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Electrical & Computer Engineering Dr. D. J. Jackson Lecture 13-
Computer Vision &
Digital Image Processing
Image Restoration and Reconstruction III
Order-Statistic filters
- Median filter
- Max and min filters
- Midpoint filter
- Alpha-trimmed mean filter
Electrical & Computer Engineering Dr. D. J. Jackson Lecture 13-
Median filter
- Replaces the value of a pixel by the median of the pixel values in the neighborhood of that pixel
- The pixel at ( x , y ) is included in the calculation
- Works well for various noise types, with less blurring than linear filters of similar size
- Odd sized neighborhoods and efficient sorts yield a computationally efficient implementation
- Most commonly used order-statistic filter
ˆ(, ) { (, )} (,) ,
f x y median gs t s t ∈ Sxy
=
Median filter example
Electrical & Computer Engineering Dr. D. J. Jackson Lecture 13-
Midpoint filter
- Replaces the value of a pixel by the midpoint between the maximum and minimum pixels in a neighborhood
- Combines order statistics and averaging
- Works best for randomly distributed noise (e.g. Gaussian or uniform)
⎥ ⎦
⎤ ⎢ ⎣
⎡ = + st ∈ Sx y st ∈ Sxy
f x y g st gs t (,) , (,) ,
max{ ( , )} min{ ( , )} 2
1 ˆ( , )
Alpha-trimmed mean filter
- If we delete the d/2 lowest and the d/2 highest intensity values from a neighborhood g ( s , t ) of size m * n and let g (^) r ( s , t ) represent the remaining mn-d pixels, the average of the remaining pixels is called an alpha-trimmed mean filter and is given by:
- d can vary from 0 to mn- 1
- If d =0 the filter becomes the arithmetic mean filter
- If d = mn- 1, the filter reduces to a median filter
st Sx y
f x y mn d gr st (,) ,
Electrical & Computer Engineering Dr. D. J. Jackson Lecture 13-
Alpha-trimmed mean filter example
Adaptive filters
- All filters considered thus far are applied to an image without regard for how image characteristics may vary from one point to another in the image
- An adaptive filter is one whose behavior can change based on statistical characteristics of an area within the image - This is typically the m * n filter region in the Sx , y window
- Generally provides superior performance at the cost of increased filter complexity
Electrical & Computer Engineering Dr. D. J. Jackson Lecture 13-
Adaptive, local noise reduction filter equation
- An adaptive expression may be written as:
- The only quantity that must be known is σ^2 η
- Everything else can be computed from Sx , y
- An assumption here is that σ^2 η≤σ^2 L
- This is generally reasonable given that the noise we are considering is additive and position independent
- If this is not true then a simple test could set the ratio of the variances to one if σ^2 η>σ^2 L
[ L ]
L
f ˆ^ ( x , y )= g ( x , y )− 2 g ( x , y )− m
2 σ
σ (^) η
Adaptive, local noise reduction filter example
Electrical & Computer Engineering Dr. D. J. Jackson Lecture 13-
Adaptive median filter
- A median filter works well in the spectral density of the impulse noise is not large - A Pa and Pb less than 0.2 is a good general rule of thumb
- An adaptive median filter can handle noise with probabilities greater than these
- An additional benefit is that the adaptive median filter attempts to preserve detail while smoothing the impulse noise
- The adaptive median filter works in a rectangular window area Sx , y - The size of Sx , y is not fixed
- The output of the filter is a single value that will be used to replace the center value of Sx , y
Adaptive median filter algorithm
- Consider the following notation
zmin = minimum intensity value in Sx , y zmax = maximum intensity value in Sx , y zmed = median intensity of values in Sx , y z ( x , y )= intensity value at ( x , y ) Smax = maximum allowed size of Sx , y
- The algorithm works in two stages (denoted A and B )
Stage A : A1= z (^) med – z (^) min A2= z (^) med – z (^) max If A1>0 AND A2<0, goto Stage B Else increase window size If window size ≤ Smax repeat Stage A Else output zmed Stage B : B1= z (^) x,y – z (^) min B2= z (^) x,y – z (^) max If B1>0 AND B2<0, output z (^) x,y Else output zmed
