Bracewell has proposed the Discrete Hartley Transform (DHT) as a substitute for the Discrete Fourier Transform (DFT), particularly as a means of convolution. Here, it is shown that the most natural extension of the DHT to two dimensions fails to be separate in the two dimensions, and is therefore inefficient. An alternative separable form is considered, corresponding convolution theorem is derived. That the DHT is unlikely to provide faster convolution than the DFT is also discussed

Poirson, A.

Watson, A. B.

English

NASA Technical Reports Server

I : 
NASA Technical Memorandum 88203 
NASA-TM-88203 19860007437 
A Separable Two - Dimensional 
Discrete Hartley Transform 
Andrew B. Watson and Allen Poirson 
December 1985 
NI\S/\ 
National Aeronautics and 
Space Administration 
J/\N 2 2 '1986 
Lf.r .C.LC ( n:C:SEf,I<CH CENTER 
LlORARY, r~ASA 
1111111111111 1111 11111 111111111111111 1111 1111 
NF01663 
https://ntrs.nasa.gov/search.jsp?R=19860007437 2020-03-20T16:15:54+00:00Z
NASA Technical Memorandum 88203 
A Separable Two - Dimensional 
Discrete Hartley Transform 
Andrew B. Watson, Ames Research Center, Moffett Field, California 
Allen Poirson, Stanford University, Palo Alto, California 
December 1985 
NI\S/\ 
National Aeronautics and 
Space Administration 
Ames Research Center 
Moffett Field, California 94035 
A SEPARABLE TWO-DIMENSIONAL DISCRETE HARTLEY TRANSFORM 
Andrew B. Watson and Allen Poirson 
Perception and Cognition Group, NASA Ames Research Center 
Moffett Field, CA 94035 
Abstract-Bracewell has proposed the Discrete Hartley Transform (DHT) as a substi-
tute for the Discrete Fourier Transform (DFT), particularly as a means of convolution 
(R. N. Bracewell, "Discrete Hartley transform," Journal of the Optical Society of Arner-
ica, vol. 73, pp. 1832-1835, 1983). Here we show that the most natural extension of the 
DHT to two dimensions fails to be separable in the two dimensions, and is therefore 
inefficient. We consider an alternative separable form, and derive a corresponding con-
volution theorem. We also argue that the DHT is unlikely to provide faster convolution 
than the DFT. 
1 
1. Introduction 
In a series of recent papers Bracewell has drawn attention to the Hartley transform 
as a substitute for the much more widely known Fourier transform. [1, 2, 3] He has 
introduced a discrete Hartley transform (OHT) as an analog to the discrete Fourier 
transform (OFT), and has developed a fast algorithm (FHT) to compete with the Fast 
Fourier Transform (FFT). The purpose of this paper is to point out certain problems 
encoun tered in the extension of the OHT from one dimension (ID) to two dimensions 
(2D). There are two possible ways in which this extension can be made. Each has its 
virtues, but only one is practical. This choice requires a new convolution theorem, 
which we provide. However, even with this theorem, the 2D DHT appears not to pro-
vide a computational advantage over the DFT. 
2. The Discrete Fourier Transform (DFT) 
Consider a discrete sequence p (x ), x = 0, ... ,N -1. The forward DFT of p (x ) 
is given by 
N-l 
PF (tl) = E p (x) exp(-i 21rUX /N) (1) 
z =0 
and the inverse DFT, 
1 N-l 
P (x ) = N E PF (u ) exp(i 21rux / N ) 
u=o 
(2) 
In our notation we depart slightly from Bracewell, who places the :: ~alar 1/ N in the for-
ward transform in both the OFT and the DHT. This differs from general practice, 
[4, 5, 6] and is less efficient if forward transforms are more common than reverse 
transforms. We therefore place the scalar in the inverse transform in our definitions of 
2 
the DFT, DHT, and SDHT (defined below). 
3. Symmetry In the DFT of a real sequence 
Although the DHT is defined for both real and complex sequences, its practical 
value arises rrom the way in which it takes advantage or the symmetry in the DFT or a 
real sequence. Therefore in what follows we consider only real input sequences. A real 
sequence, defined by N real numbers, has a DFT that contains N complex entries, or 
2N real numbers. However, only N of the 2N real transform values are independent 
because the transform is Hermitian, that is, its real part is even and its imaginary part 
is odd. The Hartley transform is one way of taking advantage of this redundancy. 
Note that the sum (or difference) of an odd and an even sequence can be easily broken 
apart again into its odd and even parts. Thus because real and imaginary parts of the 
DFT are even and odd respectively, we can add them together with the assurance that 
they can be broken apart to recover the rull complex DFT when desired. We thererore 
define the ID DHT of a real sequence p (x ) as the real part minus the imaginary part of 
the DFT, 
PH (u ) = ReP F (u ) - ImP F (u ) 
Expanding the right side by means or Eq. 1, 
N-l 
PH (u ) = E p (x ) [ cos21rUX / N + sin21rux / N 
z=o 
N-l 
PH (u ) = E p (x ) cas(21rUX / N) 
z=o 
The sequence cas defined by 
cas(21rUX / N ) = COB (21rux / N ) + sin(21rux / N ) 
(3) 
(4) 
(5) 
(6) 
therefore serves as the basis sequence for the DHT. Note that the DHT may be 
3 
expressed as the sum of its even and odd parts 
(7) 
where 
1 PHO (u ) = "2 [ PH (u )-PH (-u ) 1 (8) 
Throughout the text we interpret all sequence arguments modulo the length of the 
I sequence, so for example if P (u ) is a sequence of length N then P (-1 )=P (N -1). 
Note that the DFT is easily reconstituted from the even and odd parts of the DHT 
PF (u ) = PHE (u) - iPHO (u ) 
The inverse transform is 
1 N-l 
P (x ) = N E PH (u ) cas(21ruX / N) 
u =0 
(9) 
(10) 
The principal advantage of the DHT over the DFT, as pointed out by Bracewell, is that 
it transforms a real input into a real output. Furthermore, the convolution theorem can 
be expressed in terms of the DHT. While this expression involves four sequence pro-
ducts, rather than the single product of the DFT convolution theorem, the former are 
products of real quantities and will therefore require on the order of 1/4 the number of 
real operations. Furthermore, if either of the inputs to the convolution is even, as is 
quite common in filtering applications, the result reduces to a single product of real 
sequences, requiring on the order of 1/4 as many operations as convolution via the DFT. 
4. The DDT in two dimensions 
One of the most frequent applications of the DFT is in image processing, where it is 
frequently used to implement convolution. For the DHT to be of use in image process-
ing it must be extensible to two (and more) dimensions, and these extensions must carry 
" 
with them a simple form of the convolution theorem. In one of his papers, BracE'well 
has offered a definition or the 20 OHT. [2] But this definition does not agree with the 
general 10 definition he proposes as embodied in Eq. (9), or with the convolution 
theorem he has offered (Rer. 2, p. 1834). 
There are two natural candidates for the extension or the OHT to two dimensions. 
The basis runctions or these two possibilities are 
cas[ 21r{ tlX / N +vy / M) I and cas(21rUX / N )cas(21rvy / M ) (11) 
In what rollows, we will consider the advantages or each alternative. 
First, which definition is consistent with the definition or the Hartley as the 
difference of even and odd parts of the OFT? Recall the observation that motivated the 
construction of the 10 DHT, namely that the OFT of a real input is Hermitian. Let 
P (x ,y ) be a real 20 sequence (image) with x = O, ... ,N -1 and y = O, ... ,M -1. In two 
dimensions, odd-ness and even-ness are defined with respect to reflections about both 
axes: 
1 Even {p (x ,y )} = PE (x ,y ) = 2' [ P (x ,y ) + P (-x ,-y ) I (12) 
1 Odd {p (x ,y )} = Po (x ,y ) = "2 [ p (x ,y ) - p (-x ,-y ) ] (13) 
Under these definitions, the real part of the OFT of a real input is even, and the ima-
ginary part is odd. We can again define the OHT to be the even part minus the odd 
part of the OFT, 
PH (tl , V ) = ReP F (tl , V ) - ImP F (tl , V ) 
= P HE (tl , V ) + P HO (U , v) . 
In this case the OFT, as for 10, is given by 
6 
(14) 
(15) 
P F (U ,v ) = P HE (U , v ) - iP HO (U , V) • (16) 
To discover the basis sequences of the 20 OHT we expand Eq. (15). Let 
a = 21t'ux IN and 6 = 21t'v1/ 1M. Then 
N-l M-l 
PH(u,v}= E E p(x,1/)[cos(a+6)+sin(a+6)] (17) 
:I =0 , =0 
N-l M-l 
= E E p (x ,1/ ) cas( a +6} . (18) 
:I =0 , =0 
Thus we see that a definition of the 2D OHT that preserves the symmetries of the 10 
transform leads to the first of the two possible basis functions introduced above. 
This definition, while it forms a natural extension of the ID transform, and leads to 
the convolution theorem offered by Bracewell, [2] has a serious drawback. Unlike the 2D 
OFT; it is not separable in U and v. Fast algorithms for the 20 OFT rely upon separ-
ability, and a straightforward extension of the FHT to 2D would also require separabil-
ity. 
6. The Separable 2D Hartley Transform 
As a solution to this dilemma, we consider an alternative definition of the 20 DHT, 
which we will call the Separable Discrete Hartley Transform (SDHT), defined by 
N-I M-I 
Ps (u ,v ) = E E p (x ,1/ ) cas(21t'ux IN) cas(21t'v1/ 1M) (19) 
:I =0 , =0 
and the inverse transform 
1 N -1 M-l 
P (x,1/) = NM E E Ps (u ,v) cas(21t'ux IN )cas(21t'v1/ 1M). (20) 
u =0 ,=0 
This is essentially the 2D DHT as given by Bracewell (only the position of the scale fac-
tor is different). [2] Note that both transforms are separable in x and 1/ , and can be 
8 
executed by first applying a 10 OHT to each row and then a 10 OHT to each co!umn 
(or vice versa). Since this transform no longer obeys Eq. (9), and does not lead to the 
convolution theorem derieved therefrom, we must discover the relationship between the 
SOHT and the OFT, and derive a convolution theorem for the SOHT. 
G. Relation between SDHT and DFT 
In the SOHT we have gained separability but have lost the simple relationships 
between the OHT and the OFT, namely, DHT = ReDFT - ImDFT and 
DFT = Even DHT - i Odd DHT. However, as we show below, a similarly simple 
relation exists for the SOHT and the OFT. We begin by expanding the cas terms in the 
definition of the SOHT, 
N-I M-I 
Ps (u ,v ) = E E p (x ,1/ ) (cosa + sina ) (cosb + sinb ) (21) 
z =0 , =0 
N -I M-I 
= E E p (x ,1/ )(cosa cosb + sina sinb + cosa sinb + sina cosb ) (22) 
z =0 11 =0 
N-I M-l 
= E E p (x ,1/ ) [ cos( a -b ) + sin( a +b ) I (23) 
z =0 , =0 
N-I M-l 
= E E p (x ,1/ ){ Re[ cos( a -b )-i sin( a -b ) I (24) 
z =0 , =0 
- Im[ cos(a +b )-i sin(a +b ) I } 
= RePF (u ,-v) - ImPF (u ,v ) (25) 
Thus the SOHT is equal to the real part of the OFT reflected about the u axis, minus 
the imaginary part of the OFT. As might be imagined, it does not matter whether the 
reflection is about the u or v axis. 
'1 
Going in the other direction, we can expand the expression for the DFT, 
PF (u ,v) = RePF (u ,v ) + ; ImPF (u ,v) (26) 
= .!. { [ RePF (u ,v ) + RePF (-u ,-v) - ImPF (u ,-v) - ImPF (-u ,v ) ] (27) 2 , 
- i [~ePF (-u ,v) - RePF (u ,-v) + ImPF (u ,v) - ImPF (-u ,-v)] } 
1 i 
= -[ Ps (u ,-v) + Ps (-u ,v)] - -2 [Ps (u ,v) + Ps (-u ,-v )bold] (28) 2 . 
= PSE (u ,-v) - iPso (u ,v) (29) 
7. Summary of relationships 
I 
For future reference we summarize the relationships among the DFT, DHT, and 
SDHT: 
Ps (u ,v) = PHO (u ,v) + PHE (u ,-v) 
Ps (u ,v) = RePF (u ,-v) - ImPF (u ,v) 
PH (u ,v ) = PSO (u ,v) + PSE (u ,-v) 
PH (u ,v ) = ReP F (u ,v ) - ImP F (u ,v ) 
PF(u ,v) = PSE (u ,-v) - iPso (u ,v) 
PF (u ,v) = PHE (u ,v) - iPHO (u ,v ) 
8. SDHT Convolution Theorem 
(30) 
(31) 
(32) 
(33) 
(34) 
(35) 
To be useful, the SDHT must yield a simple convolution theorem. We can derive 
this theorem from the preceding relations between DFT and SDHT. Let r (x ,Y ) be the 
result of convolving the two real sequences p '( x ,Y ) and q (x ,Y ), 
N -1 M-l 
r (x ,Y ) = p (x ,Y ) * q (x ,Y ) = E E p (i ,j) q (x -i ,Y - j) (36) 
i =0 j =0 
From the convolution theorem for the DFT we have 
8 
RF (u ,v) = PF (U ,v )QF (U ,v ) (37) 
From Eq. (31) we can convert this to an expression for the SDHT, 
Rs (u ,v) = ReRF (u ,-v) - ImRF (u ,v) (38) 
= Re[ PF (u ,-v )QF (u ,-v) ] - Im[ PF (u ,v )QF (u ,v) ] (39) 
The next step is to substitute Eq. (34) for each DFT, 
= Re{ [PSE (u ,v) - iPso (i ,-v)][ QSE (u ,v) - iQso (u ,-v) ] } (40) 
- Im{ [PSE (u ,-v) - iPso (u ,v)][ QSE (u ,-v) - iQso (u ,v)] } 
Thus we arrive at a convolution theorem 
Rs (u ,v) = PSE (u ,v )QsE (u ,v) - Pso (u ,-v )Qso (u ,-v) (41) 
+ PSE (u ,-v )Qso (u ,v) + Pso (u ,v )QSE (u ,-v) 
G. Symmetrical Inputs 
Bracewell notes that if one of the inputs to a convolution is even, then the DHT of 
the convolution reduces to the product of the DHTs. For the SDHT, examination of Eq. 
(41) shows that if p (x ,1/ ) is even, the SDHT of the convolution reduces to 
R S (u ,v ) = P SE (u ,v ) Q SE (u ,v ) 
+ Pso (u ,-v )Qso (u ,-v )+PSE (u ,-v )Qso (u ,v) 
Thus only three products need to be computed. 
10. Complexity 
(42) 
As noted above, the SDHT is one method of exploiting the symmetry of the DFT of 
a real sequence. Other well-known methods allow one to compute the DFT of a real 
sequence in essentially half the number of operations required for a complex sequence. 
[5] Let us call this sort of transform an RDFT. Furthermore, the result of an RDFT on 
a real sequence of length N is N /2 complex values. Following Bracewell, let us assume 
for the moment that for a sequence of length N the number of operations required by 
SDHT and RDFT are equal. The convolution via the RDFT will require N /2 complex 
multiplies, or approximately 2N real multiplies. Convolution via the SDHT will require 
4N real mUltiplies in the general case. When one of the inputs is symmetrical, then as 
noted above only aN real multiplies are required. In neither case, however, does the 
SDHT confer an advantage over the DFT. 
11. Summary 
We have shown that the most natural extension of the DHT to two dimensions fails 
to be separable in the two dimensions. We have examined an alternative, separable 
form for the 2D DHT, originally proposed by Bracewell. We have determined its rela-
tion to the DFT and have derived a corresponding convolution theorem. Finally, we 
note that the DHT is unlikely to provide faster convolution than the DHT in the general 
case. 
Acknowledgements 
We thank Albert J. Ahumada, Jr., Ronald N. Bracewell, and Brian A. Wandell Cor use-
ful discussions. 
10 
References 
[1] R. V. L. Hartley, "A more symmetrical Fourier analysis applied to transmission 
problems," Proceedings 0/ the Institute 0/ Radio Engineers, vol. 30, pp. 144-
150, 1942. 
[2] R. N. Bracewell, "Discrete Hartley transform," Journal 0/ the Optical Society 0/ 
America, vol. 73, pp. 1832-1835, 1983. 
[3] R. N. Bracewell, "The fast Hartley transform," Proceedings 0/ the IEEE, vol. 72, 
pp. 1010-1018, 1984. 
[4] H. J. Nussbaumer Fast Fourier trans/ormation and convolution algorithms. New 
York: Springer-Verlag, 1982. 
[5] IEEE Acoustics, Speech, and Signal Processing Committee Programs lor digital 
signal processing. New York: IEEE Press, 1979. 
[6] A. V. Oppenheim and R. W. Schafer Digital signal processing. Englewood Cliffs, 
N.J.: Prentice Hall, 1975. 
11 
I 2 Government Accession No 1 Report No. 
NASA TM-88203 
4, Title and Subtitle 
A SEPARABLE TWO-DIMENSIONAL DISCRETE HARTLEY 
TRANSFORM 
7 Author(s) 
Andrew B. Watson and Allen Poirson (Stanford 
Univ~rsity, Palo Alto, CA) 
9 Performing Organization Name and Address 
Ames Research Center 
Moffett Field, CA 94035 
12 Sponsoring Agency Name and Address 
National Aeronautics and Space Administration 
Washington, ,DC 20546 
15 Supplementary Notes 
3 Recipient's Catalog No 
5 Report Date 
December 1985 
6 Performing Organization Code 
8 Performing Organization Report No 
, ' A-86071 
10 Work Unit No 
11. Contract or Grant No. 
13 Type of Report and Period Covered 
Technical Memorandum 
14 Sponsoring Agency Code 
506-47 
Point of contact: Andrew B. Watson, Ames Research Center, MS 239-3, 
Moffett Field, CA 94035 (415)694-5419 or FTS 464-5419 
16 Abstract 
Bracewell has proposed the Discrete Hartley Transform (DHT) as a 
substitute for the Discrete Fourier Transform (DFT), particularly as a 
means of convolution (R. N. Bracewell, "Discrete Hartley transform," Journal 
of the Optical Society of America, vol. 73, pp. 18~2-1835, 1983). Here we 
show that the most natural extension of the DHT to two dimensions fails to 
be separable in the two dimensions, and is therefore inefficient. We con-
sider an alternative separable form, and derive a corresponding convolution 
theorem. We also argue that the DHT is unlikely to provide faster convolu-
tion than the DFT. 
17 Key Words (Suggested by Author(s)) 
Image processing 
Convolution 
Fourier transform 
19 Security aasslf (of thiS report) 
Unclassified 
18 Distribution Statement 
Unlimited 
Subject category - 53 
1
20 SecUrity Classlf (of thiS page) 
Unclassified 
No of Pages 
14 A02 
"For sale by the National Technical Information Service, Springfield, Virginia 22161 
End of Document 


A separable two-dimensional discrete Hartley transform

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