image restoration - lecture 14 - uppsala universityimage restoration lecture 14 milan gavrilovic...

Image restorationLecture 14

Milan Gavrilovicmilan@cb.uu.se

Centre for Image AnalysisUppsala University

Computer Assisted Image Analysis2009-05-08

M. Gavrilovic (Uppsala University) L14 Image restoration 2009-05-08 1 / 33

Today’s lecture

Notes on chalkboard: Some properties of 2D DFT (review ofsection 4.6 in Gonzales-Woods)What is FFT?Filters in the frequency domain (a retrospective view of L04 andLab1)

General concepts of image restoration (5.1 − 5.2 and 5.5 in GW)Periodic noise reduction by frequency domain filtering (5.4 in GW)Inverse and Wiener filtering (5.6 − 5.8 in GW)

The Fast Fourier Transform (FFT)

Discrete Fourier Transform (DFT)

F(u) =1M

M−1∑x=0

f (x)e−j2πux/M for u = 0, 1, . . . , M − 1

Fast Fourier Transform is an efficient algorithm for computing the DFT.

For a 1-D signal:The ordinary DFT of M points require M2 operations(multiplications/additions).The FFT of M points require M log2 M operations.

The FFT algorithm is based on the so-called successive doublingmethod. Implementation of FFT in 2D is analogous to the 1D case.

Filtering in the Frequency Domain

The basics

1 Multiply the input image by (−1)x+y to center the transform.2 Compute F(u, v), the DFT of the image from (1).3 Multiply F(u, v) by a filter function H(u, v).4 Compute the inverse DFT of the result in (3).5 Obtain the real part of the result in (4).6 Multiply the result in (5) by (−1)x+y.

G(u, v) = H(u, v)F(u, v)

Element-by-element multiplication.

Ideal Lowpass Filters and "Ringing"

In the frequency domain, define a cut-off radius D0.

Ideal Lowpass Filter (ILPF)

H (u, v) ={

1 if D (u, v) ≤ D0;0 if D (u, v) > D0.

To find h(x, y):

1 Centering: H(u, v) ∗ (−1)u+v

2 Inverse Fourier transform3 Multiply real part by (−1)x+y

Properties of h(x, y):

1 It has a central dominant circular component (providing theblurring)

2 It has concentric circular components (rings) giving rise to theringing effect.

Example of h(x) from theinverse FT of a disc (ILPF)with radius 5.

Original image (top left) andfiltered images with ILPF ofradius 5, 15 and 30,removing 8, 5.4 and 3.6% ofthe total power.This type of artefacts arenot acceptable in e.g.medical imaging.

Reduce "Ringing" with Non-Ideal Lowpass Filters

Butterworth Lowpass Filter (BLPF)

H (u, v) =1

D(u,v)D0

n is the order of the filterA high n will cause "ringing" (approaching ILPF)No sharp discontinuity

Butterworth Lowpass Filter

BLPF"ringing"effects fordifferentvalues of n.

In general, BLPFs of order 2 are a good compromise between effectivelowpass filtering and acceptable ringing characteristics.

Gaussian Lowpass Filter

Gaussian Lowpass Filter (GLPF)

H (u, v) = e−D2(u,v)

D0 is the standard deviation (σ), or the "spread" of the Gaussian.The inverse FT of a Gaussian is also a Gaussian, meaning aGaussian smoothing in the spatial domain.Guarantees no ringing.

Highpass Filters

Ideal Highpass Filter (IHPF)

H (u, v) ={

0 if D (u, v) ≤ D0;1 if D (u, v) > D0.

Butterworth Highpass Filter (BHPF)

H (u, v) = 1 − 1

D(u,v)D0

)2n =1

D0D(u,v)

Gaussian Highpass Filter (GHPF)

H (u, v) = 1 − e−D2(u,v)

Same ringing effect as for the lowpass filters.

Highpass Filters

Spatial repre-sentation ofdifferenthighpassfilters (ideal,ButterworthandGaussian).

Ringing effect noticeable in ideal and Butterworth filters.

High-Frequency Emphasis Filtering

Hhfe (u, v) = a + b ∗ Hhp (u, v)

Result ofhigh-frequencyemphasis filtering(original, highpass;high-freq emphasis,histogramequalization).

Image Restoration

Restore an image that has been degraded in some way.Make a model of the degeneration process and use inversemethods.Image restoration is an objective method using a priori informationof the degradation.Compare with image processing which is a method to present theimage in a "better" way.

f (x, y) is the original image.H represents the system that affects our image.n(x, y) is disturbance, e.g., noise or external contribution.Obtained degraded image g(x, y) = H(f (x, y)) + n(x, y).Possible defects in the imaging system causing degradation:

I Bad focusing.I Motion.I Non-linearity of the sensor.I Noise.I etc...

Image Restoration

Possible approaches:Inverse filtering.Try to model degradation effect.Use Fourier-domain methods and identify which frequencies arerelated to the degrading effect.

Mathematical Fundamentals - Convolution

Convolution of Two Continuous Functions

f (x, y)⊗ g (x, y) =∫ ∞

−∞

∫ ∞

−∞f (α, β) g (x − α, y − β) dαdβ

Convolution by the Impulse Function∫ ∞

−∞f (x) δ (x − x0) dx = f (x0)

Note: The sifting property (L13) can be represented as convolvingwith the impulse function.

Simplified Model

Assume a model without noise n.

g (x, y) = H [f (x, y)]

Assume that the operator H isI linear: H[k1f1(x, y) + k2f2(x, y)] = k1H[f1(x, y)] + k2H[f2(x, y)]I additive: H[f1(x, y) + f2(x, y)] = H[f1(x, y)] + H[f2(x, y)]I homogeneous: H[k1f1(x, y)] = k1H[f1(x, y)]I position invariant: H[f (x − α, y − β)] = g(x − α, y − β)

then, the following is true

g (x, y) =∫ ∞

−∞

∫ ∞

−∞f (α, β) H [δ (x − α, y − β)] dαdβ

Impulse Response

Impulse Response for Degradation Function H

h (x, α, y, β) = H [δ (x − α, y − β)]

In imaging systems the impulse response is called the pointspread function (PSF).PSF describes how a point is imaged.

Fredholm Integral

Fredholm Integral or Superposition Integral of the First Kind

g (x, y) =∫ ∞

−∞

∫ ∞

−∞f (α, β) h (x, α, y, β) dαdβ

The Fredholm integral says that if the impulse response is known,the response to any input signal can be calculated.

If degradation function, H, is position invariant, the integral isreduced to the convolution integral:

Fredholm Integral of Position Invariant Degradation Function

g (x, y) =∫ ∞

−∞

∫ ∞

−∞f (α, β) h (x − α, y − β) dαdβ

Point Spread Function

Single impulse signal.

Two impulse signals.

Output blurred by the PSF.

Fourier Methods

From Fourier Analysis:sinusoidal function in time (or spatial) domain of one variable

corresponds totwo impulse functions in Fourier domain.

Therefore,periodic noise in an image (i.e., repeated noise patterns) causespeaks in the FT of the image.by supressing these peaks and inverse transforming the image, arestored image is achieved.

(The last problem in computer exercise 1)

Fourier Domain Filters

Bandreject filters (ideal, Butterworth and Gaussian).

Notch filters (ideal, Butterworth and Gaussian).

Periodic Noise Reduction

Notch Filtering Example

Original image.

Fourier spectrum and it’s partremoved with notch filter.

Inverse transform of notchfiltered image and result ofnotch filtering.

Inverse Filtering

We can thus describe our model as g(x, y) = f (x, y)⊗ h(x, y).The Fourier transform gives

G (u, v) = F (u, v) · H (u, v) ⇔ F̂ (u, v) =G (u, v)H (u, v)

By modeling the degenerating effect (h) and dividing the FT of theimage by the FT of the model, we can get the FT of a restoredimage.If noise is present, we get g(x, y) = f (x, y)⊗ h(x, y) + n(x, y)and

F̂ (u, v) =G (u, v)H (u, v)

− N (u, v)H (u, v)

The inverse transform gives the restored image.The method is called inverse filtering.

Problems with Inverse Filtering

At deconvolution, the FT of the image is divided by the FT of thedegrading effect.

Problems:I Small values of H(u, v) can cause overflow (usually small at HF).I If noise is included, it can be dominating.

Solutions:I Perform division only in a limited part of the (u, v)-plane.I Use weights to limit the effect at division with small numbers.

Inverse Filtering Examples

Original image.

Degraded image.

Full filter.

Radius of 85.

Radius of 70.

Radius of 40.

Wiener Filtering

We consider images and noise as random processes.We try to find an estimate f̂ of the uncorrupted image f such thatthe mean square error is minimized.

e2 = E{(

f − f̂)2

}If we assume the following conditions

1 Noise and image are uncorrelated.2 One or the other has zero mean.3 Greylevels in the estimate are linear functions of the graylevels in

the degraded image....then the minimum of the error function (in the frequency domain) isgiven by

F̂ (u, v) =

[H∗ (u, v) Sf (u, v)

Sf (u, v) |H (u, v)|2 + Sη (u, v)

]G (u, v)

Wiener Filtering

The Wiener filter can be written in a simpler form

F̂ (u, v) =

1H (u, v)

|H (u, v)|2

|H (u, v)|2 + Sη(u,v)Sf (u,v)

G (u, v)

Note that if the noise is zero, the Wiener filter reduces to theinverse filter.The noise power spectrum is constant when dealing withspectrally white noise.Approximation frequently used

F̂ (u, v) =

H (u, v)|H (u, v)|2

|H (u, v)|2 + K

]G (u, v)

K is adjusted interactively for best result.

Filter Comparison

See the original and degraded image in slide 30.

Full inverse. Limited inverse. Wiener filter.

Figure 5.29 - another comparison!

Drawback with Wiener Filtering

Drawback:Must know (or approximate) the power spectra of undegradedimage and noise.

Better choice:Constrained Least Squares Filter (CLSF).

I Need only knowledge of mean and variance of noise (apart fromdegradation function).

I Only one parameter, which can be iteratively computed foroptimality.

CLSF, section 5.9 in GW - non-mandatory reading assignment!

Reading InstructionsChapters for next lecture

Chapter 7 in Gonzales-Woods (only briefly).

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