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)
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
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
13
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