short GDHT sequences or to decompose a long GDHT sequence into some short GDHT sequences in the compressed domain. We address this problem in this letter. In particular, we are interested in the problem of the direct computation of the lengthtype-II GDHT coefficients, when the two adjacent sets of type-II GDHT coefficients are given. The traditional way of implementing this calculation is shown in Fig. 1. According to this scheme, two inverse GDHTs of lengthare required, followed by a length- type-II GDHT. Such a process may not be optimal in terms of the number of arithmetic operations. Inspired by a method proposed by Skodras [7], which was derived to the direct computation of -point DCT when the two adjacent -point DCT coefficients are known, we propose in this letter a new approach that is directly based on the GDHT domain for solving the above-mentioned problem. The rest of this letter is organized as follows. Section II presents two new raidx-2 algorithms for computing the -point type-II GDHT coefficients. Section III describes the way of obtaining the -point GDHT coefficients when given the two adjacent -point GDHTs. The computational complexity of the proposed approach is analyzed in Section IV. II. TYPE-II GDHT ALGORITHMS FOR BEING AN EVEN NUMBER The type-II GDHT of an input data sequence is defined as [4]
,
(1) and the corresponding inverse GDHT (IGDHT) is given by [4]
(2) with
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When the sequence length (1) can be decomposed into
IEEE SIGNAL PROCESSING LETTERS, VOL. 14, NO. 5, MAY 2007
is even, for the even index ,
needs additions and each given , multiplications. The above decomposition method is categorized as “decimation in frequency.” Its computational complexity is given by
(9) Hu [4] proposed a “decimation in time” algorithm for computing the type-II GDHT coefficients. The computational complexity of their method is (3) The above equation shows that GDHT. For the odd index, we have
is a length-
type-II (10) (4)
By using the property
(5) equation (4) can be expressed as
(6) Making the change of variable in the second term of the right-hand side of (6), we obtain (7), as shown at the , bottom of the page. Proceeding in a similar way as for we have (8), as shown at the bottom of the page, with and . is also a lengthEquation (8) shows that type-II GDHT. Moreover, the computation of , for
and are the number of where multiplications and additions for the computation of type-I . GDHT of length It can be seen from (9) and (10) that the efficiency of the proposed method is comparable to that derived by Hu. Note that in [6], Bi derived a radix-2 algorithm for computing the is calculated with (3), type-II GDHT coefficients where defined by (4) is transformed into a lengthand type-IV GDHT, and the latter can be further decomposed into two sequences of type-II GDHT with length . The computational complexity of Bi’s method is [6]
(11)
(12) It was shown [6] that Bi’s algorithm requires about the same computational complexity in terms of the total number of arithmetic operations as that needed in Hu’s algorithm.
(7)
(8)
SHU et al.: DIRECT COMPUTATION OF TYPE-II DISCRETE HARTLEY TRANSFORM
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Although the above-proposed method is efficient for computing the type-II GDHT coefficients, but for the purpose of this letter, we will derive another approach for computing defined by (4). In fact, (4) can be rewritten as
(13) Fig. 2. Flow graph of the proposed method.
where
(14)
Equation (19) shows that the computation of based on (16) and (18) requires much more arithmetic operations than that based on (8), but the advantage of the former algorithm is that it can be used to efficiently calculate the -point GDHT -point sequences of GDHT when the from two adjacent latter coefficients are given. This will be explained in the next section.
(15) As for
,
III. PROPOSED APPROACH FOR COMPUTING THE LENGTHTYPE-II GDHT COEFFICIENTS
can be decomposed into
(16) which is a length, For
type-II GDHT. , using the property , we have
Let us readdress the problem as follows. Assume an -point sequence is created by the concatenation of , i.e., , and two adjacent sequences of length , and and are their lengthtype-II GDHT coefficients, respectively. How can we efficiently comwhen pute the -point type-II GDHT coefficients and are known? To answer this question, we propose a new approach based on the algorithm derived in the previous section. A. Computation of
,
From (3), we have
(20) (17) B. Computation of
thus
,
From (16), we have
(21)
(18) type-II GDHT. Note that which is also a length. and are both -point type-II GDHTs, Since , , we can obtain the sequence from and with additions. The computational complexity of the above method is given by
C. Computation of
,
From (18), we have
(22) (19)
According to the derivation above, we can construct the whole structure in Fig. 2.
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TABLE I COMPUTATIONAL COMPLEXITY OF TRADITIONAL AND PROPOSED METHODS.
D. Computation of
If the conventional approach shown in Fig. 1 is applied, two -point IGDHTs and one -point GDHT are required. Using the algorithm proposed by Hu to compute the -point GDHT, the total numbers of additions and multiplications for the traditional approach are given by
,
Once the values and , are obtained, we can compute by simply using (13). Note that for , we have . IV. COMPUTATIONAL COMPLEXITY Fig. 2 shows that the computation of , , and requires two -point vector multiplications and -point type-II GDHTs, asadditions, respectively, and three suming that both of GDHT and IGDHT are of the same comusing plexity. On the other hand, the computation of (13) needs additions. Therefore, the computational complexity of the proposed method is
(23) For comparison purpose, we choose , where is a pos-point GDHT itive integer, so that the fast computation of can be applied. According to the algorithm proposed by Hu [4], the number of operations for computing the type-II GDHT of is given by size
(24) Using (24), (23) becomes
(25)
(26) Table I shows the number of arithmetic operations of the two methods for some specific values of . It can be seen that about . 7%–24% arithmetic operations can be saved except for V. CONCLUSION We propose in this letter an efficient approach to compute a type-II length- type-II GDHT given two adjacent lengthGDHT coefficients. The arithmetic operations can be saved varying from 16 to 64. Generally from 16% to 24% for speaking, the greater the value of , the more operations we can save. Therefore, the proposed method could find its application in signal processing tasks. REFERENCES [1] Z. Wang, “Harmonic analysis with a real frequency function, I Aperiodic case, II Periodic and bounded cases, and III Data sequence,” Appl. Math. Comput., vol. 9, pp. 53–73, 1981. [2] J. Xi and J. F. Chicharo, “Computing running discrete Hartley transform and running discrete transforms based on the adaptive LMS algorithm,” IEEE Trans. Circuits Syst. II, vol. 44, no. 3, pp. 257–260, Mar. 1997. [3] Z. Wang, “Comments on ‘Generalized discrete Hartley transform’,” IEEE Trans. Signal Process., vol. 43, no. 7, pp. 1711–1712, Jul. 1995. [4] N. C. Hu, H. I. Chang, and O. K. Ersoy, “Generalized discrete Hartley transforms,” IEEE Trans. Signal Process., vol. 40, no. 12, pp. 2931–2940, Dec. 1992. [5] G. Bi and Y. Chen, “Fast generalized DFT and DHT algorithms,” Signal Process., vol. 65, no. 3, pp. 383–390, 1998. [6] G. Bi, Y. Chen, and Y. Zeng, “Fast algorithms for generalized discrete Hartley transform of composite sequence lengths,” IEEE. Trans. Circuits Syst. II, vol. 47, no. 9, pp. 893–901, Sep. 2000. [7] A. N. Skodras, “Direct transform to transform computation,” IEEE Signal Process. Lett., vol. 6, no. 8, pp. 202–204, Aug. 1999.
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