Wiener Deconvolution: Theoretical Basis

Transcription

1 Wiener Deconvolution: Theoretical Basis The Wiener Deconvolution is a technique use to obtain the phase-velocity ispersion curve an the attenuation coefficients, by a two-stations metho, from two pre-processe traces (instrument correcte) registere in two ifferent stations, locate at the same great circle that the epicenter (Hwang an Mitchell, 1986). For it, the earth structure crosse by the waves between the two stations selecte, can be consiere as a filter that acts in the input signal (registere in the station more near to the epicenter) transforming this signal in the output signal (registere in the station more far to the epicenter). Thus, the problem can be written in convolution form as: g(t) = f ( τ)h(t - τ) τ where f(t) is the trace recore at the station more near to the epicenter, g(t) is the trace recore more far an h(t) is the time-omain response of the interstation meia.

2 Wiener Deconvolution: Theoretical Basis In frequency omain, the above-mentione convolution can be written as: g( t) = f( t) h( t) G( ω) = F( ω) H( ω) G( ω ) = F( ω) H( ω) ΦG( ω) = ΦF( ω) + ΦH( ω) amplitue spectrum phase spectrum F(ω) = input-signal spectrum,, H(ω) = meia-response spectrum G(w) = output-signal spectrum (Green Function) Then, the frequency-omain response of the inter-station meia H(ω) can be written as: G( ω) H( ω ) = ΦH( ω) = ΦG( ω) - ΦF( ω) F( ω)

3 Wiener Deconvolution: Theoretical Basis A problem arises from the spectral holes present in the input-signal spectrum F(ω). These points are small or zero amplitues that can exit for some values of the frequency ω. At these points the spectral ratio, preforme to compute H(ω), can be infinite or a big-meaningless number for these frequencies. A metho to avoi this problem is the computation of the Green Function by means of a spectral ratio of the cross-correlation ivie by the auto-correlation: * ω y(t) = g(t) f (t) Y( ω ) = G( ω ) F ( ) x(t) = f (t) f (t) X( ω ) = F( ω * ) F ( ω ) G( ω)f H( ω) = F( ω)f an the amplitue an phase spectra of H(ω) are compute by: * * ( ω) ( ω) Y( ω) = X( ω) H( ω) = Y( ω) X( ω) = G( ω) F( ω) F( ω) F( ω) Φ Φ Φ H Y X ( ω) = ΦY( ω) - ΦX( ω) = ΦG( ω) - ΦF( ω) ( ω) = Φ ( ω) = Φ G F ( ω) - Φ ( ω) - Φ F F ( ω) ( ω) = 0

4 Wiener Deconvolution: Theoretical Basis The above-mentione process is calle Wiener econvolution. The goal of this process is the etermination of the Green function in frequency omain. This function H(ω) can be use to compute the inter-station phase velocity c an the attenuation coefficients γ, by means of: c(f ) = ft f + ( Φ (f ) 0 H ± N) ln H(f ) γ(f ) = sin sin where f is the frequency in Hz (ω = πf), N is an integer number (Bath, 1974), t 0 is the origin time of the Green function in time omain, is the interstation istance, 1 is the epicentral istance for the station near to the epicenter an is the epicentral istance for the station far to the epicenter. These istances are measure over the great circle efine by the stations an the epicenter. 1

5 Wiener Deconvolution: Theoretical Basis The inter-station group velocity U g an the quality factor Q, can be obtaine from c an γ by means of the well-known relationships (Ben- Menahem an Singh, 1981): 1 U g = 1 c + T c c T Q = π γu g T where T is the perio (T = 1/f). The inter-station group velocity also can be compute applying the MFT to the Green function expresse on time omain (Hwang an Mitchell, 1986).

6 Wiener Deconvolution: An Example E S 1 S The Wiener filtering, as an example, has been applie to the traces shown above, which has been instrumentally correcte. These traces have been recore at the stations S 1 more near to the epicenter E an S more far.

7 Wiener Deconvolution: An Example The cross-correlation of the traces S 1 an S an the auto-correlation of the trace S 1 are performe. The auto-correlation can be winowe to remove noise an other unesirable perturbations, which can prouce spectral holes in the autocorrelation spectrum an the Green function on frequency omain.

8 Wiener Deconvolution: An Example The phase velocity (a), attenuation coefficients (c), group velocity (b) an quality factor (); can be calculate through the Green function in frequency omain, as it has been explaine in the previous slies.

9 Wiener Deconvolution: References Bath M. (1974). Spectral Analysis in Geophysics. Elsevier, Amsteram. Ben-Menahem A. an Singh S. J. (1981). Seismic Waves an Sources. Springer-Verlag, Berlin. Hwang H. J. an Mitchell B. J. (1986). Interstation surface wave analysis by frequency-omain Wiener econvolution an moal isolation. Bull. of the Seism. Soc. of America, 76, Wiener Deconvolution: Web Page

10 Approximation Theory Given a Hilbert space H with an inner prouct (, ) we e ne (' 1 ; ' ) = jj' 1 ' jj = p (' 1 ' ; ' 1 ' ) Given a subspace S H with a basis ( 1 ; ; ; :::; n ) P Any b' S can be written as b' = n i i. i=1 Problem: Given ' H n b' S so that jj' b'jj is a minimum Solution: Choose ' so that (' b'; v) = 0 for every v S 1

11 Wiener Filter Notation: f(x; y) = "original" picture g(x; y)= "actual" picture bf(x; y) = estimate to f(x; y) Assumption: the noise is aitive. So (*) g(x; y) = RR h(x; x ; y; y)f(x; y)xy+ n(x; y) We assume h(x; x ; y; y) is known (?) Assumption: 1. E [n(x; y)] = 0. E [f(x; y)f(x; y)] = R ff (x; y; x; y) E [n(x; y)n(x; y)] = R nn (x; y; x; y) known Problem: Given g(x; y) n f(x; b y) so that " = E f(x; y) f(x; b y) is a minimum Solution: To get a meaningful solution we nee to assume that b f(x; y) is a linear function of g(x; y). bf(x; y) = RR m(x; x ; y; y)g(x; y)xy We nee to calculate m(x; x ; y; y) as a function of g an h We wish that RR E bf(x; y) m(x; x ; y; y)g(x; y)xy be a minimum Using the orthogonality theorem we have h RR i E bf(x; y) m(x; x ; y; y)g(x; y)xy; g(ex; ey) = 0 all ex; ey Interchanging the orer of the integrals h i E bf(x; y)g(ex; ey) = RR E m(x; x ; y; y)g(x; y)g(ex; ey)xy

12 or using the cross-correlation h i R fg (x; y; ex; ey) = E bf(x; y)g(ex; ey) R gg (x; y; ex; ey) = E [g(x; y)g(ex; ey)] R fg (x; y; ex; ey) = RR R gg (x; y; ex; ey)m(x; x ; y; y)xy image This is still very i cult so we assume that all the statistics are homogeneous an invariant. So Then R fg (x ex; y ey) = RR Fourier transform yiels but from (*) we have R fg (x; y; ex; ey) = R fg (x ex; y ey) R gg (x; y; ex; ey) = R gg (x ex; y ey) R nn (x; y; ex; ey) = R nn (x ex; y ey) m(x; x ; y; y) = m(x x y y) image R gg (x ex; y ey)m(x x ; y y)xy = R gg (x ex; y ey) m(x; y) S fg (u; v) = S gg (u; v)h(u; v) g(x; y) = RR h(x; x ; y; y)f(x; y)xy+n(x; y) = RR h(x x ; y y)f(x; y)xy+n(x; y) an in Fourier space G = HF + N 3

13 Wiener-Khinchine Theorem Theorem 1 The Fourier Transform of the spatial autocorrelation function is equal to the spectral ensity jf(u; v)j Proof. The spatial autocorrelation function is e ne by R ff (ex; ey) = 1R 1 1R 1 f(x + ex; y + ey)f(x; y)xy Muliply both sies by the kernel of the Fourier transform an integrate. Then br ff (ex; ey) = De ne Then = 1R 1R 1 1 1R 1R 1 1 R ff (ex; ey)e i(exu+eyv) exey 1R 1 1R 1 f(x + ex; y + ey)f(x; y)e i(exu+eyv) xyexey s 1 = x + ex s = y + ey 1R 1R 1R 1R br ff (ex; ey) = = 1R 1 1R f(s 1 ; s )f(x; y)e i((s1 x)u+(s x)v) xys 1 s f(s 1 ; s )e i(s1u+sv) s 1 s = F(u; v)f (u; v) = jf(u; v)j 1R 1R f(x; y)e i(xu+yv) xy 1 1 4

14 De nition: A vector has a Gaussian (or normal) istribution if its joint probability ensity function has the form p(x; ; C) = 1 p () n et(c) e 1 (x )t C 1 (x ) where C is a nxn symmetric positive e nite matrix E[X] = cov(x) = C De nition: A vector has a Poisson istribution if its joint probability ensity function has the form p(x; ) = n Q i=1 xi i e i x i! x i is a nonnegative integer where C is a nxn symmetric positive e nite matrix E[X] = cov(x) = iag( 1 ; ; :::; n ) 5

15 Summary Wiener Filter The Wiener filter is the MSE-optimal stationary linear filter for images egrae by aitive noise an blurring. Calculation of the Wiener filter requires the assumption that the signal an noise processes are secon-orer stationary (in the ranom process sense). Wiener filters are often applie in the frequency omain. Given a egrae image x(n,m), one takes the Discrete Fourier Transform (DFT) to obtain X(u,v). The original image spectrum is estimate by taking the prouct of X(u,v) with the Wiener filter G(u,v):

16 The inverse DFT is then use to obtain the image estimate from its spectrum. The Wiener filter is efine in terms of these spectra: The Wiener filter is: Diviing through by makes its behaviour easier to explain:

17 Diviing through by The term can be interprete as the reciprocal of the signal-to-noise ratio. Where the signal is very strong relative to the noise, an the Wiener filter becomes - the inverse filter for the PSF. Where the signal is very weak, an.

18 For the case of aitive white noise an no blurring, the Wiener filter simplifies to: where is the noise variance. Wiener filters are unable to reconstruct frequency components which have been egrae by noise.

19 They can only suppress them. Also, Wiener filters are unable to restore components for which H(u,v)=0. This means they are unable to uno blurring cause by banlimiting of H(u,v). Such banlimiting which occurs in any realworl imaging system.

20 Steps Obtaining can be problematic. One can assume has a parametric shape, for example exponential or Gaussian. Alternately, can be estimate using images representative of the class of images being filtere.

21 Wiener filters are comparatively slow to apply, since they require working in the frequency omain. To spee up filtering, one can take the inverse FFT of the Wiener filter G(u,v) to obtain an impulse response g(n,m). This impulse response can be truncate spatially to prouce a convolution mask. The spatially truncate Wiener filter is inferior to the frequency omain version, but may be much faster.

22 Constraine Optimisation The algebraic framework can be use to evelop a family of image restoration methos base on optimisation. Imposing appropriate constraints on the optimisation allows us to control the characteristics of the resulting estimate in orer to enhance its quality. The egraation moel is the same: an LSI egraation an aitive noise: f = H s + Suppose we apply an LSI filter H to f an obtain an estimate g: g = Hf. Noting that the noise energy can be written as H s f = n = n n

23 Filter Design Uner Noise Constraints we woul expect that a goo estimate woul satisfy the conition. H g f = n A simple optimisation strategy woul be to assume that the noise is low energy an simply choose the estimate that minimises vector, we obtain: H g f. Taking the erivative with respect to the estimate T ( H g f) ( H g f) g = H T ( H g f)

24 Minimum Noise Assumption Setting to zero an solving for g, we obtain what is known as the unconstraine optimisation estimate: 1 g = H f. This is the familiar inverse filter, the eterministic solution, appropriate for the zero noise case, but exacerbating any high frequency noise. To avoi this result, the constraine optimisation approach introuces a new criterion, J ( g ) = Qg which is to be minimise subject to the noise energy constraint, H g f = n. The criterion matrix Q is an LSI system chosen to select for the unesirable components of the estimate.

25 . If Q is high pass in nature, the estimate will be smoother than an unconstraine result. Our problem can now be formalise as a classical constraine optimisation problem that can be solve using the metho of Lagrange. We want to minimise J ( g ) = Qg subject to the H g f = n

26 The metho of Lagrange augments the criterion with a term incorporating the constraint multiplie by a constant: J ( g) + λ [ ] H g f n

27 Now, we take the erivative with respect to g an set to zero, which gives us an expression for the estimate in terms of the Lagrange multiplier constant. g ( ( ) ( ) ) T T T T T T g Q Qg + λ Hg f Hg f n n = Q Qg + λh ( H g f ) Setting to zero, we obtain: g [ ] T T 1 T H H + Q Q H f = γ

28 In principle, we impose the constraint in orer to fin the multiplier λ= 1γ an thus obtain the estimate that satisfies the constraint. Unfortunately, there is no close form expression for λ. While it is possible to employ an iterative technique, ajusting λ at each step to formally meet the constraint as closely as esire, in practice, λ is typically ajuste empirically along with the criterion matrix Q.

29 After iagonalization the frequency response of the filter is: H( u) = H ( u) H ( u) +γ Q( u) Now, consier the choice of the criterion matrix Q, an the Lagrange constant γ. Obviously if either or both are zero, the filter reuces to the unconstraine, inverse filter result. There are three common more interesting choices. If we let Q= I, the criterion is just the estimate energy, g, an the frequency response is: H( u) = H H ( u) ( u) +γ.

30 This simple filter is a pragmatic alternative to the Wiener filter when estimation of the signal an noise spectra are ifficult. The constant is etermine empirically to insure that at high frequencies, the egraation moel has small frequency response, 1 H ( u) H ( u) γ rather than growing with frequency an amplifying high frequency noise like the inverse filter

31 A secon choice is to set ) ( ) ( ) ( u P u P Q u s n = γ, which results in the Wiener econvolution filter, ) ( ) ( ) ( ) ( ) ( u P u P u H u H H u s n + = For our stanar signal moel, the signal spectrum ecreases with frequency square so the enominator of grows as frequency square, causing greater smoothing than the minimum energy estimate filter. H (u)

32 A thir choice of the criterion is the secon ifference operator, with impulse response [ 1 1] q( n) = an the corresponing impulse response matrix Q. The continuous version of the secon ifference operator is the secon erivative, with frequency omain behaviour proportional to. The enominator of now grows u H (u) as an the estimate is even smoother than that of the Wiener econvolution filter.

33 Figure 15.1 shows examples of constraine optimization results. The egraation moel is Gaussian low pass filter blur an aitive Gaussian white noise. Note that the Q= I conition with γ =0. 0 oes the least smoothing, the secon ifference criterion with γ =0. 0 oes the most smoothing, an the Wiener econvolution filter is intermeiate between these two. Degrae Image Q = I, gamma =.0 Q = [n ifference], gamma =.0 gamma Q = 1/SNR

34 Aaptive Filters for Image Smoothing Funamental issues: Noise (typically high frequency) an signal (typically eges, also comprising high frequency components) overlap in frequency an cannot be separate by the simple frequency component weighting characteristic of LSI systems. Alternately, image signals are non-stationary an any global processing is likely to be sub-optimal in any local region. Early efforts to aress this problem took an approach base on local statistics, esigning operators that were optimise to the local image characteristics.

35 local linear minimum mean square error (LLMMSE) filter introuce by Lee in In this approach, an optimal linear estimator for a signal in aitive noise is forme as sˆ ( n) = α f ( n) + β f ( n) = s( n) + n( n) is the observation The noise an signal are assume to be inepenent.

36 Let the noise be a zero mean white Gaussian noise process with variance α σ n The parameters an β are chosen to minimise the mean square estimation error criterion J ( α, β ) = E ( sˆ( n) s( n) ) = E αf ( n) + β s( n) [ ] [( ) ]

37 Taking the erivative of the ( ) [ ] 0 ) ( ) ( ) ( ), ( = + = n f n s n f E J β α α β α [ ] 0 ) ( ) ( ), ( = + = n s n f E J β α β β α Solving these equations for α an β, using the results ) ( ) ( n s n f = an n s f σ σ σ + =, n s s σ σ σ α + = ) ( ) 1 ( n f β α =.

38 The resulting LLMMSE estimate is then: ˆ σs ( n) σn s( n) = f ( n) + f ( n), σ ( n) σ ( n) f f The estimate is a weighte sum of the observation an its local average, using local variance for the weighting. When the local signal variance is much greater than the constant noise variance, the estimate is just the observation no smoothing occurs. When the local variance is entirely attributable to noise, the estimate is just the local average maximum smoothing occurs The LLMMSE filter oes little near eges or high contrast texture regions an smoothes as much as it can when the signal component is constant

39 Note that we have to estimate the noise variance somehow, as we i with the conventional Wiener filter. The choice of winow size over which to estimate the local mean an variance is important. It nees to be at least 5 5 for reasonable variance estimates, but it shoul be small enough to insure local signal stationarity. Lee foun that both 5x5 an 7x7 worke well.

40 Results illustrating the effect of the LLMMSE filter for three ifferent region sizes are shown in Figure Note the resiual noise within the local neighbourhoo near strong eges. Original plus noise 3x3 5x5 7x7

41 The LLMMSE filter can be interprete as a local impulse response or mask whose weights are functions of the local statistics in the input. Specifically, α + (1 α)/ N n = 0 " h( n)" = (1 α)/ N n ( N 1)/ 0 else where = 1 ( n) an N is the winow size over which the local statistics are α σ n σ f estimate. You can also etermine a local frequency response whose banwith is a function of local signal-to-noise variance ratio.

42 Limitations of the LLMMSE filter are apparent. The nee to restrict the winow's size to achieve local stationarity restricts the amount of smoothing that can be achieve in constant signal regions. It may be esirable to allow the smoothing winow to grow as large as the signal constant region allows. Some alternate noise suppression scheme to simple averaging shoul be employe, such as a local orerstatistic filter (meian filter an the like, which we will iscuss further later). Alternately, it may be esirable to apply the LLMMSE filter iteratively, achieving repeate smoothing in constant signal regions. The winow's square shape is also a limitation.

43 Near an ege, the LLMMSE filter oes no smoothing, allowing visible noise in close proximity to the ege. The winow shape shoul also aapt, allowing the filter to smooth along but not across eges. The constant weights within the winow, except for the centre point, limit the smoothing to the box filter, or simple local averaging variety. Perhaps some sort of locally variable weighting within the winow woul improve the performance near eges.

44 An extension of the local statistics filter that aresses these limitations is the local aaptive recursive structure where the output is forme with a local recursive ifference equation whose coefficients epen on the local input statistics. A simple one imensional version is: g( n) = α f ( n) + βg( n 1) β =1 α, a similar ege- If the parameter α is chosen the same as for the LLMMSE filter an epenent smoothing behaviour occurs. In constant signal regions, the next input is ignore in favour of the last output smoothing occurs because noise is ignore In strong signal regions the last output is ignore in favour of the next input an eges an textures are preserve. The local impulse response has an exponential form: n " h( n)" = αβ u( n)

45 It is as if both the weights within the winow an the winow size are now epenent on the local signal-to-noise variance ratio (SNR). As the local SNR ecreases, β increases, the weights become more uniform an the impulse response extens over a larger interval, an more smoothing occurs.

46 β can also be interprete as the pole position in the Z plane for the local system transfer function ( = n H ( z) h( n) z ). The pole varies from 0, the all-pass case with no smoothing, at maximum SNR n an approaches 1, the maximum smoothing case at minimum SNR. Note that this is a causal, first orer filter. If we construct a two imensional version with ifferent horizontal an vertical parameters, we can accomplish irectionally epenent smoothing: g( m, n) = α f ( m, n) + β g( m, n 1) + β g( m 1, n). h v We coul even a in a iagonal term if esire. Figure 16. compares recursive an non-recursive versions of LLMMSE ege-epenent smoothing. In the next session we will explore a more general alternative to these simple local statistics filters.

47 Recursive Lee Filter 7x7 Nonrecursive Lee