Deconvolution, Deblurring and Restoration T , Biomedical Image Analysis Seminar Presentation Seppo Mattila & Mika Pollari

Overview (1/2) Linear space-invariant (LSI) restoration filters - Inverse filtering - Power spectrum equalization - Wiener filter - Constrained least-squares restoration - Metz filter Blind Deblurring

Overview (2/2) Homomorphic Deconvolution Space-variant restoration –Sectioned image restoration –Adaptive-neighbourhood deblurring –The Kalman filter Applications - Medical - Astronomical

Introduction Find the best possible estimate of the original unknown image from the degraded image. One typical degradation process has a form: g(x,y) measured image f(x,y) true (ideal) image h(x,y) point spread function (PSF) (impuse response function) n(x,y) additive random noise

Image Restoration General One has to have some a priori knowledge about the degragation process. Usually one needs 1) model for degragation, some information from 2) original image and 3) noise. Note! Eventhough one doesn’t know the original image some information such as power spectral density (PSD) and autocorreletion function (ACF) are easy to model.

Linear-Space Invariant (LSI) Restoration Filters Assume: linear and shift-invariant degrading process Random noise statistically indep. of image-generating process Possible to design LSI filters to restore the image

Inverse Filtering Consider degrading process in matrix form: Given g and h, estimate f by minimising the squared error between observed image (g) and : where and are approximations of f and g Set derivative of є 2 to zero: (see Sect for details) (if no noise) (if noise present)

Inverse Filtering Examples Works fine if no noise but... H(u,v) usually low-pass function. N(u,v) uniform over whole spectrum. High-freq. Noise amplified!! 0.4x 0.2x f' G

Power Spectrum Equalization (PSE) Want to find linear transform L such that: Power spectral density (PSD) = FT(Autocorrelation function) PSD( (u,v)) = PSD(f(u,v)) i.e....

The Wiener Filter (1/2) Degradation model: Assumtions: Image and noise are second- order-stationary random processes and they are statistically independent Optimal mean-square error (MSE) criterion Find Wiener filter (L) which minimize MSE

The Wiener Filter (2/2) Minimizing the criterion we end up to optimal Wiener filter. The Wiener filter depends on the autocorrelation function (ACF) of the image and noise (This is no problem). In general ACFs are easy to estimate.

Comparison of Inverse Filter, PSE, and Wiener Filter

Constrained Least-squares Restoration Minimise: with constraint: where L is a linear filter operator ● Similar to Wiener filter but does not require the PSDs of the image and noise to be known ● The mean and variance of the noise needed to set optimally. If = 0 inverse filter...

The Metz Filter Modification to inverse filter. Supress the high frequency noise instead of amplyfying it. Select factor so that mean-square error (MSE) between ideal and filtered image is minimized.

Motion Deblurring – Simple Model Assume simple in plane movement during the exposure Either PSF or MTF is needed for restoration

Blind Deblurring Definition of deblurring. Blind deblurring: models of PSF and noise are not known – cannot be estimated separately. Degragated image (in spectral domain) consist some information of PSF and noise but in combined form.

Method 1 – Extension to PSE Broke image to M x M size segment where M is larger than dimensions of PSF then Average of PSD of these segments tend toward the true signal and noise PSD This is combined information of blur function and noise which is needed in PSE Finaly, only PSD of image is needed

Extension to PSE Cont...

Method 2 – Iterative Blind Deblurring Assumptation: MTF of PSF has zero phase. Idea: blur function affects in PSD but phase information preserves original information from edges.

Iterative Blind Deblurring Cont... Fourier transform of restored image is Note that smoothing operator S[] has small effect to smooth functions (PSF). This leads to iterative update rule

Examples of Iterative Blind Deblurring

Homomorphic deconvolution Start from: Convert convolution operation to addition: ● Complex cepstrum: Complex cepstra related: ● Practical application, however, not simple...

Steps involved in deconvolution using complex cepstrum

Space-variant Image Restoration So far we have assumed that images are spatially (and temporaly) stationary This is (generally) not true – at the best images are locally stationary Techniques to overcome this problem: –Sectioned image restoration –Adaptive neighbourhood deblurring –The Kalman filter (the most elegant approach)

Sectioned Image Restoration Divide image into small [P x P] rectangular, presumably stationary segments. Centre each segment in a region, and pad the surrounding with the mean value. For each segment apply separately image restoration (e.g. PSE or wiener).

Adaptive-neighborhood deblurring (AND) Grow adaptive neighborhood regions: Apply 2D Hamming window to each region: Estimate the noise spectrum: Centered on (m,n) Pixel locations within the region A is a freq. domain scale factor that depends on the spectral characterisics of the region grown etc.

AND segmentation

Adaptive-neighborhood deblurring (AND) Cont… Frequency-domain estimate of the uncorrupted adaptive-neighborhood region: Obtain estimate for deblurred adaptive neigborhood region m,n (p,q) by FT -1 Run for every pixel in the input image g(x,y) Deblurred image

Comparison of Sectioned and AND-technique

Kalman Filter Kalman filter is a set of mathematical equations. Filter provides recursive way to estimate the state of the process (in non-stationary environment), so that mean of squared errors is minimized (MMSE). Kalman filter enables prediction, filtering, and smoothing.

Kalman Filter State-Space Process Eq. Observation Eq. Innovation process:

Kalman Filter in a Nutshell (1/2) Data observations are available System parameters are known –a(n+1,n), h(n), and the ACF of driving and observation noise Initial conditions Recursion

Kalman Filter in a Nutshell (2/2) 1)Compute the Kalman gain K(n) 2)Obtain the innovation process 3)Update 4)Compute the ACF of filtered state error 5)Compute the ACF of predicted state error

Wiener Filter Restoration of Digital Radiography

Astronomical applications Images blurred by atmospheric turbulence Observing above the atmosphere very expensive (HST) Improve the ground-based resolution by –Suitable sites for the observatory 4 km height) –Real time Adaptive optics correction –Deconvolution

Point Spread Function (PSF) in Astronomy Ideal PSF if no atmosphere FWHM ~ 1.22x /D < 0.1" (8m telescope) Atmospheric turbulence broadens the PSF Gaussian PSF with FWHM ~ 1" I observed = I real ⊗ PSF Easy to measure and model from several stars usually present in astro- images Determines the spatial resolution of an image Commonly used for image matching and deconvolution

Richardson-Lucy deconvolution Used in both fields: astronomy & medical imaging Start from Bayes's theorem, end up with: Takes into account statistical fluctuations in the signal, therefore can reconstruct noisy images! In astronomy the PSF is known accurately From an initial guess f 0 (x) iterate until converge

Astro-examples Observed PSF Inverse filter Richardson-Lucy