An Accurate Discrete Fourier Transform for Image Processing

Transcription

1 An Accurate Discrete Fourier Transfor for Iage Processing Norand Beaudoin' and Steven S. Beauchernint Abstract The classical ethod of nuerically coputing the Fourier transfor of digitizedfunctions in one or in d- diensions is the so-called discrete Fourier transfor (DFT), efficiently ipleented as Fast Fourier Transfor (FFT) algoriths. In ny cases the DFT is not an adequate appxiation of the continuous Fourier transfor. The ethod presented in this contribution provides accurate approxiations of the continuous Fourier transfor with siilar tie coplexity. The assuption of signal periodicity is no longer posed and allows to copute nuerical Fourier transfors in a broader doain of frequency than the usual halfperiod of the DFT. In iagepcessing this behavior is highly welcoed since it allows to obtain the Fourier transfor of an iage without the usual interferences of the periodicity of the classical DFT. The atheatical ethod is developed and nuerical exaples are presented. 1 Introduction The Fourier transfor and its nuerical counterpart. the discrete Fourier transfor (DFT), in one or in any diensions, are used in any fields such as atheatics, physics, cheistry, engineering, iage processing, and coputervision(4, , 8, 14, I]. "The DIT is of interest priarily because it approxiates the continuous Fourier transfor" [4]. In this regard, the DFT, usually coputed via a fast Fourier transfor (FFT) algorith, ust he used with caution since it is not a correct approxiation in all cases [13, IO, 15, 61. As an exaple, for a function such as h (1) = e-s0t, t E [0,1], the error on DFT { h} around f = 64 decreases roughly as N-'I3. Hence, one ust increase N by a factor of 1000 to 'Norand Beaudoin is a post-doctoral fellow at the Uepanent of Cornputer Science, Middlcsei College. University of Western Ontario. London. Canada, N6A 5B7. Phone: (519) Eail: norand@csd.uwo.ca 'Steven S. Beauchein is with the Uepanent of Coputer Science. Middlesex College, University of Western Onlario. London, Canada. N6A 5B7. Phone: (519) I.Eail: kau@csd.uwo.ca decrease the error by a factor of 10. One ay increase the accuracy of the nuerical Fourier transfor when the nuber of sapled data points is liited. This can be ipleented through the assuption that the function fro which the sapled data points are extracted and its derivatives are continuous. In a sense, the sapling process perfored through the Dirac cob [4] isolates each data point, rendering the independent fro each other. The function and its derivatives are no longer continuous. By re-establishing the continuity between sapled points. a ethod that yields a highly accurate nuerical Fourier transfor can be devised. 2 Theory Lett E W and f E W. As usual, C is the field of coplex nubers, W is the set of real nubers, N the set of nonnegative. integers - and PI* = M\.. (0).. Let us define two rectangular functions: and R (t) = x (t - 0-) x (-t + T+) S (1) = x (t) x (-t + T) I where xis Heaviside's function, and let 0- = lirn (0 -E), T: = lirn (Ta S i 0 c+n Let g : R + (R or C) he a continuous function which adits derivatives of any order for all t such that S (t) # 0. Let us now define the function h (t) = R (t) 9 (t) (3) and adopt the following definition of Fourier transfors F{h(t))=/ I~(t)e-'~"f*dt (4) R In virtue of Heaviside's and Dirac's delta functions properties 112. SI, the nth derivative of h with respect tot is h(") (t) = x (t - 0-) x (-t + T') g(") (t) + D, (t) (5) UO2 $17.00 Q 2002 IEEE 935

2 in which D, (t) is defined as sus on j and p. we obtain: D, (1) = 0 if n=o { Zb ($w(o-)6"--'(t - 0-) -g()(~+)~"-~-i(t - T+)) if n E N* (6) Eq. (5) and (6) express the fact that the nth derivative of h with respect to 1 is the ordinary nfh derivative of the function h strictly inside the rectangular box where it is continuous and differentiable, in addition to the nth derivative of h in the region where it is discountinuous. According to our definition of the Fourier transfor, we have: To nuerically copute the Fourier transfor of h, we ust evaluate it for soe discrete values of f. Let f = kaf = k/t, k E N be these discrete variables. In addition, let us define Hk as the discrete version of 7 { h(") (t)). The integral in (12) depends only on the varable f (or k) and on paraeters p and At and can be evaluated analytically, whether f is continuous or discrete, once and for all, for each value of p as: We can expand the integral in (7) into parts to for: The su of the first and last integrals of the right handside of (8) is clearly 7 { D, (t)}. Hence, (8) becoes: T F{hln)(t)} = /h(")(t)e-""ltdl+f{d. (t)} (9) n By separating the interval [0, T] into N equal At = TIN subintervals. (9) can be rewritten as: (10) Since h("), between and at 0 andt is continuous and differentiable, it can be approxiated, fort E IjAt, ( j + 1) At]. for each j E [0, N - 11, by a Taylor expansion: where hi.'"'is the fh derivative of h at the point t = jat. Merging (IO) and (1 I), using the substitution r = t - jat and perforing the adequate perutation of the integral and Since the integral in the definition of Ip is always finite and. in the context of Gaa function [9], p! = foo when p is a negative integer then I, = 0 for p < 0. The suation on j in (12). when f = kaf = k/t is the discrete Fourier transfor of the sequence hy"), j E [0, N - 11 C N [4]. We denote it as Fp+,,,k. Since At = TIN and f = k/t we have: N-1 Fp+n,k = hip+")e-'z"% (14) j=n Substituting (13) and (14) in (12). we obtain the following result:.f {h'") (t)} = 2 IpFptn,l; + F {D, (t)} (15) p=n When n = 0, (15) becoes: Hk = IpFp,k (16) p=o Now, integrating by parts the right hand side of (7) yields: 7 { h("+')} = i27rf7 { h(")} (17) Defining b, = i27rf7 { Dn} -+{D.+l},cobining (15) and (17) and reorganizing the ters yields: -i2afiofn,k+x (I(p-~) - i27rf1,) Fp+& = b, (18) p=1 With the following definition: J, = I--, - i27rfi, (19) 936

3 hg) (1 8) becoes: p=1 Given the definition of g and h. we have g(") (0-) = g(") (0) = h(") (0) andg(") (Tt) = g(") (T) = h(") (T). Using these facts in addition to the properties of Fourier transfors and those of Dirac delta functions [5], expanding b, results in the siple following for: In the discrete context, where f = k IT, (21) takes a siple for: b, = h(") (T) - h(") (0) = h$) ~ (22) Up to this point, all equations are rigorously exact since p tends towards infinity. However, in practical situations we introduce approxiations by liiting the range on p. Let us define H E N. the truncating paraeter. We refer lo it as the order of the syste. Let us expand (20) for each value of n E [O, B - 11 c N. This generates a syste of Q equations, and for each of these we let p range fro 1 to Q - n. Thereafter, the ters are reorganized to obtain the following syste, which is written in a atrix for: or ore copactly as: MhFh Y B + C The general expression for eleents of Mb is: Let us now write (24) as: (Mh)pv = J,-,+I Fh Y Mh' (B + C) Provided a nuerically known atrix B (see next section on boundaries conditions). the values of FD are then obtained fro (26). Hence, the ters in (16), for p E [O,Q]. are copletly deterined, and the truncated version of (16) can be written as: (27) Let us define a one row atrix Is = [I, I2... Is], and write (27) as follows: Hk Y 1nFn.k + lefh (28) 1 0 ' (23) (24) (26) With (28). we approxiate the Fourier transfor (or the inverse Fourier transfor) of a digitized function in one diension. The digitized Fourier transfor calculated with (28) is not band liited (as with the DFT which is periodical). Eq. (28) reains valid and accurate as an approxiation of the analytical Fourier transfor for all positive or negative values of k (or f ) [2]. Eq. (28) contains the sybolic for of Fh which can be used as is to for a single equation without having to nuerically evaluate each ter of Fb separately. On the other hand, if, for instance, (26) is used to nuerically copute each ter of F b for values of k fro 0 to N - 1, it produces Q different sequences of nubers which are approxiations of the DFT of the derivatives h:p), for values of p E [l, Q]. Thus, applying the inverse DFT operation to each of these sequences generates the corresponding sequences h:p) that are the nuerical derivatives of order 1 to Q of the initial function h:n). This iplies that one can nuerically copute the derivatives of any order of a digitized function or signal [2]. Derivatives calculated in that anner are continuous inbetween and at each data point. We obtain spline polynoials of any odd degree, with their corresponding properties, erely with the use of a classical DFT (FFT) [2]. Such high-order spline interpolation polynoials allows integrals between any liit to be accurately coputed [2]. 3 Boundary conditions Eq. (26) requires the values of B to be properly coputed. Eq. (22) shows that the values of B are directly related to the values of the derivatives of different orders at both ends (0 and T) of the function. The coputation of such derivatives based on a few data points near the boundaries of the function is known to be a crude and often inaccurate ethod. In [21 and [3], a ethod based on a11 the data points of the function has been devised, resulting in high nuerical accuracy. The drawback is that soeties a liitation on the order H of the syste is iplied. Most of the tie that liitation is harless but, in iage processing. it ay happen that Q < 1. To overcoe that inconvenient, a ore robust ethod that still uses the inforation fro all data points of the function has been devcloped.. This ethod is based on the fact that the nuerical Fourier transfor presented in this paper gives the derivatives of any order up to Q of the initial function. Furtherore, as entioned earlier, these derivatives for spline polynoials of degree H that interpolate the sapled initial function hj, Hence it is possible to copute a spline nor [7] that contains the boundary conditions (B) as paraeters. It is then possible to adjust these paraeters to iniize the spline nor. Liitation on space does not perit to go through1 all the atheatical details of the ethod. However let us briefly show the 937

4 ain operational equations. The one colun vector or the 0 x 1 atrix B that contains the boundaries conditions is given by: S is the following three-suation operator:.. In (29) we define the atrix as follows: El = - -. T. [Ellj...ElejlT and EL = EL^^,..E~ej] in which T eans the tranose (not coniuaate) and the overbar eans the coplex conjugate. In addition, Aij = D-' {-PIIJOF,~}~ and Ei,j = {qi}j. D-' stands for the inverse discrete Fourier transfor. The ter qi represents the eleent in the lth row,,nth colun of the atrix hi;'. All the eleents, except A that depends on h, depend on the paraeters (A', T) of the function and not on the actual values hj of the function. Hence, they are coputed only once. 4 Tie coplexity A close exaination of (23) and (28) reveals that the coputation of only one classical FFT is required. The other ters for a correcting operation to be applied once on each of the N values of the FFT. The tie coplexity of the entire correcting operation is 0 (N) and the tie coplexity of the FFT is 0 ( N log N ). Hence, the tie coplexity of the entire algorith is 0 ( N log N) when 0 is kept constant. The tie coplexity relatively order of the syste, is 0 (O*). However, the error E on coputed results decreases exponentially with an increase of the order 8. The ethod can be applied sequentially to copute d-diensional Fourier transfors. In the ultidiensional case, for each a E { 1,2,..., d } we have t, E [O, T,]. This interval is separated into Ne equal At, = Te/Ne suhintervals. and fa = kaaf, = k,/t,. As with the ordinary DFT (FFT). the order in which the diensions are treated is irrelevant. The nuber of ties (28) has to be applied to copletely copute the d-diensional Fourier transfor is: d 1 PQ where P = n N, and Q = - (31),=I p=1 The tie coplexity is then 0 (P log P). Let us consider N, = o,n, Va, a, as constants. Then it is easy to show that the tie coplexity is 0 (Nd log N) which is the sae as for the DFT in d-diensions. d 5 Exaples in 1 diension and on iage In this section, an exaple in one diension is used to illustrate the algorith and an iage processing exaple is presented. The one diensional function is a decaying exponential (exp(-50t)) representing a frequent situation in iages where values changes rapidly within a few pixels only. Figures 1 shows, respectively, the classical DFT and the new discrete Fourier transfor (NewDFT), both evaluated fro the N = 64 data points of the exponential decay. The usual periodical behavior of the DFT is evident and the absence of periodicity of the NewDFT is clearly seen. The sae figure also shows the percentage of error of the DFT (fairly coputed on the first half of the period only) and the error of the NewDFT coputed on the full nuber of data points. The error of the DFT is between 40% and 120% while the error of NewDFT (with 6 = 9 ) is between 0.05% and 0.25%. Figure 2 shows the iage of a siple occlusion scene involving two sine plaids, for which the Fourier transfor ust be coputed to reveal the frequency structure of the occlusion. This type of iages is relevant to theoretical and practical studies in iage processing [I]. The sae figure also shows the DFT and the NewDFT (0 = 3) of the sine plaid occlusion scene. It should be noted that the initial signal is 64x64 and the transfored signals have beencoputed on 128x128 data pointscentredon the origin (fl, f2) = (0,O). One can clearly see the repeated lines and peaks due to periodicity of the classical DFT. On the other hand, the NewDFT copletely eliinates that annoying behavior. Even on an extended nuber of data points, the NewDFT gives an excellent approxiation of what would be the analytical Fourier transfor (often difficult if not ipossible to copute) of the iage. 6 Conclusion The ethod presented in this contribution provides accurate approxiations of the continuous Fourier transfor. is no longer periodical and yields a broader frequency doain than the usual half-period of the DFT. The ethod gives accurate nuerical partial derivatives of any order and the polynonial splines of any odd degree with their optial boundary conditions. The tie coplexity is the sae as for the FFT. The tie coplexity, relatively to 6 (independent of the tie coplexity related to N) is 0 (@*) while the accuracy increase exponentially with 0. Hence, the nuerical accuracy increases uch ore rapidly than the coputational cost of the proposed ethod. Finally, results show that a significant iprovent is accoplished in iage processing throught the coplete eliination of periodicity in the coputed nuerical Fourier transfor. 938

5 ~~ DFT (top) and NewDFT (botto) Error on DFT Error on NewDFT Figure 1. Classical discrete Fourier transfor (DFT, top curve) and discrete Fourier transfo given by the new ethod (NewDPT, botto curve) and percentage of error on DFT coputed on the first half of data points only and percentage of error on NewDFT coputed on the full range of data points. The exact Fourier transfor is not shown since it is indistinguishable fro NewDFT. Occlusion DP3' of occlusion :Vrwl)l~'l' 01 occlusioii Figure 2. Occlusion of a two diensional signal by an another of different frequency and orientation and the discrete Fourier transfor of the occlusion coputed with the classical DFT and with the NewDFT. References [I] S. Beauchein and J. Baon. On the fouricr properties of discontinuous otion. Journal of Marhernuiical loging ondvision, 13: , [Z] N. Beaudoin. A high-accuracy atheatical and nuerical ethod for fourier transfor. integral, derivative, and polynoial splines of any order. Canadion Journal of Physics. 76(9):659477, [31 N. Beaudoin and S. Beauchein. Accurate nuerical fourier transfor in d-diensions. TR-S6I. Uepr. Cornpurer Science, The Universiry oj Wesrern Ontario. pages [4] E. 0. Brigha. The Jus1 Fourier rronrfonn. Prentice-Hall E. Butkov. Marheiical Physics. Addison-Wesley, Rcading, M. Froeven and L. Helleans. IDroved algorith for the discrete fourier transfor. Rev. Sci. Insrru., 56( 12): [7l K Encyclopedic Diciiony of Marhenrics. Matheatical Society of Japan, The MIT Press, [RI C. Kiltel. Physiy-srdel'hui solide. Dunod Universitl' [9] E. Kreyszig. Advanced ginewing rnurlwnrrrii~s. Wiley [IO] S. Makinen. New algorith for the calculalion of [he tourier transfor of discrete signals. H Sci. lnrtnr,n.. 53(5): [I I] L. Marchildon. Mt'coniyur Qtianriyr~r. De Bocck Unwcrsite [ I21 N. Morrison. Inrroducrion io Fourier arralvrir. Wiley- Interscience, (I31 J. Schutte. New fat fourier transfor algorith for linear syste analysis applied in uleculnr bea relaxation rpectrosc6py. Rev. Sci. Insrrutn.. 52(3):400. I YX I [I41 L. G. Shapiro and G. C. Stockan. Copurw Vision. Pren- [ice Hall, [I51 S. Sorella and S. K. Ghorh. Iproved ethod for [he discrete fast fourier transfor. Re>! Sci. Insrrira ): I34X