Boundary conditions for Young - van Vliet recursive filtering

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Boundary conditions for Young - van Vliet recursive filtering

Bill Triggs, Michaël Sdika

To cite this version:

Bill Triggs, Michaël Sdika. Boundary conditions for Young - van Vliet recursive filtering. IEEE

Transactions on Signal Processing, Institute of Electrical and Electronics Engineers, 2006, 54 (6),

pp.2365 - 2367. �10.1109/TSP.2006.871980�. �inria-00548616�

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 6, JUNE 2006 2365

Boundary Conditions for Young–van Vliet Recursive Filtering

Bill Triggs, Member, IEEE,andMichaël Sdika

Abstract—Young and van Vliet have designed computationally efficient methods for approximating Gaussian-based convolutions by running a re- cursive infinite-impulse-response (IIR) filter forward over the input signal, then running a second IIR filter backward over the first filter’s output.

To transition between the two filters, they use a suboptimal heuristic that produces significant amplitude and phase distortion for all points within about three standard deviations of the right-hand boundary. In this corre- spondence, a simple linear transition rule that eliminates this distortion is derived.

Index Terms—Bidirectional recursive filtering, boundary conditions, Gaussian smoothing.

I. INTRODUCTION

Young andvan Vliet (YvV) have developedcomputationally effi- cient forward–backward infinite-impulse-response (IIR) recursions for Gaussian filters [1], Gaussian derivatives [2], and Gabor filters [3].

(See [3] for their most recent design rules for Gaussians, and [4] for space-variant extensions anda performance comparison with other IIR Gaussian methods, including Deriche’s original method [5], [6].) Our approach also applies to the analogous recursion [7], [8] for B-spline- basedsignal processing. All of the YvV filters work forward, recur- sively calculating a running sumutas a linear combination of the input signalitandthekpreviousuvalues, then work backwards calculating a running sumvt as a linear combination ofut andthelpreviously calculatedvvalues:

ut= it+

k j=1

ajut0j; t = 1; . . . ; n (1)

vt= ut+ ^l

j=1

bjvt+j; t = n; . . . 1: (2) The final output is a scaledversion of vt, and faj;j=1...kg and fbj;j=1...lgare suitably chosen filter coefficients. For Gaussians, YvV chosek = landaaa = bbb[1], [3]. For other filters,itmay be a linear transformation of the original input signal, e.g., a discrete derivative for derivative filters [1].

II. PROBLEMWITHHEURISTICBOUNDARYCONDITIONS

To complete the specification (1), (2), we must fix initial conditions foruneart = 1andforvneart = n. Foru, we can pretendthat the signal existedandtook some nominal constant valuei (typically either 0 ori1) for allt < 1. The correct initialization att = 1is then to set allu10j;j=1...ktoi =(1 0 ^k_j=1aj), the steady-state response to an infinite stream ofi’s. Similarly, if we couldsuppose that for allt > n, uttook some constant valueu+, the correct condition att = nwould be to setvn+j;j=1...ltou+=(10 ^l_j=1bj), the steady-state response to an infinite stream ofu+’s. YvV apparently do exactly this, withi = i1

andu+= un(cf. [3, eq. (20) and(21)]). Another plausible choice for Manuscript receivedJuly 19, 2004; revisedJune 13, 2005. This work was partly supportedby European Union research projects LAVA andaceMedia.

The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Trac D. Tran.

The authors are with GRAVIR-CNRS-INRIA, 38330 Montbonnot, France (e-mail: Bill.Triggs@inrialpes.fr; Michael.Sdika@inrialpes.fr; website:

http://lear.inrialpes.fr/people/triggs).

Digital Object Identifier 10.1109/TSP.2006.871980

Fig. 1. Impulse responses for points at various numbers of standard deviations from the right boundary, for a YvV Gaussian filter, with the standard YvV boundary heuristicu = u (top) andwith our new boundary correction (14) (bottom). The correctedresponses are much closer to the desired(truncated Gaussian) form.

u+wouldbei+=(1 0 ^k_j=1aj), the steady stateuresulting from an infinite stream of constant input valuesi+abovet = n(typically,i+

wouldbe eitherinor 0).

Unfortunately, neither choice for u+ is correct. If the forward filter were continuedtot nwith inputi+, its output woulddecay smoothly fromuntoi+=(1 0 ^k_j=1aj)within a few standard devi- ations, andthe corresponding backwardfilter wouldtake all elements of this “advance warning” signal into account when calculating its response. In fact, the forward–backward processonlygives the correct overall impulse response when the full double recursion is run without truncation. Incorrect truncation causes significant amplitude and phase (geometric position) distortion for all points within about three standard deviations of the boundary. Fig. 1 illustrates the extent of the problem.

III. DERIVATION OFLINEARBOUNDARYCORRECTION

To correct for the effects of truncation, we notionally extendthe for- ward–backward pass tot ! 1assuming a constant input valuei+

abovet = nandcalculate the coefficientsfvn+j;j=1...lgthat would result from this infinite extension, giveni+andthe final forwardfilter statefun0j;j=0...k01g. The whole process is linear so thev’s must be linear functions of theu’s andi+. First suppose thati+= 0. Gathering 1053-587X/$20.00 © 2006 IEEE

2366 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 6, JUNE 2006

theu’s,v’s into runningk,lvectorsuuu,vvv, the forwardandbackward passes become

uuut= AuAuAut01= AAA^t0nuuun; t > n; i+ 0 (3) vvvt= III1uuut+ BvBvBvt+1; t n (4)

whereAAA^k = AAA 1 AAA 1 . . . 1 AAA(kterms) is thek^thpower ofAAAandsee (5), shown at the bottom of the page

uuut ut

... ut0k+1

AA A

a1 1 1 1 ak01 ak

1 1 1 1 0 0 ... . .. ... 0 1 1 1 1 0

(6)

vvvt vt

... vt+l01

BBB

b1 1 1 1 bl01 bl

1 1 1 1 0 0 ... . .. ... 0 1 1 1 1 0

(7)

andIII1 = (1 0 . . . 0)^>(1 0 . . . 0)is anl 2 kmatrix with a 1 in the top-left corner andzeros elsewhere. Combining these equations for all t n, we havevvvn = ( ¹_i=0BBBⁱIII1AAAⁱ)uuun. We needto calculate the l 2 kmatrixMM M ¹_i=0BBBⁱIII1AAAⁱthat links the initial backwardstate vvvnto the final forwardoneuuun. ByMMM’s recursive definition

MM

M = III1+ BMABMABMA: (8)

WritingMMM = mmm¹

... mm m^l

by rows as akl-element row vectorMM =M^$ (mmm¹; . . . ; mmm^l)and, similarly forIII1, converts (8) to

MMM =$ ^$III1+MMM^$

bbb1AAA AAA 1 1 1 0 ... ... . .. ... bbbl01AAA 0 1 1 1 AAA

bbblAAA 0 1 1 1 0

: (9)

This sparse system is easily solvedto give(eee¹= (1; 0; . . . ; 0)):

mmm¹= eee¹ III 0 ^l

j=1

bbbjAAA^j

01

; mmmⁱ= mmm¹AAAⁱ⁰¹; i = 2; . . . ; l (10)

Alternatively, if l k, it may be preferable to write MMM = (mmm₁; . . . ; mmm_k)by columns as akl-element column vectorMMM^l, so that (8) becomes

MMM^l= III^l₁+

aaa1BBB BBB 1 1 1 0 ... ... . .. ... aaak01BB 0 1 1 1 BB BB

aaakBBB 0 1 1 1 0

MMM^l (11)

with solution(eee1 (1 0 . . . 0)^>):

m m

m1= III 0 ^k

j=1

aaajBBB^j

01

eee1 (12)

mmm_i=

k j=i

aaajBBB^j0i+1 mmm₁; i = 2; . . . ; k: (13)

As an example, theMMM of the (k = l = 3,aaa = bbb) Gaussian filter recommended by YvV is given in (5), shown at the bottom of the page.

Finally, to handle nonzeroi+, we can simply reduce to thei+ = 0 case by subtracting the constant-uresponseu+= i+=(1 0 ^k_j=1aj) from each component ofuuun, applyMMM, then add back the corresponding constant-vresponseu+=(1 0 ^l_j=1bj)to getvvvn.

IV. SUMMARY OFMETHOD

In summary, Young andvan Vliet recursive filters suffer from se- vere amplitude and phase distortion at the right boundary unless the backwardrunning coefficients are initializedfrom the forwardones as follows, whereMMMis given by (5), (8), (10) or (12) and(13):

vn

... vn+l01

= MMM

un0 u+

... un0k0 u+

+ v+

... v+

(14)

u+= i+

1 0 ^k_j=1aj v+= u+

1 0 ^l_j=1bj: (15) An implementation for two-dimensional Gaussian image filtering is available on the author’s website. J.-M. Geusebroek has also incor- poratedthe technique in his IIR filtering package, available from his website http://www.science.uva.nl/~mark/downloads.html

ACKNOWLEDGMENT

The authors wouldlike to thank J.-M. Geusebroek for pointing out an error in the submittedversion of (14).

M

MM = 1

(1 + a10 a2+ a3)(1 0 a10 a20 a3) 1 (1 + a2+ (a10 a3)a3) 2

0a3a1+ 1 0 a²₃0 a2 (a3+ a1)(a2+ a3a1) a3(a1+ a3a2) a1+ a3a2 0(a20 1)(a2+ a3a1) 0(a3a1+ a²₃+ a20 1)a3

a3a1+ a2+ a²₁0 a²₂ a1a2+ a3a²₂0 a1a²₃0 a³₃0 a3a2+ a3 a3(a1+ a3a2)

(5)

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 6, JUNE 2006 2367

REFERENCES

[1] I. Young andL. van Vliet, “Recursive implementation of the Gaussian filter,”Signal Process., vol. 44, pp. 139–151, 1995.

[2] L. van Vliet, I. Young, andP. Verbeek, “Recursive Gaussian deriva- tive filters,” inProc. Int. Conf. Pattern Recognition, Brisbane, 1998, pp.

509–514.

[3] I. Young, L. van Vliet, andM. van Ginkel, “Recursive Gabor filtering,”

IEEE Trans. Signal Process., vol. 50, no. 11, pp. 2799–2805, Nov. 2002.

[4] S. Tan, J. Dale, andA. Johnston, “Performance of three recursive algo- rithms for fast space-variant Gaussian filtering,”Real-Time Imag., vol.

9, pp. 215–228, 2003.

[5] R. Deriche, “Recursively implementing the Gaussian andits deriva- tives,” in Int. Conf. Image Processing, Singapore, Sep. 1992, pp.

263–267.

[6] , “Recursively implementing the Gaussian andits derivatives,”

INRIA Sophia-Antipolis, Montbonnot, France, Research Rep. 1893, 1993.

[7] M. Unser, A. Aldroubi, and M. Eden, “B-spline signal processing: Part I—Theory,”IEEE Trans. Signal Process., vol. 41, no. 2, pp. 821–833, Feb. 1993.

[8] M. Unser, A. Aldroubi, and M. Eden, “B-spline signal processing: Part II—Efficient design and applications,”IEEE Trans. Signal Process., vol.

41, no. 2, pp. 834–848, Feb. 1993.