Convolution Correlation

7/28/2019 Convolution Correlation

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Todays Lecture :

Convolution and Correlation

Impulse response

Convolution in linear time-invariant systems

Properties of linear time-invariant systems

Convolution in images: the Gaussian filters

Break

Implementation of linear time-invariant systems

Correlation of discrete-time signals

Correlation in images

Local image enhancement

image and signal processing

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Method for Analysis of a Discrete-Time

Linear Time-Invariant (LTI) System

Any input signal can be decomposed into the weighted sum of

unit sample sequences.

Analyze the linear, time-invariant (LTI) system by its response to

a unit sample signal system impulse response h(n).

The LTI system output for an arbitrary input signal is the con-

volution sum between the input signal and the system impulse

response h(n).


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Decompose a Discrete-Time Signal x(n)

Into a Sum of ImpulsesSelect elementary signal: time shifted impulses xk(n) = (n k).

Multiply x(n) and (n k): x(n)(n k) = x(k)(n k).


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Decompose a Discrete-Time Signal x(n)

Into a Sum of Impulses

The value of x(n) at position k can be obtained from

x(n)(n k).

In order to obtain the entire signal x(n) we simply sum the product

sequences for all possible k:

x(n) =

k=

x(k)(n k).


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The Convolution Sum

Define the response of a Linear, Time Invariant (LTI) system to the

unit sample signal (n):

h(n) [(n)]

Now use the arbitrary input signal represented as a sum of impulses

x(n) =

k=

x(k)(n k).

The LTI system output signal y(n) from the input signal x(n) is

y(n) = [x(n)].

Use the x(n) sum and the linearity condition of the LTI system

y(n) = [

k=

x(k)(n k))] =

k=

x(k)h(n k).


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The Convolution Sum Continued

Thus, the relaxed Linear, Time Invariant (LTI) system is completely

characterized by a single function h(n), which is the systems response

to the unit sample signal (n).

The LTI systems response to any input signal x(n) is the convolution

sum

y(n) =

k=x(k)h(n k) = x h.


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A Closer Look at the Convolution at Time

n0Compute the LTI system output at time n = n0,

y(n0) =

k=

x(k)h(n0 k).

From this we observe that:

The input signal x(k) and the impulse response h(n0k) are both

functions of k.

The signals x(k) and h(no k) are multiplied together to form a

new signal.

The output signal y(n0) is the sum of the elements of this product

signal.


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The Four Steps in Convolution

Computing the convolution between x(k) and h(k) to obtain the out-

put signal value y(n0), at time n = n0, involves the following four

steps

1. Folding Fold h(k) about k = 0 to obtain h(k).

2. Shifting Shift h(k) by n0 to obtain h(n0 k).

3. Multiplying Multiply x(k) by h(n0k) to obtain vn0 x(n)h(n0

k).

4. Summing Sum all the values of the product signal vn0(k) to obtain

the system output value y(n0) at time n = n0.

The complete system output signal y(n) for < n < is computed

by repeating Step 2, 3 and 4 for all time instants n.


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Convolution Example 2.3.2


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Properties of Convolution

Commutative x(n) h(n) = h(n) x(n)

y(n) =

k= x(k)h(n k) =

k= x(n k)h(k)

It does not matter which of the two signals that is folded and

shifted.

Associative [x(n) h1(n)] h2(n) = x(n) [h1(n) h2(n)]The input can be convolved with first one than the other impulse

response, or the impulse responses can be convolved with each

other first.

Distributive x(n) [h1(n) + h2(n)] = x(n) h1(n) + x(n) h2(n)If two LTI systems with impulse responses h1(n) and h2(n) are

excited by the same input, the sum of the responses is identical to

that of a LTI system with impulse response h(n) = h1(n) + h2(n).


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Causality of Linear Time-Invariant Systems

The output signal of a causal LTI system at time n0 does only depend

of the values of x(n) for n n0. What does that mean for the systems

impulse response h(n)?

y(n0) =

k=

h(k)x(n0 k)

=

k=0

h(k)x(n0 k) +1

k=

h(k)x(n0 k)

= (h(0)x(n0) + h(1)x(n0 1) + h(2)x(n0 2) + ) +

(h(1)x(n0 + 1 ) + h(2)x(n0 + 2 ) + h(3)x(n0 + 3 ) + ).

Now, if y(n0) is only dependent of past and present inputs then

h(n) = 0 for n < 0.

Thus the impulse response h(n) is 0 for n < 0. Nothing happens

before the input signal is applied.


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Causality of Linear Time-Invariant Systems

Now use a causal input x(n) to a LTI system h(n), then the limits in

the convolution sum is further restricted:

y(n) =n

k=0

h(k)x(n k)

=

nk=0

x(k)h(n k).

Notice that

the upper limit is growing with time, and

the output y(n) of a causal system, in response to a causal input,

is causal.


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Stability of Linear Time-Invariant Systems

A system is BIBO (bounded input, bounded output) stable if and

only if its output signal y(n) is bounded for every bounded input

x(n).

A linear time-invariant system is stable if its impulse response is

absolutely summable

k=

|h(k)| <


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Systems with Finite-Duration and

Infinite-Duration Impulse Responses FIR system:

Finite Impulse Response system such that h(n) = 0 for n < 0 and

n M.

y(n) =M1k=0

h(k)x(n k)

The system acts as a window, that views the most recent M inputsignal samples in forming the output. The FIR system has a finite

memory length of M samples.

IIR system: Infinite Impulse Response system.

y(n) =

k=0

h(k)x(n k)

The output uses the present and all past input samples, and thus

it has an infinite memory.


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Discrete 2D (Spatial) Convolution

f(x, y) h(x, y) =1

N M

M1m=0

N1n=0

f(m, n)h(x m, y n)

which is equivalent to our original definition of discrete 2D filtering.

The only difference is a flipping of the filter h about the origin.

g(x, y) =a

s=a

b

t=b

w(s, t)f(x + s, y + t)

This is a direct extension from the 1D convolution.


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Properties of convolutions

Besides the commutative, distributive and associative properties,

f g = g f

f (g + h) = f g + f h

f (g h) = (f g) h

the convolution result is limited and differentiable if one operand is.

In this case we have

D(f g) = Df g = f Dg

where D denotes differentiation.


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Gaussian Filter and Its Derivatives

This means we can get a smoothed differentiated image by convolving

the image with the derivative of a Gaussian function.

Important filters:

G(x, ) = 1(2)

12

e x2

22

Gx(x, ) = x

2G(x, )

Gxx(x, ) = (1

2

+x2

4

)G(x, )

Gxxx(x, ) = (3x

4

x3

6)G(x, )

. . .


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Gaussian Filters

From left to right: Original image, smoothed image by convolutionwith a Gaussian, blurred derivative in x-direction, burred derivative in

y-direction.


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Recursive Discrete-Time Systems

The output of a LTI system can be described explicitly in terms of

present and past values of the input. However past output values can

also be used in the description.

The cumulative average of a signal is y(n) = 1n+1 nk=0 x(k). Thecomputation of y(n) requires storage for samples x(k), 0 k n.

However we can use the previous output y(n 1):

(n + 1)y(n) =n1

k=0 x(k) + x(n) = ny(n 1) + x(n).

Hence y(n) = nn+1y(n 1) +

1n+1x(n).

The recursive system requires only one memory location.


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LTI Systems Described by Difference

Equations

Systems described by constant-coefficient linear difference equations

are a subclass of the recursive and nonrecursive systems.

The general form is

y(n) = N

k=1

aky(n k) +M

k=0

bkx(n k)

Recursive systems described by linear constant-coefficient differenceequations are linear and time invariant.


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Implementation of Linear Time-Invariant

Systems.

First order LTI system

y(n) = a1y(n 1) + b0x(n) + b1x(n 1).

Viewed as two LTI systems in cascade

FIR (Finite Impulse Response):

v(n) = b0x(n) + b1x(n 1)

and IIR (Infinite Impulse Response):

y(n) = a1y(n 1) + v(n).

We can interchange the order of two LTI filters without affecting the

overall system response (direct form II).

w(n) = a1w(n 1) + x(n)

y(n) = b0w(n) + b1w(n 1)


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General Linear Time-Invariant Recursive

System.

y(n) = N

k=1

aky(n k) +M

k=0

bkx(n k)

This can be written as a recursive system

w(n) =N

k=1

akw(n k) + x(n)

followed by a nonrecursive system (direct form II)

y(n) =M

k=0

bkw(n k)


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Direct Form II Structure.


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Direct Form II Structure.

The Direct Form II Structure requires:

The number of multiplications: M + N + 1.

The number of delays: max(M, N).

The Direct Form II structure requires a minimum number of delays

for the realization of the system, and is denoted a canonic form.


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Recursive and Nonrecursive Realizations of

FIR Systems

A FIR system is described by

y(n) = N

k=1

aky(n k) +M

k=0

bkx(n k).

A nonrecursive system has ak = 0 and the impulse response is the

filter coefficients bk.

A FIR system can always be realized both recursively and nonrecur-

sively.

FIR and IIR are general characteristics of a LTI system, recursive and

nonrecursive are structures for implementation of the system.


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Correlation of Discrete-Time Signals

Distance from radar to target:

x(n): Sampled version of the signal transmitted by the radar.

y(n): Sampled version of the signal reflected by the target (f.inst.

airplane, ship) and received by the radar.

Now assume the received signal y(n) is a delayed version of x(n)

y(n) = x(n D)

is an attenuation and D is the delay in samples out and home.

How can we determine the distance to the target?

Use correlation between the transmitted signal x(n) and the received

signal y(n)!


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Correlation of Discrete-Time Signals

The cross correlation between the finite energy signals x(n) and y(n)

rxy(k) =

n=

x(n)y(n k) k = 0,1,2, . . .

or equivalently y(n)

rxy(k) =

n=

x(n + k)y(n) k = 0,1,2, . . .


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Correlation of Discrete-Time Signals III

What happens if the order of x(n) and y(n) is reversed?

ryx(k) =

n=

y(n)x(n k)

=

n=

x(n k)y(n)

=

m=

x(m)y(m + k)

= rxy

(

k)


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Correlation

Cross correlation of x(n) and y(n): rxy(k) =

n= x(k)y(n k).

Shifting into y(n k).

Multiplying x(n) and y(n k).

Summing of product terms.


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Energy and Power of a Signal

The energy of a signal x(n) is defined as

E

n=

|x(n)|2

The average power of a signal is

P limx

1

2N + 1

N

n=N

|x(n)|2


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Autocorrelation

Correlation of a signal x(n) with itself.

rxx(k) =

n=

x(n)x(n k)

The signal energy is the autocorrelation for k = 0.

rxx(0) =

n=

x2(n)


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Properties of Auto- and Crosscorrelation

How are rxx(k), rxy(k) and ryy (k) related?

Look at the energy E(k) in the signal ax(n) + by(n k)

E(k) =

n=

(ax(n) + by(n k))2

= a2

n=

x2(n) + b2

n=

y2(n k) +

2ab

n=

x(n)y(n k)

= a2rxx(0) + b2ryy ( 0 ) + 2abrxy(k)

Now use that E(k) 0 k = 0,1,2, . . .

a2rxx(0) + b2ryy ( 0 ) + 2abrxy(k) 0

and dividing by b2, which is assumed = 0 gives

rxx(0)(a

b)2 + 2rxy(k)(

a

b) + ryy(0) 0


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Properties of Auto- and Cross-Correlation

E(k) is quadratic and nonnegative ina

b with coefficients rxx(0), 2rxy(k),and ryy(0)

rxx(0)

a

b

2+ 2rxy(k)(

a

b) + ryy (0) 0.

The discriminant

D = (2rxy(k))2 4rxx(0)ryy (0) = 4(rxy(k) rxx(0)ryy (0)) 0.

Condition on crosscorrelation

|rxy(k)| rxx(0)ryy(0) = ExEy.Condition on autocorrelation for y(n) = x(n)

|rxy(k)| rxx(0) = Ex


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Example of Autocorrelation


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Autocorrelation of Periodic Signals.

The autocorrelation of a periodic signal (power signal) is periodic.

rxx(k) = limx

1

2M + 1

Mn=M

x(n)x(n k)

Assume that x(n) is periodic with period P, then

rxx(k + P) = limx

1

2M + 1

Mn=M

x(n)x(n k P).

Because x(n k P) = x(n k)

rxx(k + P) = rxx(k)


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Periodicity in Sunspot Numbers.


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Spatial Correlation

Correlation can be used to find matches of a subimage w(x, y) in an

image f(x, y):

c(x, y) =

s

t

f(s, t)w(x + s, y + t)

The summation is taken over the region where the images overlap.

Compare it to filtering:

g(x, y) =a

s=a

bt=b

w(s, t)f(x + s, y + t)


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Local Enhancement by Histogram

Equalization

For each pixel, compute histogram in the neighborhood and obtain

histogram equalization. Map the gray level of the center pixel.


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Use of Histogram Statistics for Image

Enhancement

Find the local mean and variance of the gray levels in a neighborhood.Enhance areas where the local mean and variance differs a lot from

the global estimates.


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Documents

Convolution Correlation