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Noise models presented by Nisha Menon K

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Noise models presented by Nisha Menon K

Noise models presented by Nisha Menon K

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• 1. 1 NISHA MENON K MTECH ROLL NO: 16 15-Nov-13 NOISE MODELS & RESTORATION OF NOISY IMAGES
• 2. NOISE AND IMAGES 15-Nov-13 The sources of noise in digital images arise during image acquisition (digitization) and transmission Imaging sensors can be affected by ambient conditions  Interference can be added to an image during transmission  2
• 3. NOISE MODEL 15-Nov-13 Consider a noisy image is modelled as follows: g ( x, y)  f ( x, y)   ( x, y) f(x, y) η(x, y) g(x, y) original image pixel noise term resulting noisy pixel 3
• 4. NOISE MODELS Spatially independent noise models        15-Nov-13  Gaussian noise Rayleigh noise Erlang (Gamma) noise Uniform noise Exponential noise Impulse (salt-and-pepper) noise Spatially dependent noise model 4  Periodic noise
• 5. Gaussian NOISE MODELS Rayleigh  Gaussian       15-Nov-13 There are many different models for the image noise term η(x, y): Most common model Rayleigh Erlang Exponential Uniform Impulse  Erlang Exponential Uniform Salt and pepper noise Impulse 5
• 6. Noise cannot be predicted but can be approximately described in statistical way using the probability density function (PDF) 15-Nov-13 Gaussian noise   Electronic circuit noise, sensor noise due to poor illumination and/or high temperature The PDF of a Gaussian random 1 variable, z,  ( z   ) 2 / 2 2 p( z )  e 2  μ  Mean:  Standard deviation:  Variance: σ 2 σ 6
• 7. RAYLEIGH NOISE Range imaging    The PDF of Rayleigh noise, 15-Nov-13  2  ab ( ) z 2 za  / for ( )  e za  p (  z b )  0 for za   Mean:  a  /4  b b4  ( ) Variance: 2  7 4
• 8. ERLANG (GAMMA) NOISE The PDF of Erlang noise, integer, a  0, b is a positive 15-Nov-13  bb z a a z1  e for z0  p)   (  b1 z ( )!  0 for z0  b   Mean:  Variance: a b σ = 2 a 2 8
• 9. Exponential noise 15-Nov-13  p z) aeaz ( Uniform noise 1   p( a z b )  0  for b az   otherwise Impulse (salt & pepper) noise P a  p ) b ( P z  0  for a z  for b z  otherwise 9
• 10. NOISE EXAMPLE This is a suitable pattern to use because it is composed of simp1e, constant areas that span the gray scale from black to near white in only three increments 15-Nov-13 The test pattern to the right is ideal for demonstrating the addition of noise Image Histogram to go here 10 Histogram
• 11. NOISE EXAMPLE (CONT…) 15-Nov-13 11 Gaussian Rayleigh Erlang
• 12. NOISE EXAMPLE (CONT…) 15-Nov-13 Histogram to go here 12 Exponential Uniform Impulse
• 13. PERIODIC NOISE 15-Nov-13 Typically arises due to electrical or electromagnetic interference Gives rise to regular noise patterns in an image Frequency domain techniques in the Fourier domain are most effective at removing periodic noise Periodic noise can be reduced by setting frequency components corresponding to noise to zero 13
• 14. HOW TO ESTIMATE NOISE PARAMETERS?   noise Analyze the FT  Noise  15-Nov-13  Periodic PDFs Sensor specification Crop the vertical strips and obtain the histogram. μ=  z p(z ) S i i zi σ = 2  (z S zi i  μ) p(zi ) 2 14
• 15. 15-Nov-13 We cannot use the image histogram to estimate noise PDF. It is better to use the histogram of one area of an image that has constant intensity to estimate noise PDF. 15
• 16. RESTORATION IN PRESENCE OF NOISE BY SPATIAL FILTERING      15-Nov-13 Spatial filters that are based on ordering the pixel values that make up the neighbourhood operated on by the filter Useful spatial filters include Mean filter Median filter Max and min filter Midpoint filter Alpha trimmed mean filter 16
• 17. MEAN FILTERS 15-Nov-13 Arithmetic mean filter (AMF, average) Geometric mean filter Harmonic mean filter Contraharmonic mean filter 17
• 18. Original image subimage 15-Nov-13 Statistical parameters Mean, Median, Mode, Min, Max, Etc. Moving window 18 Output image
• 19. MEAN FILTERS 15-Nov-13  Arithmetic mean filter ˆx ) 1 g,) f ,   (y (t s mn (,)S s t xy  Geometric mean filter 1 mn   ˆx )  g , ) f(,y t ( s ( s ,) S  xy  t  19
• 20. Harmonic mean filter  15-Nov-13  Works well for salt noise, but fails for pepper noise mn 1  ( ,t) s ( ,) S g st  xy ˆx ) f( ,y  20
• 21.  Contraharmonic mean filter : eliminates pepper noise Q0 15-Nov-13 Q0 : eliminates salt noise gst Q1 (, )   ( xy ˆ xy s,t) S f( , ) Q g, (  st) ( ,) S st xy For Q = 0, the filter reduces to an arithmetic mean filter. For Q = -1, the filter reduces to a harmonic mean filter. 21
• 22. Image Corrupted by guassian noise 15-Nov-13 Filtering with Arithmetic mean filter Filtering with geometric mean filter 22
• 23. 15-Nov-13 Corrupted by pepper noise Corrupted by salt noise Contraharmonic filter Q= -1.5 Contraharmonic filter Q=1.5 23
• 24.  Usage  Image obtained using a contraharmonic mean filter With Q=-1.5 15-Nov-13  Arithmetic and geometric mean filters: suited for Gaussian or uniform noise Contraharmonic filters: suited for impulse noise but necessary to know whether noise is dark or light for selecting Q value Image obtained using a contraharmonic mean filter With Q=1.5 24
• 25. ORDER-STATISTICS FILTERS 15-Nov-13 Median filter Max and min filters Midpoint filter Alpha-trimmed mean filter 25
• 26. MEDIAN FILTER 15-Nov-13 ˆ f ( x, y )  median{g ( s, t )} ( s ,t )S xy Excellent at noise removal, without the smoothing effects that can occur with other smoothing filters Particularly good when salt and pepper noise is present. 26
• 27. MIDPOINT FILTER 15-Nov-13 Midpoint Filter: ˆ ( x, y )  1  max {g ( s, t )}  min {g ( s, t )} f  ( s ,t )S xy 2 ( s ,t )S xy   Good for random Gaussian and uniform noise 27
• 28. MAX AND MIN FILTER ˆ f ( x, y )  max {g ( s, t )} 15-Nov-13 Max Filter: ( s ,t )S xy Min Filter: ˆ f ( x, y )  min {g ( s, t )} ( s ,t )S xy Max filter is good for pepper noise and min is good for salt noise 28
• 29. ALPHA-TRIMMED MEAN FILTER ˆ f ( x, y)  1 S gr (s, t ) mn  d ( s ,t ) xy 15-Nov-13 Alpha-Trimmed Mean Filter: We can delete the d/2 lowest and d/2 highest grey levels So gr(s, t) represents the remaining mn – d pixels Useful in situations involving multiple types of noise, such as a combination of salt-and-pepper and Gaussian noise 29
• 30. 15-Nov-13 Corrupted by salt and pepper noise After passing through Median filter 30
• 31. 15-Nov-13 max filter min filter 31
• 32. Image corrupted by additive uniform noise Arithmetic mean filter Geometric mean filter 15-Nov-13 Median filter Image additionally corrupted by additive salt-andpepper noise Alpha trimme mean filter 32
• 33. ADAPTIVE FILTERS 15-Nov-13 The filters discussed so far are applied to an entire image without any regard for how image characteristics vary from one point to another The behaviour of adaptive filters changes depending on the characteristics of the image inside the filter region 33
• 34. General concept: -Filter behavior depends on statistical characteristics of local areas inside mxn moving window Statistical characteristics: Local mean: 1 mL =  g(s,t) mn (s,t)S xy Noise variance:  2 Local variance: 1 σ = (g(s,t)  mL )2  mn (s,t)S xy 2 L 34 15-Nov-13 - More complex but superior performance compared with “fixed” filters
• 35. Adaptive, Local Noise Reduction Filter Purpose: want to preserve edges 15-Nov-13 Concept: If 2 is zero,  No noise the filter should return g(x,y) because g(x,y) = f(x,y) If L2 is high relative to 2,  Edges (should be preserved), the filter should return the value close to g(x,y) If L2 = 2,  Local area has same properties as overall image. The filter should return the arithmetic mean value mL Formula: x ˆ,)g ) (y  f y (y  ,) (  g m x x ,  2 2 L L 35
• 36. Adaptive Median Filtering 15-Nov-13 zmin = minimum gray level value in Sxy zmax = maximum gray level value in Sxy zmedian = median of gray levels in Sxy zxy = gray level value at pixel (x,y) where Smax = maximum allowed size of Sxy 36
• 37. Algorithm Return zxy  to preserve original details Else return zmedian  to remove impulse 15-Nov-13 Level A: Determine A1= zmedian – zmin whether zmedian A2= zmedian – zmax is an impulse or not If A1 > 0 and A 2 < 0,  zmedian is not an impulse goto level B Else  Window is not big enough increase window size If window size <= Smax repeat level A Else return zxy Level B: Determine B1= zxy – zmin whether zxy B2= zxy – zmax is an impulse or not If B1 > 0 and B2 < 0  zxy is not an impulse 37
• 38. Purposes of the algorithm Remove salt-and-pepper (impulse) noise  Provide smoothing of other noise  15-Nov-13  Reduce distortion, such as excessive thinning or thickening of object boundaries 38
• 39. ADAPTIVE FILTERING EXAMPLE 15-Nov-13 Image corrupted by salt and pepper noise with probabilities Pa = Pb=0.25 Result of filtering with a 7 * 7 median filter Result of adaptive median filtering with Smax = 7 39
• 40. CONCLUSION   Different types of noise pdfs have been discussed Estimation of noise parameters Restoration of images using Order- Statistic filters Adaptive filters 15-Nov-13  40
• 41. 15-Nov-13 41