(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.

1. SPATIAL FILTERING
ANUJ ARORA
B-TECH 2nd YEAR
ELCTRICAL ENGG.

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

3. SPATIAL FILTERING
NEIGHBORHOOD
• Typically, the neighborhood is rectangular and its size
is much smaller than that of f(x,y)
- e.g., 3x3 or 5x5

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

7. HANDLING PIXELS CLOSE
TO BOUNDARIES
pad with zeroes
or
0 0 0 ……………………….0
000……………………….0

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.

9. LINEAR SPATIAL FILTERING
METHODS
• Two main linear spatial filtering methods:
• Correlation
• Convolution

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

12. CORRELATION AND
CONVOLUTION
Correlation:
Convolution:

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

16. SMOOTHING FILTERS: AVERAGING
(LOW-PASS FILTERING)

17. SMOOTHING FILTERS:
AVERAGING
• Mask size determines the degree of smoothing and
loss of detail.
3x3 5x5 7x7
15x15 25x25
original

18. SMOOTHING FILTERS:
AVERAGING (CONT’D)
15 x 15 averaging image thresholding
Example: extract, largest, brightest objects

19. SMOOTHING FILTERS:
GAUSSIAN
• The weights are samples of the Gaussian function
mask size is
a function of σ :
σ = 1.4

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

21. SMOOTHING FILTERS:
GAUSSIAN (CONT’D)

22. AVERAGING VS GAUSSIAN
SMOOTHING
Averaging
Gaussian

23. SHARPENING FILTERS
(HIGH PASS FILTERING)
• Useful for emphasizing transitions in image intensity
(e.g., edges).

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

25. SHARPENING FILTERS:
UNSHARP MASKING
• Obtain a sharp image by subtracting a lowpass filtered
(i.e., smoothed) image from the original 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.

29. SHARPENING FILTERS:
DERIVATIVES (CONT’D)
• The gradient is a vector which has magnitude and
direction:
| | | |
f f
x y
or
(approximation)

30. SHARPENING FILTERS:
DERIVATIVES (CONT’D)
• Magnitude: provides information about edge strength.
• Direction: perpendicular to the direction of the edge.

31. SHARPENING FILTERS: GRADIENT
COMPUTATION
• Approximate gradient using finite differences:
sensitive to horizontal edges
sensitive to vertical edges
Δx

32. SHARPENING FILTERS: GRADIENT
COMPUTATION (CONT’D)
• We can implement and using masks:
• Example: approximate gradient at z5

33. SHARPENING FILTERS: GRADIENT
COMPUTATION (CONT’D)
• A different approximation of the gradient:
•We can implement and using the following masks:

34. SHARPENING FILTERS: GRADIENT
COMPUTATION (CONT’D)
• Example: approximate gradient at z5

35. EXAMPLE
f
y
f
x

36. SHARPENING FILTERS:
LAPLACIAN
The Laplacian (2nd derivative) is defined as:
(dot product)
Approximate
derivatives: