fourier transform – chapter 13. image space cameras (regardless of wave lengths) create images in...
TRANSCRIPT
Image space
• Cameras (regardless of wave lengths) create images in the spatial domain
• Pixels represent features (intensity, reflectance, heat, etc.) from the “real 3D world”
Image space
• All operations we’ve looked at so far are applied in the spatial domain– Histogram (statistical) operations
– Point operations
– Filter (convolution) operations
– Edge operations
– Corner (feature) operations
– Line (curve) detection
– Morphological operations
– Region operations
– Color space
Frequency domain
• As it turns out, all the spatial domain “signals” can be represented in the frequency domain
• And, some of the previously mentioned operations can be performed more efficiently in the frequency domain– Convolution– Filtering – Compression
The Fourier Transform
• The Fourier Transform provides the means from moving between the spatial and frequency domains
• Developed in the area of sound processing– Decomposition of sound waves into
elementary harmonic functions
Sine and Cosine functions• You’ve seen these many times before
)cos()( xxf )2cos()2cos()cos()( kxxxxf
Sine and Cosine functions• Frequency – cycles on horizontal axis
)3cos()( xxf
frequencyangularperiodTT 2
Sine and Cosine functions• Orthogonality
– We can combine sine and cosine waveforms with varying frequency, amplitude, and phase parameters to create other sine and cosine waveforms
)cos()sin()cos( xCxBxA
A
BC BA tan
122
Sine and Cosine functions• Vector representation
• Complex number representation
• Euler notation (complex numbers on the unit circle)
biaz
71828.2)sin()cos( eiz ei
A
B C
)cos( x
)sin( x )cos()sin( xBxA
vector length ≡ amplitude
Euler notation• Euler notation
• This brings us to the “complex-valued sinusoid”
• Since it’s on a unit circle the amplitude is
• And the phase is
• That is, multiplying by a real value alters the amplitude, multiplying by a complex value alters the phase
71828.2)sin()cos( eiz ei
)sin(}Im{
)cos(}Re{
eei
i
aaaso eeeiii
1
eeeiii )(
Fourier Series
• Not only can sinusoidal functions [of varying frequency, amplitude, and phase] be combined to create other sinusoidal functions but…
• They can be combined to create almost and periodic function
frequencylfundamenta
xkxkxgk
kk BA
0
000)sin()cos()(
Fourier Series
• Frequencies kω0x are harmonics (multiples) of the fundamental
• Ak and Bk are derived via Fourier Analysis
frequencylfundamenta
xkxkxgk
kk BA
0
000)sin()cos()(
Fourier Integral
• But that wasn’t enough…Fourier wanted to cover non-periodic functions too
• Requires more than just integer multiples (harmonics) of the fundamental frequency
• Requires infinitely many frequencies
dxxxg BA )sin()cos()(0
Fourier Integral
• To solve for the amplitudes Aω and Bω we need the following integrals
• Aω and Bω form continuous functions of coefficients (corresponding to infinitely many, densely spaced frequencies)
• Aω and Bω form the Fourier Spectrum
dxxxgB
dxxxgA
B
A
)sin()(1
)(
)cos()(1
)(
Fourier Transform
• Apply the Fourier Series to complex-valued functions using Euler’s notation to get the Fourier Transform
• And the inverse Fourier Transform
dxxgG exi
)(
2
1)(
dxxGg exi
)(
2
1)(
Fourier Transform
• In general– A real-valued function yields a complex-
valued Fourier Transform– A complex-valued function yields a read-
valued Fourier Transform
Fourier Transform
• Properties– There are a bunch of properties that you
can read about, but only one is “surprising”
• Convolution Property– Convolution in the spatial domain is point-
by-point multiplication in the frequency domain
Convolution Property
• Spatial domain– Perform a slide/multiply-accumulate
operation with a kernel and the image
• Frequency domain– Fourier Transform kernel → spectrum– Fourier Transform image → spectrum– Multiply the two spectrums– Inverse Fourier Transform product →
filtered image