Filtering

Course web page:vision.cis.udel.edu/cv

March 5, 2003 Lecture 9

Announcements

• Read Forsyth & Ponce, Chapter 8-8.3.1, 9.2-9.2.1 for Friday

• Homework 2 will be assigned on Friday and due Monday, March 17

Outline

• Systems, filtering• Linear shift invariance =

convolution• Discrete vs. continuous

convolution• Low-pass filters• Image pyramids

Imaging Systems

• An imaging system describes a functional transformation f of an image due to…– Physics: A real-world phenomenon such as

blurring from defocus or fish-eye lens distortion– Filtering: A transformation we apply in order

to• Undo or mitigate the bad effects of a physical system

(e.g., deblur, undistort, etc.)• Emphasize or highlight particular image properties

(e.g., color similarity, edges, etc.)I I’

f

Linear Shift Invariance

• Possible properties of f– Superposition: f(I1 + I2) = f(I1) + f(I2)

– Scaling: f(®I) = ®f(I)– Shift invariance:

f(Shift(I, k)) = Shift (f(I), k)A system with these properties is performing convolution

What’s Not a Convolution?

• Nonlinear systems– E.g., radial distortion of fish-eye lens

is not LSI because geometric transformation depends on pixel location

f

courtesy of M. Fiala

Impulse Response

• One way to look at what a convolution is doing is to measure its effect on a ± function– Reasonable ± approximation is a black image

containing a single bright dot of light at the origin

• The output is f’s impulse response (aka a point-spread function)

• Convolving an image consists of measuring f’s response to the brightness (a different impulse) at every image location

Continuous Convolution

• Definition:

• Sifting property: If I is the delta function ±, then the integral is 0 everywhere but at x = 0, y = 0, and I’(u, v) = f(u, v) (because ± integrates to 1). Different values of u, v thus map out the point spread function

• An arbitrary I can be represented as an infinite collection of shifted, scaled impulses

Convolution Notes

• Notation: I’ = f ¤ I (book uses ¤¤ for 2-D convolution)

• Note the assumption that both the impulse function and the images are continuous and defined everywhere

• Properties– Commutative– Associative

Discrete Convolution

• Sum instead of integral: I’ = K ¤ I is defined by:

I’(u, v) = §x, y K(u ¡ x, v ¡ y) I(x, y) where the kernel K approximates the impulse function f by

sampling from it where it is non-zero – Think of the last part of HW 1—e.g., if f is a Gaussian, the matrix

of Z-values would be K• Convolution: Correlation with kernel rotated 180

– This distinction goes away for isotropic kernels• Some ways to interpret what the kernel is doing

– As a template being matched by correlation (non-normalized)– As simply a set of weights on the corresponding image pixels

111-121-1-11

Dealing with Image Edges

• Only convolve with interior– Shrinks image

• Zero-padding– Results in spurious gradients

• Border replication• Symmetric: Reflect image at

border b so that I(b + i) = I(b ¡ i)– Results in spurious 2nd-derivatives

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Filtering Example 1:

1 -1 -1

1 2 -1

1 1 12 2 2 3

2 1 3 3

2 2 1 2

1 3 2 2

Rotate

1-1-1

12-1

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(using zero-padding)

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and so on…

Final Result

2 2 2 3

2 1 3 3

2 2 1 2

1 3 2 2 12

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I’I

1-1-1

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Why is I’ large in some places and small in others?

Smoothing (Low-Pass) Filters

• Problem: How to suppress noise, aliasing?– If object reflectance changes slowly and noise

at each pixel is independent, then we want to replace each pixel with something like the average of neighbors

• Disadvantage: Sharp (high-frequency) features lost

• Types– Mean filter (box)– Median (nonlinear)– Gaussian

111

111

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3 x 3 box filter

Example: Gaussian Noise

from Forsyth & Ponce

Box Filter: Smoothing

7 x 7 kernelOriginal image

Filtering in Matlab

• imfilter(I, K)filters image I with kernel K– Default filtering is correlation (no kernel

rotation)– Can set options on border handling

• corr2, conv2 are the generic versions• Kernel creation

– Custom (like you did in the last part of HW 1)

– fspecial function