LINEAR SYSTEMCONVOLUTIONLINEAR FILTER

CS 111: Digital Image ProcessingAditi Majumder

Outline

• Linear System• Properties• Response

• Convolution• Concept• Properties

• Linear Filter• Low pass, High pass, Aliasing…

Properties of Linear System

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1. Homogeneity:

2. Additivity:

3. Shift Invariance:

Other Properties of Linear Systems1. Commutative:

2. Superposition: If each generates multiple outputs, Then the addition of inputs generates an addition of outputs.

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Decomposition - Synthesis

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Response of Linear System• Impulse: Signal with only one non-zero sample. • Delta (δ[t]) is an impulse with non-zero sample at t = 0

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Response of Linear System• Impulse response h[t]

• output of the system to the input δ[t].

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Response of Linear System• Impulse response h[t]

• output of the system to the input δ[t].• Convolution: Response of a linear system with impulse

response, h, to a general signal

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Convolution – Input side

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a1 a2 a3 a4 a5 a6 a7

i=1

Input

Kernel

Output

k1 k2 k3

i=2

k1 k2 k3

i=3

Convolution – Output side

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a1 a2 a3 a4 a5 a6 a7Input

Kernel

Output

k3 k2 k1

i=1 i=2

k3 k2 k1

i=3

Convolution

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s[m]

h[m]

s[0].h[n]

s[1].h[n-1]

s[2].h[n-2]

2D Convolution

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Properties of Convolution

• All pass system

• Amplifier (k>0) / attenuator (k

Properties of Convolution• Commutative

• Associative

• Distributive

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Properties of Convolution• Cascading convolutions

• Combination of parallel convolutions

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Blurring filters

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• More blurring implies widening the base and shortening the height of the spike further.

• What does it look like?• Box filters are not best blurring filters but the easiest to

implement.

Duality

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Spatial Domain Frequency Domain

Duality

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Spatial Domain Frequency Domain

Widening in one domain is narrowing in another and vice-versa.

Duality• Convolution of two functions in time/spatial domain is a

multiplication in frequency domain• Vice Versa

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All Pass Filter

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Low Pass Filter

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t

k[t]

F

f

K[f]

A(f)X

t

a(t)

Low Pass Filtering• Box filter is known as low pass filter.

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Box Filter

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• Effect of increasing the size of the box filter

Gaussian Pyramid

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Gaussian Pyramid

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Box is not the only shape• Gaussian is a better shape• Any thing more smooth is better

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t

x[t]

F

f

X[f]

Hierarchical Filtering

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1/4 1/4

1/4 1/4

N x NN/2 x N/2

N/4 x N/4

1 x 1

Issue of Sampling• As an image undergoes low pass filtering, its frequency

content decreases • Minimum number of samples required to adequately sample

the low pass filtered image is less.• Low pass filtered image can be at a smaller size than the

original image.

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Subsampling

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Simple subsampling

Pre-filtering and subsampling

Aliasing Artifact

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Input (256 x 256)

Subsampled(128 x 128) Subsampled from filtered image(128 x 128)

Insufficient sampling.Hence, aliasing.

Filtering reduces frequency content.Hence, lower sampling is sufficient.

Filtered (256 x 256)ANTI-ALIASING

High Pass Filter•

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High Pass Filter

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Original Image Low pass filtered High pass filtered

Band-limited Images (Laplacian Pyramid)

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Gn

Gn-1

Gn-2

fnfn-1fn-2

fn-2

Band-limited Images (Laplacian Pyramid)

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2D Filter Separability• Visualizing 2D filters from their 1D counter part

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Box Filter Gaussian Filter High Pass Filter

2D Filter Separability•

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2D Filter Separability• Advantage

• Separable filters can be implemented more efficiently

• Convolving with h• Number of multiplications = 2pqN

• Convolving with a and b• Number of multiplications = 2(p+q)N

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Linear system�Convolution�Linear FilterOutlineProperties of Linear SystemOther Properties of Linear SystemsDecomposition - SynthesisResponse of Linear SystemResponse of Linear SystemResponse of Linear SystemConvolution – Input sideConvolution – Output sideConvolution2D ConvolutionProperties of ConvolutionProperties of ConvolutionProperties of ConvolutionBlurring filtersDualityDualityDualityAll Pass FilterLow Pass FilterLow Pass FilteringBox FilterGaussian Pyramid �Gaussian Pyramid �Box is not the only shapeHierarchical FilteringIssue of SamplingSubsamplingAliasing Artifact High Pass FilterHigh Pass FilterBand-limited Images (Laplacian Pyramid)Band-limited Images (Laplacian Pyramid)2D Filter Separability2D Filter Separability2D Filter Separability