Wiener Filter for Deterministic Blur Model

Transcription

1 Wiener Filter for Deterministic Blur Model Based on Ch. 5 of Gonzalez & Woods, Digital Image Processing, nd Ed., Addison-Wesley, 00 One common application of the Wiener filter has been in the area of simultaneous de-blurring and de-noising of an image. From Section 5.5: ff(xx, yy) is the original image (-D signal of xx, yy spatial variables) Observed image is g xy, = H f xy, + nxy (, ) Observed Image Blur Operator Original Image -D Random Noise If there were only blurring seek to find inverse of H If there were only noise seek a filter that passes image & removes some noise The Wiener filter seeks to optimally balance these two issues! 1

2 LSI Blur Model (Section 5.5) Common model for the blur operator is Linear Shift-Invariant (LSI): (, ) + (, ) = (, ) + (, ) Haf xy af xy ahf xy ahf xy Linearity (, ) = (, ) Hf xy f xy ( α, β) = ( α, β) in out H fin x y fout x y Shift Inv. An LSI system can be described by its impulse response: (, ) (, ) H δ xy hxy Also called Linear Position-Invariant (LPI) Then the output is expressed as a -D convolution: (, ) = (, ) (, ) H f xy hx αy β f αβ dαdβ This is a continuousspace version it is possible to do the same for discrete-space version

3 Now our blurred image model is: blur (, ) = (, ) (, ) f xy hx αy β f αβ dαdβ Taking the -D Fourier transform of the above model gives blur (, ) = (, ) (, ) F uv H uv F uv Where the -D Fourier transforms are given by (, ) (, ) jπ ( ux+ vy) = (, ) (, ) F u v f x y e dxdy jπ ( ux+ vy) H u v h x y e dxdy = Frequency Response of Blur Note that in principle if we know HH uu, vv then we can use the inverse filter to recover the original image (, ) F uv = Fblur ( uv, ) (, ) H uv 3

4 Blur Caused by Planar Motion (see Section 5.6.3) A common blur model is that due to planar motion while the camera s aperture is open during time interval [0,T]. This means that during [0,T] the image moves in the x and y directions according to functions xx 0 (tt) and yy 0 (tt) Then our observed image model is: (, ) = ( (), ()) + (, ) g xy f x x t y y t dt nxy 0 0 Converting to frequency domain gives (see Gonzalez & Woods for steps): with (, ) (, ) = (, ) (, ) + (, ) G uv H uv F uv N uv T [ ] j ux0() t vy0() t H u v e π + dt = 0 4

5 Blur Caused by Uniform Linear Motion (see Section 5.6.3) This is planar motion with constant speed in each direction: xx 0 tt = aaaa TT and yy 0 (tt) = bbtt TT Then the blur s frequency response is two sincs: ( ua + vb) ( ua + vb) sin π j ( ua vb) π + H( uv, ) = T e π So this is like a -D lowpass filter! Original Image Blurred Image w/ aa = bb = , TT = 11 5

6 Blurred and Noisy Image Now our observed image model is: -D WSS Noise Process (, ) = ( α, β) ( αβ, ) α β+ (, ) g xy hx y f d d nxy Taking the -D Fourier transform of the above model gives (, ) = (, ) (, ) + (, ) G uv H uv F uv N uv Solving this using the Inverse Filtering Viewpoint gives G uv, N uv, F( uv, ) = = F( uv, ) + H uv, H uv, This can exacerbate the noise when HH(uu, vv) is 0 or small at some frequencies! The inverse filter focuses on the de-blurring The Wiener filter seeks to optimally balance these two issues. We will solve the Wiener filter in this Frequency-Domain view 6

7 Wiener Filter for Blurred & Noisy Images (see Sect. 5.8) Without going into the details we will borrow the frequencydomain view we saw for the IIR Wiener smoother. Recall. x[n] = s[n] + w[n] No Blurring Pss ( f ) + S f = X f Pss f Pww f Accounting for the blurring and the change to -D gives (, ) = (, ) (, ) + (, ) G uv H uv F uv N uv Must Know! * H ( uv, ) Pff ( uv, ) F( uv, ) = H( uv, ) Pff ( uv, ) + Pnn ( uv, ) PP ffff (uu, vv) = Power Spectral Density of Image PP nnnn (uu, vv) = Power Spectral Density of Noise Bayesian!! Thing we are estimating is modeled as random! 7

8 In practice our knowledge needed varies: HH(uu, vv) might be reasonably well known PP nnnn (uu, vv) might be quite well known PP ffff (uu, vv) might NOT be known at all! E.g., white noise PP nnnn uu, vv = NN oo (constant) Then re-arranging gives F( uv) * H ( uv, ), = H( uv, ) + No Pff ( uv, ) A common trick is to replace this term by a constant K: * H ( uv, ) F( uv, ) = H( uv, ) + K then the value of K is chosen interactively to give the best result. (So this is an off-line approach). With enough attention to detail, this approach can be implemented using the FFT algorithm and applied to discrete images. 8

9 Results for Wiener Filter (sect. 5.8 of Gonzalez and Woods) Blurred & Noisy Inverse Filter Wiener Filter Original Image Increasing Noise Variance Uniform Blurred Image w/ aa = bb = , TT = 11 9

10 Iterative Wiener Filter (Not in Gonzalez and Woods) Start by using the above solution: * H ( uv, ) F0 ( uv, ) = H( uv, ) + K Now use that estimate image to estimate the PSD of the image F uv, P uv, 0 ff,0 Re-Estimate image using F( uv) Iterate: F uv, P uv, n ff, n Lots of literature on how to estimate the PSD of a WSS process * H ( uv, ) 1, = H( uv, ) + No Pff,0 ( uv, ) * H ( uv, ) Fn + 1 uv, = H( uv, ) + No Pff, n ( uv, ) 10

11 Alternate View (not in Gonzalez and Woods) An alternate view is a discrete-image, space-domain approach: H is blurring matrix formed to provide equivalent of D conv g = Hf + n f is vector formed by stacking columns of discrete image RR ffff = Correlation Matrix of Image s Vector f RR nnnn = Correlation Matrix of Noise s Vector n Recall our general Wiener smoothing with vector parameter: x = s+ w s = R ss ( R ss + R ww [ N N][ N N][ N 1] Modifying this to handle the blurring matrix gives T T = ff ff + nn ) 1 x 1 f R H HR H R g g & n vectors defined similarly Compare to previous Freq. Domain result: * H ( uv, ) Pff ( uv, ) F( uv, ) = H( uv, ) Pff ( uv, ) + Pnn ( uv, ) 11

12 Iterative Version of Alternate View (not in Gonzalez and Woods) Initial Filtering: Use g to get estimate RR gggg Use it to approximate RR ffff,00 = RR gggg Use that in place of RR ffff in filter to get ff 00 Iteration for n = 1,, 3, : Use ff nn 11 to estimate RR ffff,nn+11 Use RR ffff,nn+11 in filter to get ff nn Lots of literature on how to estimate the Correlation Matrix of a WSS process 1