Computer Vision & Digital Image Processing
Image Restoration and Reconstruction II
Electrical & Computer Engineering Dr. D. J. Jackson Lecture 12-1
Outline
• Periodic noise • Estimation of noise parameters • Restoration in the presence of noise only –spatial filtering
Electrical & Computer Engineering Dr. D. J. Jackson Lecture 12-2 Periodic noise
• Periodic noise typically arises from interference during image acquisition • Spatially dependent noise type • Can be effectively reduced via frequency domain filtering
Electrical & Computer Engineering Dr. D. J. Jackson Lecture 12-3
Sample periodic images and their spectra
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Electrical & Computer Engineering Dr. D. J. Jackson Lecture 12-4 Sample periodic images and their spectra
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Electrical & Computer Engineering Dr. D. J. Jackson Lecture 12-5
Estimation of noise parameters
• Noise parameters can often be estimated by observing the Fourier spectrum of the image – Periodic noise tends to produce frequency spikes • Parameters of noise PDFs may be known (partially) from sensor specification – Can still estimate them for a particular imaging setup – One method • Capture a set of “flat” images from a known setup (i.e. a uniform gray surface under uniform illumination) • Study characteristics of resulting image(s) to develop an indicator of system noise
Electrical & Computer Engineering Dr. D. J. Jackson Lecture 12-6 Estimation of noise parameters (continued)
• If only a set of images already generated by a sensor are available, estimate the PDF function of the noise from small strips of reasonably constant background intensity • Consider a subimage (S) and let ps(zi), i=0,1,2,…L-1 • denote the probability estimates of the intensities of the pixels in S. • L is the number of possible intensities in the image • The mean and the variance of the pixels in S are given by:
L−1 L−1 2 2 z = ∑ zi ps (zi ) and σ = ∑(zi − z) ps (zi ) i=0 i=0
Electrical & Computer Engineering Dr. D. J. Jackson Lecture 12-7
Estimation of noise parameters (continued)
• The shape of the noise histogram identifies the closest PDF match – If the shape is Gaussian, then the mean and variance are all that is needed to construct a model for the noise (i.e. the mean and the variance completely define the Gaussian PDF) – If the shape is Rayleigh, then the Rayleigh shape parameters (a and b) can be calculated using the mean and variance – If the noise is impulse, then a constant (with the exception
of the noise) area of the image is needed to calculate Pa and Pb probabilities for the impulse PDF
Electrical & Computer Engineering Dr. D. J. Jackson Lecture 12-8 Histograms from noisy strips of an area of an image
Electrical & Computer Engineering Dr. D. J. Jackson Lecture 12-9
Restoration in the presence of noise only –spatial filtering • When only additive random noise is present, spatial filtering is commonly used to restore images • Common types
– Mean filters – Order-Statistic filters – Adaptive filters
Electrical & Computer Engineering Dr. D. J. Jackson Lecture 12-10 Mean filters (arithmetic)
• Arithmetic mean filter – Computes the average value of a corrupted image g(x,y) in the area defined by a window (neighborhood) 1 fˆ(x, y) = ∑ g(s,t) mn (s,t)∈Sxy – The operation is generally implemented using a spatial filter of size m*n in which all coefficients have value 1/mn – A mean filter smoothes local variations in an image – Noise is reduced as a result of blurring
Electrical & Computer Engineering Dr. D. J. Jackson Lecture 12-11
Mean filters (geometric)
• Geometric mean filter – A restored pixel is given by the product of the pixels in an area defined by a window (neighborhood), raised to the power 1/mn
1 ⎡ ⎤ mn fˆ(x, y) = ⎢ ∏ g(s,t)⎥ ⎣⎢(s,t)∈Sxy ⎦⎥
– Achieves smoothing comparable to the arithmetic mean filter, but tends to loose less detail in the process
Electrical & Computer Engineering Dr. D. J. Jackson Lecture 12-12 Arithmetic and geometric mean filter examples
Electrical & Computer Engineering Dr. D. J. Jackson Lecture 12-13
Mean filters (harmonic)
• Harmonic mean filter – A restored pixel is given by the expression mn fˆ(x, y) = 1 ∑ g(s,t) (s,t)∈Sxy
– Works well for salt noise (fails for pepper noise) – Works well for Gaussian noise also
Electrical & Computer Engineering Dr. D. J. Jackson Lecture 12-14 Mean filters (contraharmonic)
• Contraharmonic mean filter – A restored pixel is given by the expression
∑ g(s,t)Q+1 fˆ(x, y) = (s,t)∈Sxy ∑ g(s,t)Q (s,t)∈Sxy
– Q is the order of the filter – Works well for salt and pepper noise (cannot do both simultaneously) – +Q eliminates pepper noise, -Q eliminates salt noise – Q=0 → arithmetic mean filter – Q=-1 → harmonic mean filter
Electrical & Computer Engineering Dr. D. J. Jackson Lecture 12-15
Contraharmonic mean filter examples
Electrical & Computer Engineering Dr. D. J. Jackson Lecture 12-16 Contraharmonic mean filter examples
Electrical & Computer Engineering Dr. D. J. Jackson Lecture 12-17