advanced transform methods
TRANSCRIPT
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Advanced Transform MethodsProfessor Sir Michael Brady FRS FREng
Department of Engineering ScienceOxford University
Hilary Term 2006
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Outline
• Recap on Fourier domain• Limitations of the Fourier transform• Gabor’s great idea• Analytic signal, local phase & features• Bases for spaces of functions• Wavelets• The Riesz transform, monogenic signal &
features again
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Recap on Fourier domain
• Frequency for signal/image processing– “Frequency freaks” vs “feature creatures”
• Fourier’s idea– Was using sine/cosine lucky or insightful?
• Efficiency: the FFT• The importance of phase• Homomorphic filtering• Image enhancement & restoration
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Frequency for signal/image analysis
+ “noise” is often high frequency– Unfortunately, so are transients such as edges, corners, …– The human visual system ignores constant signals but detects
signal changes– Filtering out high frequencies can damage edges
+ Textures often have repeated patterns that correspond to peaks in Fourier spectrum– Unfortunately, this is too simple for texture analysis in practice
+ Frequency codes for position in MRI+ Sadly, Fourier theory is often ignored in image analysis –
at cost of poorer results than necessary– Example of rotational invariance of feature detection
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Frequency freaks
Sinusoidal grating
Campbell and Robson’s contrast sensitivity function CSF
Example of a CSF: threshold contrast vs spatial frequency (cycles/degree or lp/mm)
Output from the monogenic signal
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a. Corners detected by a conventional algorithm as a shape rotates;
c. Edges detected by the Canny edge detector as the shape rotates
b. Corners detected after Fourier analysis of the filter response is analysed as a function of orientation, leading to a filter whose response is anisotropic
d. Edges after similar analysis
Source: DPhil thesis by Jason Merron, 1997
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Fourier’s idea (1807)Fourier presented a memoir to the Institut de France in which he claimed that all continuous functions of a real variable f(t) can be represented as series of harmonically related sines and cosines. The complete mathematical story was not worked out until 1960!
Definition: Let f(t) be a continuous function of a real variable t, then the Fourier Transform of f is defined to be:
∫∞
∞−
−→ℑ dtetfuf jut)( :)(
ℑNote that is an operator that transforms a function f of one variable into another:
)()(: 11 ℜ→ℜℑ LL∫ +∞<ℜ dtL f(t) :is that functions, integrable of space (Hilbert) theis )( where 1
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Inverse FT usually exists
An important result is that for most functions f and for most t the inverse transform exists too:
ff ˆby )( denote :otation ℑN
∫∞
∞−
=
ℜ∈ℜ∈
dueuftf
tLf
jut)(ˆ21)(
, allalmost for then ,)( If 1
π
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Power and phase
)(ˆ)(ˆ
)(tan
)(ˆ)(ˆ)(ˆ
where
)(ˆ)(ˆ :form lexponentiain Express
)(ˆ)(ˆ)(ˆfunction luedcomplex va a is )(ˆ that Note
22
)(
ufufu
ufufuf
eufuf
ufjufufuf
r
i
ir
uj
ir
=
+=
=
+=
φ
φ
The first of these is the square root of the power spectrum of f, the second is the phase angle of f.
Most of signal processing is concerned with the power spectrum; but (as we shall see) phase dominates in image analysis
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Decomposing an image into its spatial frequency components
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Rectangle function and sinc
)(sinc)(
)sin()(
21||,021||,1
)(
XuXXu
XuXuF
x
xxf
πππ
==
⎪⎩
⎪⎨
⎧
≥
<=
Rectangle function plays a key role in image/signal processing. In the frequency domain it defines a perfect bandpass filter, in the space domain a perfect windowing operator.
Sinc interpolation plays an important role in MRI image analysis, and geometric alignment of data sets.
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Fourier transform properties
)(ˆ)(derivativeFrequency )(ˆ)()(derivative Time
)(ˆ)( Scaling
)(ˆModulation)(ˆ)(nTranslatio
)(ˆ)(ˆ21))((nConvolutio
)(2)(ˆInverse)(ˆ)(property
0
00
0
ufjtufjutf
sufsstf
uufeufettf
ufuhtfh
uftfuftf
pp
pp
tju
ujt
−
−−
∗
−
−π
π
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Examples, and a question
( )
1)(2
sinc)(
)()(21cos
t
t
t
δ
ω
λωδλωδλ
Π
−++
A reminder from the FT course:
OK, so “all continuous functions of a real variable f(t) can be represented as series of harmonically related sines and cosines”
But was Fourier's observation just lucky? Or, is there a deeperreason why cosines and sines - complex exponentials - should play a key role?
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Linear, time invariant filtering
βαβαβα
, constantsany for )()()( iflinear isoperator An gfgf LLL L +=+
Linearity is a simplifying assumption that is made throughout much of engineering science, even if it rarely holds exactly.
))(( ))(( then),((t) if is,That .by delayed also isoutput then the
, timeaby delayed is input to theifinvariant timecalled is operator An
tftftff
f
τ
τ
τττ
τ
LL
L
=−−=
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linear, time-invariant operators: impulses and convolution
)()( denote and1011
)(
function Dirac the)(by Denote
utttt
t
t
u −=⎩⎨⎧
≠=
=
δδ
δ
δ
Linear, time-invariant systems are characterised by their response toDirac functions. Assume for the moment that the followingintegrations are defined: ∫
∞
∞−
= dttuftf u )()()( δ
dttuftf u )()()( impliesLinearity δLL ∫∞
∞−
=
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Impulse response
The impulse response h of the linear, time-invariant operator L is
)()( tth δL=
))((
)()(
)()()(
thatso ),()( that proves invariance time
tfh
duutfuh
duuthuftf
uthtu
∗=
−=
−=
−=
∫
∫∞
∞−
∞
∞−
L
Lδ
The impulse response function of a linear time-invariant operator is a convolution
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complex exponentials are eigenfunctions of convolution operators
ωfrequency at the of FT theis )()(ˆ eigenvalue The
)(ˆ
)(
)( )(
hdueuhwh
ewh
dueuhe
dueuhe
jwu
jwt
jwujwt
utjwjwt
∫
∫
∫
∞
∞−
∞
∞−
∞
∞−
−
=
=
=
=L
… so linearity impulse response convolution sines & cosines
So Fourier's insight was deep, though it may also have been lucky!
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Convolution theorem and FFTWe have seen that linear, time-invariant operators correspond precisely with those defined by convolution. That is, L is a linear, time-invariant operator if, and only if, there is a function h(t), called the impulse-response function of L so that Lf = h*f.
The convolution theorem is the core of the application of Fourier theory to signal and image processing, since the naïve shift-and-multiply implementation of convolution is intrinsically expensive, havingcomplexity O(N2) where N is the desired number of values of u.
( ) fhfh
LfhLhLf
ˆˆ*
and )( then ,)( and )( If :Theoremn Convolutio^
111
=
ℜ∈∗ℜ∈ℜ∈
Convolution in the time/space domain = multiplication in Fourier domain
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Practical importance of the convolution theorem
),(),(),(),( vuGvuFyxgyxf ⇔∗
Smoothing an image with a Gaussian kernel. Could do it in the space domain, using separability of the Gaussian, OR1. Compute the FT of the image and of the
Gaussian kernel
2. Multiply their FTs
3. Compute inverse FT of productThis only makes sense because there is a FAST FT algorithm
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The FFT
∑∑−
=
−
=
−==
1
0
1
0)(1)(1)(ˆ
N
t
utN
N
t
Njut
WtfN
etfN
tf
Assume N = 2n for some n, so that N = 2M for some M. Substituting:
⎥⎦
⎤⎢⎣
⎡++= ∑∑
−
=
−
=
uM
M
t
utM
M
t
utM WWtf
MWtf
Muf 2
1
0
1
0)12(1)2(1
21)(ˆ
So that:
[ ]uMWufufuf 2oddeven )(ˆ)(ˆ
21)(ˆ +=
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FFT algorithmGiven a signal {f(0), f(1), …., f(63)}, it is broken down in to:
• {f(0), f(2), … , f(62)} and
• {f(1), f(3), …, f(63)
That is, the FT of a signal of length 64 can be computed from two FTs of length 32. These in turn are broken down in to their odd and even bits
{f(0), f(4), …}; {f(2), f(6), …}; {f(1), f(5), …}; {f(3), f(7),..}
So it can be computed from 4 of length 16. In general, For a signal of length N, the algorithm takes time N log N instead of N2 by the naïve method.
There is nothing magical about this process, it follows from simple properties of sines and cosines. Exactly, the same idea works for wavelets
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2D FT
∫ ∫
∫ ∫∞
∞−
∞
∞−
+
∞
∞−
∞
∞−
+−
=
=
dudvevufyxf
dxdyeyxfvuf
vufyxf
vyuxj
vyuxj
)(
)(
),(ˆ),(
),(),(ˆ
definecan then we ,integrable is ),(ˆ if and ,integrable and continuous is ),( If
We can define the power and phase as for 1D FT
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FT pair in 2Dv
)()(),( YvsncXuXYsncvuF ππ=
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FT of a circular disk
ρρπρ /)()(),(||,0||,1
),(
1 aaJFvuFarar
yxf
==⎩⎨⎧
≥<
=
2D version of a sinc function
Important in ultrasound image analysis
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2D FT of an X shape
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Discrete FT
∑
∑
∑
∞
−∞=
∞
−∞=
−
∞
−∞=
−=
=
−=
±
nd
n
jnuTd
nd
Tnuf
Tuf
Tf
enTfuf
nTtnTftf
nTfnTtftf
)2(ˆ1)(ˆ
:is intervalsat uniformly samplingby obtained signal discrete theof FT The Theorem
)()(ˆ
:find we1, is delta a of FT thefact that theand ,on theorem translati theUsing
)()()(
:sum Dirac weighteda toscorrespond )( sample discreteA ).( into sampled is )(function The 0
π
δ
This shows that sampling f at intervals T is equivalent to making its Fourier transform periodic by summing all its translations.
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Discrete image filteringThe properties of 2D space-invariant operators are essentially the same as in 1D. A linear operator L is space-invariant if, for any
),(),(
then),,(),(
,
,
qmpnfmnf
qmpnfmnf
qp
qp
−−=
−−=
LL
An image can be decomposed as a sum of discrete Diracs and then linearity and space-invariance proves that any L has an associated impulse response function h(n,m) so that application of L is equivalent to a 2D convolution with h
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SeparabilityThe impulse response function h(n,m) is separable if it is a product of two one-dimensional functions:
)()(),( 21 mhnhmnh =In the case of a separable impulse response function, the two-dimensional convolution can be rewritten as a 1D convolution along the columns followed by a 1D convolution along the rows (or viceversa).
( ) ∑ ∑∞
−∞=
∞
−∞=⎟⎟⎠
⎞⎜⎜⎝
⎛−−=∗
p qqpfqmhpnhmnhf ),()()(),( 21
Reduces compute complexity to O(2N); the discrete FT can be computed by column then by row.Example: Gaussian convolution and pipeline architectures
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FT of a Gaussian
2222
2222/2
,)(),(
,2
1)(222
22
vueFvuF
yxrerf r
+===
+==
−
−
ρρπσ
σρπ
σ
• FT of a Gaussian is a Gaussian
• Note the inverse scale relation
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Power spectrum of FT of an image
f(x,y)
|F(u,v)|
original Low pass High pass
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Image with periodic structure
f(x,y) |F(u,v)|
FT has peaks at spatial frequencies of repeated textures
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Forensic application
|F(u,v)|
Remove peaks, equals periodic background
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Energy and Phase
Seems like phase carries less information about the image …
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The importance of phaseOppenheim’s demonstration
I1 I2
FFT FFT
FFT-1
I3
Do you see: I1, I2, neither, or both?
),(11
1),(),(ˆ vujevuPvuI Φ= ),(22
2),(),(ˆ vujevuPvuI Φ=
),(13
2),(),(ˆ vujevuPvuI Φ=
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Keble Another dinosaur
Keble phases the dinosaur
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Keble Another dinosaur
the dinosaur phases Keble
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Step edge detection by differentiation
inte
nsity
Signal position, x
Profile of a step intensity change superimposed on a ramp
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Basic feature detection -conventional approach
inte
nsity
Signal position, x
Profile of a step intensity change superimposed on a ramp
d in
tens
ity/ d
x
Constant gradient + max at the step. Note amplified noise
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Differentiation by the central difference is equivalent to convolution with the odd symmetric kernel
Differentiation as convolution
2)1()1()( −−+
≈=
ififdx
xdf
ix
Approximate differentiation as central difference:
⎥⎦⎤
⎢⎣⎡−
21,0,
21
Problem is that differentiation amplifies noise, suggesting pre-filtering the image with a noise reducing, smoothing filter
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Convolution with the derivative of a Gaussian
)()( xfdx
xdG∗σ
Step detection by finding “significant” maxima of the convolution of the signal with a derivative of a Gaussian
Step superimposed on a ramp signal… take second derivative
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Gaussian derivatives in 2D
),(21
2
),(2
),(),(
2222
222
2/2/3
2/)(4
yxfeex
yxfexyxfyxGx
yx
yx
∗⎟⎠
⎞⎜⎝
⎛∗⎟⎠
⎞⎜⎝
⎛ −=
∗⎟⎠⎞
⎜⎝⎛ −
=∗∂∂
−−
+−
σσ
σ
σπσπ
πσGaussian is separable: equivalent to convolution in the x-direction and then in the y-direction
Directional derivative of Gaussian shown as a surface
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Example
X-derivative Y-derivative
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Gradient magnitude
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Basic feature detection –zero crossingsin
tens
ity
Signal position, x
Profile of a step intensity change superimposed on a ramp
d2in
tens
ity/ d
x2
Zero crossing at step. But doubly amplified noise
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Gaussian scale space
This shows a Gaussian blurred signal Gσ*S(t) as σ increases (from bottom to top)
The positions of the zero-crossings of the second derivative of the blurred signal are marked. This produces a “fingerprint”
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Homomorphic filteringIllumination model: i(x,y)= illum(x,y)reflec(x,y) where i(x,y)is the image, illum(x,y) is the spatially slowly-varying illumination, and reflec(x,y) is the reflectance of visible surfaces, assumed piecewise constant.
Taking logs to convert to a sum (indicated by Capitalised names),then apply the FT:
)),(),(()),(),((),(
:FT inverse then takeand ),(filter suitable aApply ),(),(),(ˆ
^1
^1
^
^^
vuReflecvuHvuIllumvuHyxEnh
vuHvuReflecvuIllumvuI
−− ℑ+ℑ=
+=
Then the enhanced image is obtained by taking exponents of this equation.
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Homomorphic filtering: example
Left: Original image. Right: processed using homomorphic filteringto achieve contrast enhancement and dynamic range compression.
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Homomorphic filtering in practice
• The choice of the filter H(u,v) is critical– it must affect low and high frequencies in different
ways – ``illumination varies spatially slowly'' suggests low-
pass– H(u,v) is typically ``1-Butterworth''
• Note that noise was ignored …• Worse, reflectance also has a low frequency
component, so separation is often hard– ``bias field'' in MRI image enhancement
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Signal/image restoration using the Fourier domain
The objective is to restore a degraded image to its original form. An observed image can often be modelled as:
),(),(),(),( yxnyxfyxhyxg +∗=
where f is the signal/image to be restored, h is the point spread function of the imaging process, n is additive noise, and g is the signal/image with which we are confronted. The aim is to restore f given g and some knowledge of n and h. Evidently: ),(ˆ),(ˆ),(ˆ),(ˆ vunvufvuhvug +=
ˆ~
-ˆˆ~ :noise gintroducin-re or,ˆ
ˆ~
:bygiven is ˆ of ~ estimatean moment, for the ˆ Ignoring
hn
hgf
hgf
ffn
==
Evidently, problems if or is small, so noise dominates0ˆ ≈h g
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Wiener filter
signal theofdensity spectralpower theis ),(~),(
noise theofdensity spectralpower theis),(ˆ ),(
),(/),(),(
where),(),(ˆ
),(ˆ),(ˆ
where),,(ˆ),(ˆ),(~
2
2
2
*
vufvuS
vunvuS
vuSvuSvuk
vukvuhvuhvuw
vuwvugvuf
f
f
=
=
=
+=
=
η
η
constant suitable a toequalset is or ely,approximatknown are ,Often
attenuated are sfrequenciehigh then , largefor ),(ˆ),( if
filter) inverse (ie ),(ˆ/1),(~ then ,0),( If
kSS
vuvuhvuk
vuhvufvuk
f η
>>
==
![Page 52: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/52.jpg)
Wiener filter example
Here, K is set to the constant value indicated
![Page 53: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/53.jpg)
Inadequacies of the FT & attempts to overcome them
• The discrete world, signal changes, localisation
• Localisation vs SNR• Gaussian “scale space”• Isotropic and anisotropic diffusion for
image analysis• Estimating orientation in images :
steerable filters
![Page 54: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/54.jpg)
The transient world
A sine function with a small discontinuity.The Fourier Transform consists of two delta functions, indicating the single dominant signal. Note the infinite tiny wiggles –across the entire spectrum.It detects that there is a discontinuity; but cannot say where
![Page 55: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/55.jpg)
The basic problem with the FT for signal/image processing
Using the FT of the Rect function, we see that the FT of any function that is bounded in time, has frequencies over an infinite range. Conversely, the inverse FT of a perfect band-pass filter has temporal components over an infinite range.
We may detect that a transient has occurred; but we cannot detect when! If we are to treat transients in signal/image processing, we must find a way to overcome this problem. (The great idea in MRI is to add a gradient field to provide localisation; but that is infeasible for most signal and image processing.)
upperlower tttttf ≥≤= or for ,0)(
![Page 56: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/56.jpg)
Space/time & frequency
• Shannon (1943) • Fourier (1807)
– the FFT– Human visual system “Frequency freaks” vs “feature
creatures”• Gabor (1950)
– The Heisenberg map• Wavelets (1978)
– the transient world
![Page 57: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/57.jpg)
Heisenberg plane
t
ω
This is a visual representation which aims to convey the way in which sampling is done between time (space) and frequency/scale
![Page 58: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/58.jpg)
Shannon or time-sampled representation
t
ω
The signal is sampled in the time domain, and the FT of each time sample covers all frequencies
The diagram attempts to capture the idea that Shannon/Nyquist theory samples signals in time; but that a rectangle function in time has infinite extent in frequency
![Page 59: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/59.jpg)
Representation of the Fourier Transform
t
ω The signal is sampled in the frequency domain, and the inverse FT of each frequency sample is spread over all times
The practical import of the Fourier Transform* is that global structure of a signal (eg a sinusoid) is converted to a single localised point.
*This is the key property of ALL transforms: global is mapped to local, so is easier to process.
![Page 60: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/60.jpg)
Time/frequency trade-offCan we construct a function f whose energy is both well localised in time and whose FT has energy concentrated in a small frequency neighbourhood?Note that, in general, to try to reduce the time spread of f, we might scale time by some s<1 while keeping its energy constant:
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that so ,)(1)( 22
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Since the FT of the scaled function is inevitably dilated by the scale parameter, we lose in frequency what we gain in time. In short, scaling doesn’t address the problem.
Is there a way to beat this problem? If not, what is the best trade-off we can manage?
![Page 61: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/61.jpg)
Gabor’s great idea
• The windowed FT• Gabor “atoms”
– sinusoids multiplied by a Gaussian “window”• Uncertainty principle & Heisenberg map• Owens-Morrone-Kovesi theory of feature
detection– Analytic signal, Hilbert transform and
quadrature filters
![Page 62: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/62.jpg)
The windowed FT
The windowed FT:
The function g is called a “window” function – it localises the Fourier integral in the neighbourhood of t = t0.
Various windowing functions have been proposed; but the most common is the Gaussian, as used by Denis Gabor(1950)
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![Page 63: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/63.jpg)
Examples of the Windowed FT
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![Page 64: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/64.jpg)
Gabor functions
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Gabor functions are sometimes called Morlet wavelets, after the French mathematician who independently discovered them
![Page 65: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/65.jpg)
Gabor functions have uniform resolution in the
Heisenberg plane
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![Page 66: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/66.jpg)
Gabor Transform in Heisenberg space
t
ω
σξ
σtThe time and frequency samples are windowed by a Gaussian whose standard deviations are
The Heisenberg box of a Gaborfunction has a size (σt,,σξ) that is independent of the space time point (t0,ξ0)
They have constant resolution across scale space, so limited usefulness
![Page 67: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/67.jpg)
Uncertainty principle
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These modulated Gaussians are called Gabor chirps
![Page 68: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/68.jpg)
Linear chirp
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The thickness of the line is intended to indicate Gaussian window
![Page 69: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/69.jpg)
Spectogram of a more complex signal
Top: the signal
Spectogram of the signal, which can be seen to contain a linear chirp, a quadratic chirp whose frequency decreases, and two modulated Gaussian functions (the black spots)
The lower figure shows the complex phase of the WFT of the signal in regions where the modulus of the spectogram is non-zero.
t
![Page 70: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/70.jpg)
Earlier, we looked at step edge detection
However, even a moderately complex real world image has a wide variety of intensity change types
![Page 71: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/71.jpg)
Feature detection
• There’s more to images than steps and corners– ramps, thin lines, and complex composite changes, ...
• The performance of feature detectors based on idealised models of a step/ridge degrade poorly on other kinds of features to which the Human Visual System is evidently sensitive
• Feature detectors based on amplitude of the gradient (local energy) are not invariant to contrast or brightness
![Page 72: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/72.jpg)
Robyn Owens’ analysis of the Fourier components of “features”
Step up: all Fourier components have phase
zero
Ridge up and down: all Fourier components have phase π/2 or 3π/2
a feature is defined as a point where there is local phase congruency
![Page 73: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/73.jpg)
Phase congruencyLocally, a one-dimensional signal can be expressed as:
PC is amplitude-weighted, mean local
phase
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Note that, as we saw earlier, the concept of local phase depends on “windowing” the signal. It also requires that we be able to estimate the local phase at each time point t. We discuss that on the next slide
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![Page 74: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/74.jpg)
Phase “congruency”
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The ratio of the amplitude-weighted sum of phasors (the red arrow) to the sum of amplitudes (the green arrow) is a measure of “congruency”
![Page 75: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/75.jpg)
(a) original image
(b) PC feature map
(c) Canny edge strength image
(d) raw PC image
![Page 76: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/76.jpg)
Boat image. Note the faithful reconstruction of the rigging
(a) original image
(b) PC feature map
(c) Canny edge strength image
(d) raw PC image
![Page 77: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/77.jpg)
Estimating local phase• Suppose that we have “windowed” the signal at
time point t for example by localling blurring with a Gaussian
• To estimate phase, we need to calculate something like sin, cos locally – this suggests using even and odd Gabor functions
• However, Gabor filtered signals have negative frequencies and non-zero DC – these cause problems in practice
• Generally, we seek a quadrature pair of filters –pairs of filters whose Fourier transforms are rotated by π/2
![Page 78: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/78.jpg)
Estimating phase
• Define the Hilbert Transform• Hilbert transform corresponds to rotation
by π/2 in the Fourier domain – Just like sine, cos – which is why the Hilbert
Transform is key to estimating local phase• Define the Analytic Signal• Finally, show how to estimate (local)
phase
![Page 79: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/79.jpg)
Hilbert transform)( ansformFourier tr uf
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![Page 80: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/80.jpg)
Hilbert transform rotates by π/2 in the Fourier domain
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The case of interest is when θ=-π/2
Recall rotation in the plane:
![Page 81: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/81.jpg)
Analytical Signal & Hilbert Transform
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![Page 82: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/82.jpg)
Analytic signal
)()(ˆ)(ˆ Evidently, usignufufH ⋅=
Original signal FT of f
Hilbert transform of f
Analytic signal for f
![Page 83: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/83.jpg)
Analytic signal and sine/cosineThe Fourier representation of a signal f(x) can be defined as :
∫∞
+=0
)(cos()()( ωωφωω dxAxf
Now include the imaginary part:
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ωωφωωωωφωω
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Evidently, the imaginary part is a phase shifted version of the real part. Again, the imaginary part is the Hilbert Transform of f(x)
![Page 84: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/84.jpg)
Implementation of Phase Congruency (Kovesi)
Need to estimate/correct for noise and estimate frequency spread
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![Page 85: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/85.jpg)
Calculation of phase congruency via the FFT
![Page 86: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/86.jpg)
(a) original image
(b) PC feature map
(c) Canny edge strength image
(d) raw PC image
![Page 87: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/87.jpg)
(a) original image
(b) PC feature map
(c) Canny edge strength image
(d) raw PC image
![Page 88: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/88.jpg)
(a) original image
(b) PC feature map
(c) Canny edge strength image
(d) raw PC image
![Page 89: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/89.jpg)
Demonstrating invariance to contrast. Canny disappears in shadow regions
(a) original image
(b) PC feature map
(c) Canny edge strength image
(d) raw PC image
![Page 90: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/90.jpg)
PC: from signals to images
So far, we have described how to compute the phase congruency PC(x) at each point x of a signal. Now we ask how to extend the idea to images.
The immediate problem is that the Hilbert transform is only defined for signals, not for images.
![Page 91: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/91.jpg)
Phase congruency of imagesAt each image location (x,y), we can choose a direction, θ, and interpolate the signal S(θ;x,y) in direction θpassing through x,y. We can then compute the PC(θ;x,y)
Finally, we can define the phase congruency for the image point (x,y) from some quantity computed from
{PC(θi;x,y):i=1..n}
Kovesi suggested using the averageof the directional phase congruences. That is what is shown in the results presented thus far.
![Page 92: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/92.jpg)
Aerials are not steps, and are hard to detect
![Page 93: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/93.jpg)
Output from a PC implementation developed by Brady and Liang in which directional phase congruency responses are combined in a rule-based manner
![Page 94: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/94.jpg)
Phase congruency• Good news:
– responds to a set of diverse features– geometrically and photometrically invariant
• Bad news: – Being a normalised quantity, PC responds strongly to noise– PC is only meaningful if there is a spread of frequencies in a
signal; but it is not obvious how to define this spread– PC seems to require a different interpretation of scale,
suggesting that high-pass filtering rather than low-pass filtering should be used.
– PC can be defined in 1D, but, until recently it was not obvious how to calculate it in 2D, 3D, ...
![Page 95: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/95.jpg)
Noise sensitivity of phase congruency
![Page 96: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/96.jpg)
The Windowed FT seems to solve all our problems of time/space trade-off
…. or does it?
![Page 97: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/97.jpg)
Sine & discontinuity
The Fourier Transform OR THE WINDOWED FOURIER TRANSFORM consists of two delta functions, indicating the single dominant signal. The wiggles show that it detects thatthere is a discontinuity; but cannot say where
![Page 98: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/98.jpg)
Wavelets find the Sine & discontinuity
scal
e
timeThe wavelet transform indicates thatthere is a discontinuity and shows where
![Page 99: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/99.jpg)
Wavelets enable the discontinuity to be localised accurately
Note that the discontinuity appears at all scales, with higher SNR at coarse scales; better localisation at finer scales. Evidently, the feature exhibits congruency
This is a Scalogram. The vertical axis shows the scale nof analysis. The horizontal axis is time t. Dark means low values, light means high.
scal
e
time
![Page 100: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/100.jpg)
Example wavelets
db10
A wavelet is a function of effectively limited duration whose average is zero
Wavelet analysis is breaking up a given signal into shifted and scaled versions of a basic “mother” wavelet
db2 Biorthogonal2.6
Meyer
![Page 101: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/101.jpg)
Wavelet definition
A wavelet is a function with zero average
which is dilated with a “scale parameter” s, and translated by some t0.
The function shape ω(t) is called the mother wavelet
∫∞
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tts
tst−
= ψψ
![Page 102: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/102.jpg)
Wavelet transformThe wavelet transform of some signal f(t) at scale s, and at position t0 is defined from the correlation:
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![Page 103: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/103.jpg)
Continuous Wavelet Transform
∫∞
∞−
= dttpositionscaletfpositionscaleCWT ),,()(),( ψ
![Page 104: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/104.jpg)
Wavelet scaling
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a
ttf ψ
![Page 105: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/105.jpg)
Five easy steps to a CWT1. Choose a wavelet and compare it to a section at the start of a signal
2. Calculate a number C that represents how closely correlated the wavelet is with this section of the signal. More precisely, if the signal energy and wavelet energy are both equal to 1, C may be interpreted as a correlation coefficient. Note that this depends on the wavelet.
![Page 106: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/106.jpg)
Five easy steps to a CWT3. Shift the wavelet to the right and repeat steps 1, 2 until you have covered the whole signal.
4. Scale (stretch) the wavelet and repeat 1-3.
5. Repeat steps 1-4 for all desired scales
![Page 107: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/107.jpg)
CWT scalogram
![Page 108: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/108.jpg)
CWT scalogram as a surface
![Page 109: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/109.jpg)
CWT of the noisy signal shown at the top, scalogram in the second row, final approximation signal in third row, and loci of maxima of scalogram over scale at bottom.
This uses the db4 wavelet
![Page 110: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/110.jpg)
CWT of the noisy signal shown at the top, scalogramin the second row, final approximation signal in third row, and loci of maxima of scalogram over scale at bottom.This uses the Haar wavelet
![Page 111: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/111.jpg)
Scale and frequency
• Low scale a compressed wavelet rapidly changing details high frequency ω
• high scale a stretched wavelet slowly changing, coarse features low frequency ω
![Page 112: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/112.jpg)
Discrete wavelet transform
• Problem is that the CWT is a fair amount of work – it is impractical in most circumstances
• It turns out – remarkably – that if we limit ourselves to positions & scales based on powers of two – so called dyadic scales – we obtain the discrete wavelet transform (DWT)
• There is a fast DWT equivalent to the FFT• The first Fast DWT was developed by Mallat
(1987). Several have been developed since.
![Page 113: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/113.jpg)
Approximation & detail
• approximations are high scale (low frequency) components of a signal
• details are the low scale (high frequency) components
• do this successively at several scales, as defined by DWT
![Page 114: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/114.jpg)
Downsampling
Using complementary filters doubles the amount of data, so over a range of scales you drown in data.
Downsampling by 2 maintains the same amount of data at each scale. With a proper choice of filter pair, this yields the DWT.
We remember that this is the key to the FFT!
![Page 115: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/115.jpg)
Downsampling II
![Page 116: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/116.jpg)
Discrete multiresolution approximations aj[n] at scales 2j computed with cubic splines
![Page 117: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/117.jpg)
Approximation Components
Detail Components
Original Signal
![Page 118: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/118.jpg)
Discrete Wavelet Transform
We’ll see later that this can be efficiently extended to 2, 3, …dimensions
![Page 119: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/119.jpg)
DWT of the signal shown in first row. The approximation at level 5 is shown in the second row, and the five detail signals are shown in rows 3 to 7.
Here we use the Haar wavelet.
![Page 120: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/120.jpg)
DWT of the signal shown in first row. The approximation at level 5 is shown in the second row, and the five detail signals are shown in rows 3 to 7.
Here we use the db4 wavelet.
![Page 121: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/121.jpg)
Extension of the FWT to 2 dimensions
The extension to 3D, 4D is straightforward.
![Page 122: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/122.jpg)
Mallat’s depiction of the 2D FWT
![Page 123: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/123.jpg)
![Page 124: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/124.jpg)
Decomposition & reconstruction
This is the basis for efficient wavelet coding and decoding of signals, used increasingly in JPEG2000 and MPEG2000
![Page 125: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/125.jpg)
Coding fidelity
Flushing the first several D’s can reduce the coding length, without sacrificing significant information
![Page 126: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/126.jpg)
Discrete Wavelet Transform
We’ll see later that this can be efficiently extended to 2, 3, …dimensions
![Page 127: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/127.jpg)
Wavelet bases & the fast WT1. Function spaces, orthonormal bases of functions, linear
approximation & projection 2. An approach is developed to multiresolution
approximations of a signal. This uses a sequence of subspaces of .
3. We show how approximation can be summarised by a single function θ.
4. This leads to the development of a scaling function ø(t) from which everything else is derived!
5. ø(t) yields a low-pass filter h(t), from which a high-pass filter g(t) is developed. Mallat calls the pair g,h a conjugate mirror pair.
6. finally the wavelet ψ(t) is derived from ø,g,h and the scheme leads directly to the Fast Wavelet Transform.
{ }Zjj ∈
V )(2 ℜL
![Page 128: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/128.jpg)
Example of g, h
Solid line is the FT of h on [-π, π] and the dotted line is the FT of g. Note how h is approximately low pass and g is high pass. These are the filters from a cubic spline multiresolution wavelet.
![Page 129: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/129.jpg)
Function spacesThe familiar idea of a vector space can be generalised to functions.For present purposes it is possible to skip most of the mathematics(of Banach, Hilbert, and Riesz spaces).
We need to define an inner product of two functions f(t), g(t). This is usually taken to be: ∫
∞
∞−
>=< dttgtfgf )()(, *
with the resulting norm: 21
, >=< fff
00, all ,0,
,,,,,
2121
=⇔=
>≥<><+><>=+<
>>=<<
ffgfgf
gfgfgfffggf
µλµλ
![Page 130: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/130.jpg)
Orthonormal function bases{ }
1 ,0,
if lorthonorma is )( functions ofset A
=
≠>=<
= ∈
n
nm
Znn
nmt
φφφφB
nn
nnff
αφα s"coordinate"for :as expressed becan class in the function every if functions of class afor basis a is
∑=B
∑ ><=n
nnff φφ, Evidently,
{ } Znjnue ∈ is basis lorthonormaFourier The
![Page 131: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/131.jpg)
Linear approximation { }
∑−
=
∈
><=
=
1
0
,
:bygiven is basis theof functions first over the function a ofion approximatlinear A
basis.function lorthonormaan is Suppose
M
nnnM
Znn
ff
Mf
φφ
φB
Since the basis is orthonormal, the approximation error is given by the sum of the remaining squared inner products:
∑∞
=
><=−=Mn
nM fffM 22 ,)( φε
the Fourier basis yields efficient linear approximations of uniformly smooth signals, with a projection over the M lower frequency sinusoids.in a wavelet basis, the signal is projected over the M larger scale wavelets, equivalent to approximating the signal at a fixed resolution.
![Page 132: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/132.jpg)
Multiresolution approximationSuppose that the wavelet basis that we seek is:
{ } 22
21)( thatrecall weand )( ,),(, 2 ⎟⎟
⎠
⎞⎜⎜⎝
⎛=
∈ j
j
jnjZnjnjnt-tt ψψψ
Then, for any signal f(t) that we wish to analyse using the waveletbasis, we are interested in the partial sums of wavelet coefficients:
∑∞
−∞=
><n
njnjf ,,, ψψ
The jth term of this sum can be interpreted as the difference between the approximations of f(t) at the resolutions 2-(j+1) and 2-j.
Multiresolution approximations compute the approximation of thesignal f(t) as orthogonal projections on different spaces Vj. The multiresolution approximation is composed of embedded grids ofapproximation.
![Page 133: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/133.jpg)
Approximation spaces
jV1−jV
1+jV
{ }
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theall contains scale aat tion (approxima
)2()(2
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jj
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jj
jj
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VV
VV
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V
{ }0 subspace resolutionfinest for the basis a are
n)-(t ons translatiits that so )(function a is ThereV
Znt ∈θθ
![Page 134: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/134.jpg)
Examples of approximation spaces
• Piecewise constant approximations: the space Vj is the set of functions f(t) that are constant in the interval [n2j,(n+1)2j]. The approximation at resolution 2j is the closest piecewise constant function on intervals of size 2j . In this case θ(t) is the constant box function.
• Shannon approximations: the space Vj is the set of frequency band limited functions whose FT has a support included in [-2-jπ , 2-jπ ]. It turns out that the spaces can be defined by sinc functions of various frequencies, indeed, θ(t)= (sin πt)/ πt.
• Spline approximations: the provide smooth approximations with fast decay. The maths is a little more complex.
![Page 135: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/135.jpg)
Scaling function øSuppose that we have been given the function θ(t). We use it, rather we use its FT, to define the scaling function ø(t) whose dilations and translations form a basis for the sequence of approximation spaces Vj.
{ } jZnnj
jjnj
k
ntt
ku
uu
V of basis lorthonormaan is shown that becan It 22
1)(
)2(ˆ
)(ˆ)(ˆ Define
,
,
2
∈
∞
−∞=
⎟⎠⎞
⎜⎝⎛ −
=
+
=
∑
φ
φφ
πθ
θφ
For piecewise constant and Shannon approximations, ø=θ. However, this is rare …
![Page 136: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/136.jpg)
Approximation using øWith this basis, the approximation of the signal f(t) over Vj is given by:
∑∞
−∞=
><=n
njnjffPj ,,,)( φφV
Denote the inner products by aj[n] since they provide an approximation of f at the scale 2j Given the scaling function ø, the approximation can be rewritten as a convolution:
)2(
22
21)(][
, nf
dtnttfna
jnj
j
j
jj
φ
φ
∗=
⎟⎟⎠
⎞⎜⎜⎝
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∞
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j
j
tf
nau
2 intervalsat sampled ,)( of version filtered pass-low a is
][ion approximat that theso ,],[in edconcentrat typicallyis )(ˆ FT theofenergy The ππφ −
![Page 137: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/137.jpg)
Low pass filter h
{ }
)(ˆ)(ˆ2
1)2(ˆ :domainFourier In the
)(,22
1][ where
)(][22
1
:for basis a is )( Since22
1 that and )(hat Remember t
0
010
uuhu
nttnh
ntnhtnt
tt
n
Zn
φφ
φφ
φφ
φ
φφ
=
>−⎟⎠⎞
⎜⎝⎛=<
−=⎟⎠⎞
⎜⎝⎛
−
⊆∈⎟⎠⎞
⎜⎝⎛∈
∑∞
−∞=
∈ V
VVV
This relates a dilation of ø by 2 to its integer translations. The sequence h[n] can be regarded as a discrete filter.
![Page 138: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/138.jpg)
Examples of hPiecewise constant approximations (eg. Haar)
⎪⎩
⎪⎨⎧ ==
>−⎟⎠⎞
⎜⎝⎛=<=
otherwise0
1,02
1][
thatfollowsit ,)(,22
1][ sinceThen .]1,0[
nnh
nttnh φφφ 1
Polynomial cubic splines
2cos
cos212
cos212
)(ˆ)2(ˆ2)(ˆ
2
21
2
2
uu
u
uuuh
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
+
+=
=φφ
![Page 139: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/139.jpg)
Orthogonal complements
0, and
:satisfy , on to sprojection its ,function any for Then,
: of complement orthogonal
thedefine now We., scaleeach at that know We
1
1
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>=<+=
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fPfPfPfPf
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jjj
jjjjj
jj
WVWV
WVV
WVVVW
VV
![Page 140: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/140.jpg)
Wavelet bases & the fast WT1. Function spaces, orthonormal bases of functions, linear
approximation & projection 2. An approach is developed to multiresolution
approximations of a signal. This uses a sequence of subspaces of .
3. We show how approximation can be summarised by a single function θ.
4. This leads to the development of a scaling function ø(t) from which everything else is derived!
5. ø(t) yields a low-pass filter h(t), from which a high-pass filter g(t) is developed. Mallat calls the pair g,h a conjugate mirror pair.
6. finally the wavelet ψ(t) is derived from ø,g,h and the scheme leads directly to the Fast Wavelet Transform.
{ }Zjj ∈
V )(2 ℜL
![Page 141: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/141.jpg)
Low pass filter h
{ }
)(ˆ)(ˆ2
1)2(ˆ :domainFourier In the
)(,22
1][ where
)(][22
1
:for basis a is )( Since22
1 that and )(hat Remember t
0
010
uuhu
nttnh
ntnhtnt
tt
n
Zn
φφ
φφ
φφ
φ
φφ
=
>−⎟⎠⎞
⎜⎝⎛=<
−=⎟⎠⎞
⎜⎝⎛
−
⊆∈⎟⎠⎞
⎜⎝⎛∈
∑∞
−∞=
∈ V
VVV
This relates a dilation of ø by 2 to its integer translations. The sequence h[n] can be regarded as a discrete filter.
![Page 142: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/142.jpg)
High pass filter g
passlow is since pass, high is Evidently,]1[)1(][
:show thattorward"straightfo" isIt 1
hgnhng n −−= −
The high pass filter g is constructed directly from the low pass filter hfrom its FT.
)(*ˆ)(ˆ :domain Fourier thein )( define We π+= − uheugtg ju
![Page 143: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/143.jpg)
Example of g, h
Solid line is the FT of h on [-π, π] and the dotted line is the FT of g. Note how h is approximately low pass and g is high pass. These are the filters from a cubic spline multiresolution wavelet.
![Page 144: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/144.jpg)
wavelet ψ
{ } jZnnj
j
j
jnjntt
uugu
t
W of basis lorthonorma an is that proved be canIt 22
21)(
2ˆ
2ˆ
21)(ˆ
:by )( function thedefine ely)(suggestivNext
,
,
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⎟⎟⎠
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⎛ −=
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛=
ψ
ψψ
φψ
ψ
![Page 145: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/145.jpg)
High pass filter g and wavelet ψ
>−⎟⎠⎞
⎜⎝⎛=<
−=⎟⎠⎞
⎜⎝⎛ ∑
∞
−∞=
)(,22
1][
)(][22
1
][ :tscoefficien thedefinecanwe together,all thisGathering
nttng
ntngtng
n
φψ
φψ
![Page 146: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/146.jpg)
Schematic of the fast wavelet transform (Mallat)
g is the high pass filter constructed from the mother wavelet ω and the scaling function υ. h is the corresponding low-pass filter
![Page 147: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/147.jpg)
Representation of the Wavelet Transform
t
s
Ψ/s
sΨNote how the sampling in time scales with frequency: as frequency decreases, temporal sampling increases, and conversely
![Page 148: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/148.jpg)
Shannon Fourier
Gabor / STFT Wavelet
Heisenberg “boxes”
![Page 149: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/149.jpg)
Zoom Zoom Zoom
Debauchies’ wavelets: compact support and orthonormal
Debauchies’ wavelets are approximately fractal, hence their use in texture analysis
![Page 150: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/150.jpg)
Segmenting an aerial image on the basis of texture: db4
Xie and Brady, 1994
![Page 151: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/151.jpg)
Coifman’s wavelets
Almost symmetric, good frequency and space localisation, compact support, orthogonal. Excellent for picking out sharp, spatially localised features
![Page 152: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/152.jpg)
Anisotropic Diffusion
Wavelet Analysis
Gaussian Scale-Space
Calcifications & vessel are much better preserved
![Page 153: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/153.jpg)
Recall Phase Congruency• We used the Hilbert Transform to define local
phase• From that we defined phase congruency at each
point f(t) of a signal• We extended the definition to phase congruency
at each point I(x,y) of an image, by computing the directional phase congruency in a set of directions θi and then combining the results
The problems are (a) that this is intrinsically inefficient; and (b) is prone to interpolation errors in forming the directional signals.
Can we compute Phase Congruency of a 2- or 3-dimensional image without having to resort to one dimensional signal-like slices?
![Page 154: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/154.jpg)
Monogenic signalPhase congruency
[ ]
[ ])(),( from phase
Localbandpass. is )( where)(*)()(
signal thesmooth practice, Inamplitude phase, local Define
)(),( signal Analytic)( ansformHilbert tr
tftf
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from then),,(is image smoothed bandpass theIf
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21
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yxIhyxIhyxIyxI
yxhyxh
vuHvuH
bbb
b
∗∗
![Page 155: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/155.jpg)
Monogenic signal
( )
( )
sinea is
and domain, Fourier thein cosinea is that Note
2),(ly equivalent,),(
2),(ly equivalent ,),( :where
),(),(
),(
22
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23
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yyxhvu
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uvuH
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vuM
+
+
+=
+=
+=
+=
⎥⎦
⎤⎢⎣
⎡=
π
π
![Page 156: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/156.jpg)
Computing the monogenic signal),( Image yxI
),(),(),(filter bandpass
yxbyxIyxI b ∗=
),(),(1 yxIyxh b∗ ),(),(2 yxIyxh b∗
phase local )()(),(tan
2
1
b
b
IhIhyx
∗∗
=θ
),(),(),( yxbyxIyxI b ∗=
energyor amplitude local )()(),( 2
22
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bbb IhIhIyxE ∗+∗+=
image thein norientatio local )()(
),(tan2
22
1 bb IhIhIyx
∗+∗=φ
![Page 157: Advanced Transform Methods](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c64ccf237e1331c514972/html5/thumbnails/157.jpg)
Monogenic triple of filters
Bandpass filter is a difference of Gaussians
h1*B h2*B
Though this generally works well, the difference of Gaussians is not the only bandpass filter that can be used …
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Mammogram segmentation
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Local phase from log Gabor filters and monogenic signal
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orientation
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Local energy, with breast edge marked
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Phase < 90 degrees
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Motion at points in the heart
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Displacement errors for two successive ultrasound images of the heart, prior to registration (left) and post registration (right). Near “features”, the error is less than a pixel.
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Affine invariance
An image with the interesting points added: blue = high symmetry; red = high antisymmetry
Affine transformed house image with interest points superimposed (circles) and matched (stars).
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Local symmetry
Points of local symmetry Locally “interesting” points
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