Two Dimensional Digital Signal Processing

8/8/2019 Two Dimensional Digital Signal Processing

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Two Dimensional Digital Signal

Processing


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Many signals are inherently two dimensional

Photographic data, weather photos, X-rays

Generally spatial signals

Basic ideas of 1D signal processing can be

extended to 2D case


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2D Signals

Notation used to represent 2D variables

where are integer variables

is short hand version of 2D signal

Perspective plots to represent 2D signalsgraphically

Third dimension represents magnitude


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2D Signals


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2D Signals

Useful 2D sequences


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2D Signals

2D Digital Impulse Or Unit Sample


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2D Signals

2D Digital Step


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2D Signals

2D Step is related to 2D Impulse by the

following relation

For 1D signal the relation is


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2D Signals

Exponential and Sinusoidal signals


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2D Signals

2D Exponential signal


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2D Systems

Convolution Theorem

For LTI Systems Convolution theorem is valid

Relation Input & Impulse response to output is


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2D Systems

For a system with Impulse response and input

Output is


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2D Systems

Direct evaluation of Convolution is difficult since

cannot be factored into


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2D Systems

Causality and Separability

A 2D filter is causal or realizable if its impulse

response satisfies the property

And separable if its impulse response can be

factored into a product of 1D responses


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2D Systems

Advantage of separable filters - 2D convolution can

be carried out as a sequence of 1D convolutions

[ ] for each value of


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2D Systems

If input sequence is also separable

where


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2D Systems

A 2D filter is stable only if the impulse

response satisfies the constraint


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2D Systems

2D linear Difference Equations

To describe an LTI 2D filter

The constant coefficients

plus a set of initial conditions specify the filter


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2D Systems

Recursion relation for assuming

is

-


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1D Fourier series relations

With sinusoidal input to a 2D system


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Fourier series & frequency domain analysis

Output of the system is


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Frequency response of the system is a Fourier

series representation

with the coefficients given by


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The frequency response is doubly periodic

Find the Fourier series coefficients, of

the filter with frequency response


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Frequency response = 1 in hatched

area


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If the frequency response can be decomposed

to the product of a term inimpulse response will be a product of a term

in

Two Dimensional Digital Signal Processing

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