(1) Spatial filtering is defined as operations performed on pixels within a neighborhood of an image using a mask or kernel. (2) Filters can be used to blur/smooth an image by reducing noise or sharpen an image by enhancing edges. (3) Common linear filtering methods include averaging, Gaussian, and derivative filters which are implemented using various mask patterns to modify pixels in the filtered image.
2. SPATIAL FILTERING
(CONT’D)• Spatial filtering is defined by:
(1) An operation that is performed on the pixels inside the
Neighborhood
(2)First we need to create a N*N matrix called a
mask,kernel,filter(neighborhood).
(3)The number inside the mask will help us control the
kind of operation we are doing.
(4)Different number allow us to blur,sharpen,find edges.
output image
4. SPATIAL FILTERING -
OPERATION
1 1
1 1
( , ) ( , ) ( , )
s t
g x y w s t f x s y t
Assume the origin of the
mask is the center of the
mask.
/ 2 / 2
/ 2 / 2
( , ) ( , ) ( , )
K K
s K t K
g x y w s t f x s y t
for a K x K mask:
for a 3 x 3 mask:
5. • A filtered image is generated as the center of the
mask moves to every pixel in the input image.
output image
6. STRANGE THINGS HAPPEN
AT THE EDGES!
Origin x
y Image f (x, y)
e
e
e
e
At the edges of an image we are missing
pixels to form a neighbourhood
e e
e
8. LINEAR VS NON-LINEAR
SPATIAL FILTERING METHODS
• A filtering method is linear when the output is a
weighted sum of the input pixels.
• In this slide we only discuss about liner filtering.
• Methods that do not satisfy the above property are
called non-linear.
• e.g.
10. CORRELATION
• TO perform correlation ,we move w(x,y) in all possible
locations so that at least one of its pixels overlaps a
pixel in the in the original image f(x,y).
/ 2 / 2
/ 2 / 2
( , ) ( , ) ( , ) ( , ) ( , )
K K
s K t K
g x y w x y f x y w s t f x s y t
11. CONVOLUTION
• Similar to correlation except that the mask is first flipped
both horizontally and vertically.
Note: if w(x,y) is symmetric, that is w(x,y)=w(-x,-y), then
convolution is equivalent to correlation!
/ 2 / 2
/ 2 / 2
( , ) ( , ) ( , ) ( , ) ( , )
K K
s K t K
g x y w x y f x y w s t f x s y t
13. HOW DO WE CHOOSE THE
ELEMENTS OF A MASK?
• Typically, by sampling certain functions.
Gaussian
1st derivative
of Gaussian
2nd derivative
of Gaussian
14. FILTERS
• Smoothing (i.e., low-pass filters)
• Reduce noise and eliminate small details.
• The elements of the mask must be positive.
• Sum of mask elements is 1 (after normalization)
Gaussian
15. FILTERS
• Sharpening (i.e., high-pass filters)
• Highlight fine detail or enhance detail that has been
blurred.
• The elements of the mask contain both positive and
negative weights.
• Sum of the mask weights is 0 (after normalization)
1st derivative
of Gaussian
2nd derivative
of Gaussian
20. SMOOTHING FILTERS:
GAUSSIAN (CONT’D)
• σ controls the amount of smoothing
• As σ increases, more samples must be obtained to represent
the Gaussian function accurately.
σ = 3
24. SHARPENING FILTERS
(CONT’D)
• Note that the response of high-pass filtering might be
negative.
• Values must be re-mapped to [0, 255]
sharpened imagesoriginal image
26. SHARPENING FILTERS:
HIGH BOOST
• Image sharpening emphasizes edges .
• High boost filter: amplify input image, then subtract a
lowpass image.
• A is the number of image we taken for boosting.
(A-1) + =
27. SHARPENING FILTERS: UNSHARP
MASKING (CONT’D)
• If A=1, we get a high pass filter
• If A>1, part of the original image is added back to the
high pass filtered image.
28. SHARPENING FILTERS:
DERIVATIVES
• Taking the derivative of an image results in sharpening
the image.
• The derivative of an image can be computed using the
gradient.