1

Analog Signals

Both independent and dependent variables can assume a continuous range of valuesExists in nature

Digital Signals

Both independent and dependent variables are discretizedRepresentation in computersSampling

Discrete independent variableSample and hold (S/H)

QuantizationDiscrete dependent variableAnalog to Digital Converter (ADC)

2

Digital Signal

Sampled Signal

3

Digitized Signal

•Depends on number of bits•12 bits = 4095 levels• 0.0 ≤ Voltage ≤4.096•2.56 and 2.5601 TO 2560•Each level (LSB) = 0.001•Error ≤ ±½ LSB•Called Quantization Error

Digitized Signal

•Depends on number of bits•12 bits = 4095 levels• 0.0 ≤ Voltage ≤4.096•2.56 and 2.5601 TO 2560•Each level (LSB) = 0.001•Error ≤ ±½ LSB•Called Quantization Error

4

Quantization Error

Usually like random noiseNoise is present in most signal acquisition systemsRandom uncorrelated samples added to the original signal

Proper Sampling

If the original signal can be reconstructed unambiguously from the sampled signal

Cycles/ Sample =Number of cycles per second

Number of samples per second

=Analog Frequency

Sampling Rate

5

Is it Proper Sampling?DC signalFreq = 0.0 x Sampling RateProper

Is it Proper Sampling?

Freq = 0.09 x Sampling RateEach sample covers 0.09 cyclesProper

6

Is it Proper Sampling?

Freq = 0.31 x Sampling RateLarger fraction of cycles per sampleProper

Is it Proper Sampling?Freq = 0.95 x Sampling RateMuch larger parts of cycles per sampleNot ProperAliasingChanges frequency and phase

7

Sampling Theorem

Proper Sampling: At least one sample per half cycleFreq ≤ 0.5 x Sampling RateSampling Rate ≥ 2 x FrequencyNyquist Rate

Time (Spatial) Domain vs. Frequency Domain

Any one dimensional analog signal can be represented as a linear combination of sine waves of different frequencies

8

1D SignalExample: Once scan line of an imageAmount of each sine wave defined by its amplitude and phase

Representation in Both Domains

Frequency

Am

pli

tud

e

2

1

0

Time Domain

Frequency Domain

Frequency

Ph

ase

180

0

-180

9

Representation in Both Domains

Time Domain Frequency Domain

Extending it to 2D

Am

plitu

dePh

ase

10

AmplitudeAmplitude

How much details?Sharper details signify higher frequenciesWill deal with this mostly

Phase

Where are the details?Though we do not use it much, it is important, especially for perception

11

Low Pass Filtering

Hierarchical Image Filtering

SPATIAL DOMAIN FREQUENCY DOMAIN

x

h1(x)

x

h2(x) *f

A1(f)

f

A2(f)x

12

Hierarchical Filtering

1/4 1/4

1/4 1/4

N x NN/2 x N/2

N/4 x N/4

1 x 1

Gaussian Pyramid

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Band-limited Images (Laplacian Pyramid)

Gn

Gn-1

Gn-2

fnfn-1fn-2

fn-2<fn-1<fn Bn= Gn-Gn-1

Bn-1= Gn-1-Gn-2

Band-limited Images (Laplacian Pyramid)

14

Edge Crispening

Second Order Edge Detection

A. The imageB. Image after

convolutionC. Segmented

convolved imageD. Edge detected

image

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Scaling Problem

Edges in coarser level do not disappear in finer levelsNew edges are addedCoarser level edges are most importantAdvances like a hierarchy