Bài giảng Chapter 6: Linear filters

The lecture presents the contents: Using a simpler operation to generate a higher order operation, Multiple summations, General formula, Doing the same with functions, Multiplication and addition of two functions, Operation that uses addition and multiplication...

Department of Mechatronics

Chapter 6 Linear Filters

Prof Fei Fei Li, Stanford University

• 2D Filter

– Convolution

– Linear Systems

Department of Mechatronics

Multiplication

• Using a simpler operat ion to generate a higher

order operation

• Mult iple summat ions

• General form ula

Doing the same with functions

• Applying the operations on

each value separated.

• The result is a function.

Department of Mechatronics

Multiplication and addition of

two functions

• Operation that uses addition and multiplication.

• Result is a function.

• It is a way to combine to functions.

• It is like weighting one function with the other.

• Flipping one function and then summing up the products for each positions for a given offset n.

( ) ( ) (n )



Department of Mechatronics

Flipping the function

Multiply and add

Department of Mechatronics

Multiply and add

Multiply and add

Department of Mechatronics

Multiply and add

Multiply and add

Department of Mechatronics

Multiply and add

Multiply and add

Department of Mechatronics

Multiply and add

Multiply and add

Department of Mechatronics

Result

Comparison

Department of Mechatronics

Continuous Convolution

• Summation becomes an integral

Department of Mechatronics

Filter

• Signal y(n) is a

convolution of u(n) with

the Transfer function

h(n)

• Filtering can be done by

the convolution of two signals

2D Filtering

A 2 D in 1 age f[i,j] can be fi l tered by a 2D kernel h[u,v] to

produce an output image g[i,j]

T hi s i s called a cross-correlation operation and written :

h i s called t l 1e " filter ", " kernel " or " mask ".

Department of Mechatronics

2D Filtering

Example finding edges

Differentiating to find the steep parts of t he picture

Department of Mechatronics

Example Simple NoiseReduct i on

• Cutting of the high frequency noise by low pass

filtering with the sine like kernel

• For not changing the aspect of the image the sum

of the kernel must be 1

1/13

Images asfunctions

• An Image as a function f from R 2 to R M :

 f( x,y) gives the intensity at position ( x, y )

 Defined over a rectangle, with a finite range:

Domanin Support range :[ , ]x[ , ] [0, 255]

f a b c d 

Department of Mechatronics

Images asfunctions

• A color imag e :

• An Image as a function f from R 2 to R M :

 f( x,y) gives the intensity at position ( x, y )

 Defined over a rectangle, with a finite range:

Domanin Support range

:[ , ]x[ , ] [0, 255]

f a b c d 

( , ) ( , ) ( , )

Images as discrete functions

• Images are usually digital (discrete):

- Sample the 20 space on a regular grid

• Represented as a matrix of integer values

Department of Mechatronics

Images as discrete functions

Cartesian coordinates

Images as discrete functions

Department of Mechatronics

Systems and Filters

• Filtering:

original pixel values

Goals:

• Extract useful information from the images

 Features (edges, corners, blobs )

 Super-resolution ; in-pain t ing ; de-noising

De -noi s i ng

Ori g in a l S a lt and p e pp e r n o i se

Super -re s olut i on

In-painting

Department of Mechatronics

2D discrete- space systems (filters)

Filter

Department of Mechatronics

Moving average

Moving average

Department of Mechatronics

Moving average

Moving average

Department of Mechatronics

Moving average

Moving average

Department of Mechatronics

Moving average

• Replaces each pi x el w it h an

• Achie v e smoothing effect

Moving average

Department of Mechatronics

Shift-invariance

Is the moving avera ge system is shift inva riant?

Department of Mechatronics

Is the moving average system is shift invariant?

Linear Systems (filters)

• Linear filtering:

- Fo rm a new i m age whose p i xe l s are a we i ghted

su m of orig i na l p i xe l va l ues

- Use t h e same set of weig h ts at eac h poi n t

• S is a linear sys t em ( f unc t ion) i ff i t S satisfies

Department of Mechatronics

Linear Shift Invariant System - LSI

The Dirac delta function as the limit (in the sense

of distributions) of the sequence of Gaussians

Kronecker delta

Department of Mechatronics

LSI (linear shift invariant ) Systems

Impulse response

Example: i mpu l se response of t h e 3 by 3 mov i ng

ave r age fi lte r :

LSI (linear shift invariant ) Systems

Department of Mechatronics

An LSI system is completely specified by its impulse

response.

LSI (linear shift invariant ) Systems

sift i ng property of the de l ta f unction

Discrete convolution

• Fold h[n,m] about or igin to form h[-k,-l]

• Shift the folded results by n,m to form h[n - k,m - I]

• Multiply h[n - k,m - I] by f[k,I]

• Sum over all k,I

• Repeat for every n, m

Department of Mechatronics

Discrete convolution

• Fold h[n,m] about or igin to form h[-k,-l]

• Shift the folded results by n,m to form h[n - k,m - I]

• Multiply h[n - k,m - I] by f[k,I]

• Sum over all k,I

• Repeat for every n, m

Discrete convolution

• Fold h[n,m] about or igin to form h[-k,-l]

• Shift the folded results by n,m to form h[n - k,m - I]

• Multiply h[n - k,m - I] by f[k,I]

• Sum over all k,I

• Repeat for every n, m

Department of Mechatronics

Convolution in 2D - Examples

Original

Original Filtered

Convolution in 2D - Examples

Department of Mechatronics

Original

Convolution in 2D - Examples

Origina l S hifted l eft

Convolution in 2D - Examples

Department of Mechatronics

Origin al

Convolution in 2D - Examples

Convolution in 2D - Examples

Department of Mechatronics

Original

Convolution in 2D - Examples

Convolution in 2D – Sharpening Filter

Or ig in al

Department of Mechatronics

Department of Mechatronics

Image support and edge effect

• A computer w ill only c onvolv e finite support signals.

ngula r region

fi ni t e support signa Is.

• A computer will only convolve finite support signals.

• What happens at the edge?

Image support and edge effect

Department of Mechatronics

Zero-Padding

Boundary Padding Options

See the reference page for imfilter for details.

Replicated Boundary Pixels

Cross correlation

Department of Mechatronics

Matlab: filter2 imfilter Matlab: conv2

Filtering: Boundary Issues

• What is the size of the output?

• MATLAB: filter2(g,f,shape)

– shape = ‘full’: output size is sum of sizes of f and g

– shape = ‘same’: output size is same as f

– shape = ‘valid’: output size is difference of sizes of f and g

valid

g g

g g