Introduction to Image Processing #5/7 · Thierry Geraud´ Introduction to Image Processing #5/7....
Transcript of Introduction to Image Processing #5/7 · Thierry Geraud´ Introduction to Image Processing #5/7....
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Outline
Introduction to Image Processing #5/7
Thierry Geraud
EPITA Research and Development Laboratory (LRDE)
2006Thierry G eraud Introduction to Image Processing #5/7
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Outline
Outline
1 Introduction
2 DistributionsAbout the Dirac Delta FunctionSome Useful Functions and Distributions
3 Fourier and Convolution
4 Sampling
5 Convolution and Linear Filtering
6 Some 2D Linear FiltersGradientsLaplacian
Thierry G eraud Introduction to Image Processing #5/7
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Outline
Outline
1 Introduction
2 DistributionsAbout the Dirac Delta FunctionSome Useful Functions and Distributions
3 Fourier and Convolution
4 Sampling
5 Convolution and Linear Filtering
6 Some 2D Linear FiltersGradientsLaplacian
Thierry G eraud Introduction to Image Processing #5/7
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Outline
Outline
1 Introduction
2 DistributionsAbout the Dirac Delta FunctionSome Useful Functions and Distributions
3 Fourier and Convolution
4 Sampling
5 Convolution and Linear Filtering
6 Some 2D Linear FiltersGradientsLaplacian
Thierry G eraud Introduction to Image Processing #5/7
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Outline
Outline
1 Introduction
2 DistributionsAbout the Dirac Delta FunctionSome Useful Functions and Distributions
3 Fourier and Convolution
4 Sampling
5 Convolution and Linear Filtering
6 Some 2D Linear FiltersGradientsLaplacian
Thierry G eraud Introduction to Image Processing #5/7
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Outline
Outline
1 Introduction
2 DistributionsAbout the Dirac Delta FunctionSome Useful Functions and Distributions
3 Fourier and Convolution
4 Sampling
5 Convolution and Linear Filtering
6 Some 2D Linear FiltersGradientsLaplacian
Thierry G eraud Introduction to Image Processing #5/7
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Outline
Outline
1 Introduction
2 DistributionsAbout the Dirac Delta FunctionSome Useful Functions and Distributions
3 Fourier and Convolution
4 Sampling
5 Convolution and Linear Filtering
6 Some 2D Linear FiltersGradientsLaplacian
Thierry G eraud Introduction to Image Processing #5/7
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IntroductionDistributions
Fourier and ConvolutionSampling
Convolution and Linear FilteringSome 2D Linear Filters
Filtering
Filtering images
is a low-level process
can be linear or not (!)
is often useful
either to get “better” datae.g., with enhanced contrast, less noise, etc.
or to transform data to make it suitable for furtherprocessing
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Convolution and Linear FilteringSome 2D Linear Filters
New Approach
An image is a function.
We have
sampling : image values are only known at given pointsIp with p = (r , c) ∈ N2
quantization : image values belong to a restricted setfor instance [0, 255] for a gray-level with 8 bit encoding
An image is a digital signal(contrary: analog).
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IntroductionDistributions
Fourier and ConvolutionSampling
Convolution and Linear FilteringSome 2D Linear Filters
Background
This lecture background is
digital signal processing.
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IntroductionDistributions
Fourier and ConvolutionSampling
Convolution and Linear FilteringSome 2D Linear Filters
About the Dirac Delta FunctionSome Useful Functions and Distributions
Outline
1 Introduction
2 DistributionsAbout the Dirac Delta FunctionSome Useful Functions and Distributions
3 Fourier and Convolution
4 Sampling
5 Convolution and Linear Filtering
6 Some 2D Linear FiltersGradientsLaplacian
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IntroductionDistributions
Fourier and ConvolutionSampling
Convolution and Linear FilteringSome 2D Linear Filters
About the Dirac Delta FunctionSome Useful Functions and Distributions
Dirac Delta Function (1/2)
The Dirac delta function, denoted by ↑, is defined by:∫ ∞
−∞s(t) ↑(t) dt = s(0)
where s is a test function.
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IntroductionDistributions
Fourier and ConvolutionSampling
Convolution and Linear FilteringSome 2D Linear Filters
About the Dirac Delta FunctionSome Useful Functions and Distributions
Dirac Delta Function (2/2)
Please note that:
↑ is not a function,
it is a distribution (or generalized function).
We have: ∫ ∞
−∞↑(t) dt = 1.
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IntroductionDistributions
Fourier and ConvolutionSampling
Convolution and Linear FilteringSome 2D Linear Filters
About the Dirac Delta FunctionSome Useful Functions and Distributions
Weird Definition (1/2)
Just think of ↑ being something like:
↑(t) =
{1 × ∞ if t = 00 if t 6= 0.
but it cannot be a proper definition!
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IntroductionDistributions
Fourier and ConvolutionSampling
Convolution and Linear FilteringSome 2D Linear Filters
About the Dirac Delta FunctionSome Useful Functions and Distributions
Weird Definition (2/2)
Here α is a constant in R or C.
We do not have α↑ = ↑, but:
↑(t) =
{α × ∞ if t = 00 if t 6= 0.
Indeed we should have∫∞−∞ α↑(t)dt being equal to α.
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Fourier and ConvolutionSampling
Convolution and Linear FilteringSome 2D Linear Filters
About the Dirac Delta FunctionSome Useful Functions and Distributions
Dirac Representation
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IntroductionDistributions
Fourier and ConvolutionSampling
Convolution and Linear FilteringSome 2D Linear Filters
About the Dirac Delta FunctionSome Useful Functions and Distributions
Outline
1 Introduction
2 DistributionsAbout the Dirac Delta FunctionSome Useful Functions and Distributions
3 Fourier and Convolution
4 Sampling
5 Convolution and Linear Filtering
6 Some 2D Linear FiltersGradientsLaplacian
Thierry G eraud Introduction to Image Processing #5/7
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IntroductionDistributions
Fourier and ConvolutionSampling
Convolution and Linear FilteringSome 2D Linear Filters
About the Dirac Delta FunctionSome Useful Functions and Distributions
Foreword
Understand that the Dirac delta function is the most importantdistribution we can think of.
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Fourier and ConvolutionSampling
Convolution and Linear FilteringSome 2D Linear Filters
About the Dirac Delta FunctionSome Useful Functions and Distributions
Sinc (1/2)
The (unnormalized, historical, mathematical) sinc function is:
sinc(t) =sin(t)
t.
The normalized sinc function is:
sinc(t) =sin(πt)
πt.
so that: ∫ ∞
−∞sinc(t) dt = 1.
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Fourier and ConvolutionSampling
Convolution and Linear FilteringSome 2D Linear Filters
About the Dirac Delta FunctionSome Useful Functions and Distributions
Sinc (2/2)
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IntroductionDistributions
Fourier and ConvolutionSampling
Convolution and Linear FilteringSome 2D Linear Filters
About the Dirac Delta FunctionSome Useful Functions and Distributions
Sign Function (1/2)
The sign function is:
sgm(t) =
−1 if t < 00 if t = 01 if t > 0.
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IntroductionDistributions
Fourier and ConvolutionSampling
Convolution and Linear FilteringSome 2D Linear Filters
About the Dirac Delta FunctionSome Useful Functions and Distributions
Sign Function (2/2)
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About the Dirac Delta FunctionSome Useful Functions and Distributions
Heaviside Step Function (1/2)
u(t) =12(1 + sgn(t)) =
0 if t < 01/2 if t = 01 if t > 0.
We have:
u(t) =
∫ t
−∞↑(τ) dτ.
Put differently u = ↑.
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Fourier and ConvolutionSampling
Convolution and Linear FilteringSome 2D Linear Filters
About the Dirac Delta FunctionSome Useful Functions and Distributions
Heaviside Step Function (2/2)
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About the Dirac Delta FunctionSome Useful Functions and Distributions
Rect (1/2)
The rectangular function is:
rect(t) =
1 if |t | < 1/21/2 if |t | = 1/20 if |t | > 1/2.
We have:
rect(t) = u(t + 1/2) u(1/2− t).
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About the Dirac Delta FunctionSome Useful Functions and Distributions
Rect (2/2)
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About the Dirac Delta FunctionSome Useful Functions and Distributions
Tri (1/2)
The triangular function is:
tri(t) =
{1− t if |t | < 10 if |t | ≥ 1.
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About the Dirac Delta FunctionSome Useful Functions and Distributions
Tri (2/2)
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IntroductionDistributions
Fourier and ConvolutionSampling
Convolution and Linear FilteringSome 2D Linear Filters
About the Dirac Delta FunctionSome Useful Functions and Distributions
Dirac Delta Function as a Limit
We can define ↑ as a limit of functions dα in the sense that:
limα→0
∫ ∞
−∞s(t) dα(t) dt = s(0).
We can choose dα in:t → N (0;α)(t) normal distributiont → rect(t/α)/α rectangular functiont → tri(t/α)/α triangular function... ...
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IntroductionDistributions
Fourier and ConvolutionSampling
Convolution and Linear FilteringSome 2D Linear Filters
About the Dirac Delta FunctionSome Useful Functions and Distributions
Dirac Comb (1/2)
The Dirac comb is:
↑↑T (t) =∞∑
k=−∞↑(t − kT ).
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Dirac Comb (2/2)
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URLs (1/2)
Distributionhttp://en.wikipedia.org/wiki/Distribution_(mathematics)
Dirac deltahttp://en.wikipedia.org/wiki/Dirac_delta_function
Dirac combhttp://en.wikipedia.org/wiki/Dirac_comb
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URLs (2/2)
Signhttp://en.wikipedia.org/wiki/Sign_function
Sinchttp://en.wikipedia.org/wiki/Sinc_function
Trihttp://en.wikipedia.org/wiki/Triangularfunction
Recthttp://en.wikipedia.org/wiki/Rectangular_function
Heaviside stephttp://en.wikipedia.org/wiki/Heaviside_step_function
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Fourier Series
You know that:
s(x) =12
a0 +∞∑
n=1
(an cos(nx) + bn sin(nx))
=∞∑
n=−∞Sn einx .
its generalization is...
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Discrete Fourier Transform
...the discrete Fourier transform of a discrete function s:
sk =N−1∑n=0
an e−i2πnk/N
where an =1N
N−1∑k=0
fk ei2πnk/N
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Fourier Transform
In the continuous case, S is the Fourier transform of s:
S(f ) =
∫ ∞
−∞s(t) e−i2πft dt
s(t) =
∫ ∞
−∞S(f ) ei2πft df .
where f is a frequency.
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Notations
We will denote with capital letters the Fourier transforms.
Considering that F is the Fourier operator on the set ofcomplex-valued functions:
S = F(s).
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Some Nice Properties
Parseval’s theorem:∫ ∞
−∞|s(t)|2 dt =
∫ ∞
−∞|S(f )|2 df .
F2(s)(t) = s(−t)
F∗ = F−1
Amazing:g(t) = αe−(t−β)2/2
is an eigenvalue of F .Thierry G eraud Introduction to Image Processing #5/7
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Convolution (1/2)
The convolution of two functions h and s is the function:
s′(t) = h(t) ∗ s(t) =
∫ ∞
−∞h(τ) s(t − τ) dτ.
The convolution operator is denoted by “∗”.
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Convolution (2/2)
FIXME: figure.
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Exercise
Show that:
rect ∗ rect = tri .
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Convolution Properties
We have:
commutativity a ∗ b = b ∗ aassociativity a ∗ (b ∗ c) = (a ∗ b) ∗ cdistributivity a ∗ (b + c) = (a ∗ b) + (a ∗ c)associativity with scalar α(b ∗ c) = (αb) ∗ c = b ∗ (αc)
Differentiation rule:
D(a ∗ b) = D(a) ∗ b = a ∗ D(b).
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Dirac and Convolution (1/2)
We have:
(↑ ∗ s) (t) =
∫ ∞
−∞↑(τ) s(t − τ) dτ = s(t) ∀t .
So ↑ ∗ s = s:
↑ is the neutral element for the ∗ operator.
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Let us note by:↑t ′(t) = ↑(t − t ′)
the translation to t ′ of the Dirac delta functionput differently it is a Dirac impulse centered at t ′
We have:
(↑t ′ ∗ s) (t) =∫∞−∞ ↑(τ − t ′) s(t − τ) dτ
= s(t − t ′).
Convolving a function with a Dirac delta function centered at t ′
means shifting this function at the value t = t ′.
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Convolution and Fourier = a Theorem
The convolution theorem is:
F(a ∗ b) = F(a)F(b)
We also have:
F(a b) = F(a) ∗ F(b)
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∫ ∞
−∞1(t) e−i2πft dt = ↑(f ).
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Some Fourier Transforms (1/2)
Frect(αt) 1
|α|sinc( fα)
sinc(αt) 1|α| rect( f
α)
sinc2(αt) 1|α| tri(
fα)
tri(αt) 1|α|sinc2( f
α)
1(t) ↑(f )↑(t) 1(f )cos(αt) 1/2(↑(f − α
2π ) + ↑(f + α2π )
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A remarkable Fourier transform:
F(↑↑T )(t) =1T
∞∑k=−∞
δ(f − k/T ) =1T↑↑1/T (f ).
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URLs
Fourier serieshttp://en.wikipedia.org/wiki/Fourier_series
Discrete Fourier transformhttp://en.wikipedia.org/wiki/Discrete_fourier_transform
Fourier transformhttp://en.wikipedia.org/wiki/Fourier_transform
Parseval’s theoremhttp://en.wikipedia.org/wiki/Parseval’s_theorem
Convolutionhttp://en.wikipedia.org/wiki/Convolution
Convolution theoremhttp://en.wikipedia.org/wiki/Convolution_theorem
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Analog Function
Consider an analog function:
s :
{R → Rt 7→ s(t)
FIXME: figure.
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From Analog to Digital (1/2)
A discrete function sd is sampled from s with the samplingfrequency fd = 1/T :
sd(t) =∑∞
k=−∞ s(kT ) ↑(t − kT )= s(t) × ↑↑T (t).
So:Sd(f ) ∝ S(f ) ∗ ↑↑fd (f )
∝ S(f ) ∗∑∞
k=−∞ ↑(f − kfd)∝
∑∞k=−∞ S(f ) ∗ ↑(f − kfd)
∝∑∞
k=−∞ S(f − kfd)
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FIXME: figure.
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From Digital to Analog (1/2)
An analog function sa is reconstructed from the digital one sd :
Sa(f ) = Sd(f ) × rect(f
2fd).
So:
sa(t) ∝ sd(t) ∗ sinc(t/T )∝
(∑∞k=−∞ s(kT ) ↑(t − kT )
)∗ sinc(t/T )
∝∑∞
k=−∞ ( s(kT ) ↑(t − kT ) ∗ sinc(t/T ) )
∝∑∞
k=−∞
(s(kT ) sinc( t−kT
T ))
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FIXME: figure.
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Shanon Sampling Theorem (1/2)
The minimum sampling frequency to be able to perfectlyreconstruct an analog signal is twice the maximum signal
frequency.
So we should have:fd > 2 fmax.
Practically this condition is (usually) never satisfied.
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FIXME: figure.
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Aliasing
The right image is anti-aliased:
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An image with aliasing presence (left) and a detail (right):
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The original Fourier spectrum (left) and after removing foldedpeaks (right):
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The result (right) compared to the original (left):
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URLs
Signal processinghttp://en.wikipedia.org/wiki/Signal_processing
Samplinghttp://en.wikipedia.org/wiki/Sampling_(signal_processing)
Shannon’s sampling theoremhttp:
//en.wikipedia.org/wiki/Nyquist-Shannon_sampling_theorem
Aliasinghttp://en.wikipedia.org/wiki/Aliasing
Anti-aliasinghttp://en.wikipedia.org/wiki/Anti-aliasing_filter
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Discrete Convolution
The convolution between a and b:
s′(t) = h(t) ∗ s(t) =
∫ ∞
−∞h(τ) s(t − τ) dτ.
can be turned into a discrete convolution:
s′[t ] = (h ∗ s)[t ] =∞∑
τ=−∞h[τ ] s[t − τ ] =
∞∑τ=−∞
h[t − τ ] s[τ ]
where t and τ are now ∈ Z.
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s′[r ][c] = (h ∗ s)[r ][c]=
∑∞r ′=−∞
∑∞c′=−∞ h[r ′][c′] s[r − r ′][c − c′]
=∑∞
r ′=−∞∑∞
c′=−∞ h[r − r ′][c − c′] s[r ′][c′]
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FIXME: figure.
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Linear Filters (1/2)
A filter φ is linear if
φ(α s1 + β s2) = α φ(s1) + β φ(s2)
for all α and β scalars, and s1 and S2 functions.
φ is a linear filter ⇔ hφ exists such as φ = hφ ∗
Put differently:
we can write φ(s) = hφ ∗ s
convolutions are the only linear filters.
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Linear Filters (2/2)
Consider that a filter φ is a black box.
If this filter is linear, we want to know hφ.
When you input the Dirac delta function (an impulse) into theblack box, the resulting function (signal) is:
hφ ∗ ↑ = hφ.
hφ is the impulse response of φ.
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Dirac Delta Function
Before all:
↑[r ][c] =
{1 if r = 0 and c = 00 otherwise.
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Outline
1 Introduction
2 DistributionsAbout the Dirac Delta FunctionSome Useful Functions and Distributions
3 Fourier and Convolution
4 Sampling
5 Convolution and Linear Filtering
6 Some 2D Linear FiltersGradientsLaplacian
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Gradients as Linear Filters (1/3)
Consider the gradient of a 2D function s:
∇s =
δsδx
δsδy
The gradient is linear; with φ∇(s) = ∇s , we have:
φ∇(αs1 + βs2) = α φ∇(s1) + β φ∇(s2)
...so it can be expressed with convolutions!
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Assuming that sampling is isotropic (Tx = Ty = 1), somediscrete approximations of the gradient are:
∇s [r ][c] ≈ ∇antes [r ][c] =
(s[r ][c] − s[r ][c − 1]s[r ][c] − s[r − 1][c]
)or:
∇s [r ][c] ≈ ∇posts [r ][c] =
(s[r ][c + 1] − s[r ][c]s[r + 1][c] − s[r ][c]
)and even:
∇s [r ][c] ≈ ∇antes [r ][c] + ∇posts [r ][c]
2=
(s[r ][c+1]− s[r ][c−1]
2s[r+1][c]− s[r−1][c]
2
)
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Gradients as Linear Filters (3/3)
With:
∇s =(
h/x∇ ∗ sh/y
∇ ∗ s)
and considering the “post” version:
∇/x s [r ][c] ≈ s[r ][c + 1] − s[r ][c]
≈ h/x [0][−1] s[r − 0][c − (−1)]
+ h/x [0][0] s[r − 0][c − 0]
we have:
h/x [r ][c] =
1 if r = 0 and c = −1
−1 if r = 0 and c = 00 otherwise
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Gradients Illustrated (1/2)
LENA (left) and the x gradient (right):
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Gradients Illustrated (1/2)
crops of resp. the x gradient (left) and the y gradient (right):
Please note that, to better view images, contrast is enhanced and values are inverted
(the lowest values are now the brightest).
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Graphical Representation
To depict functions we use a graphical representation:
h/x =
0 0 01 -1 00 0 0
In such representations, the origin is always centered and wedo not represent null values that lay outside the window.
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Gradient Magnitude (1/2)
The magnitude is often approximated with a L1 norm, so:
|∇s | = | δsδx| + | δs
δy|.
For instance with the “post” version:
|∇posts | [r ][c] = | s[r +1][c] − s[r ][c] | + | s[r ][c+1] − s[r ][c] |
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Gradient Magnitude (2/2)
Warning: magnitude is also video inverted here.
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Other Versions of Gradient Magnitude (1/3)
Many versions of the gradient magnitude exist... for instancethis one (the Roberts filter):
|∇robertss | [r ][c] = | s[r+1][c+1]− s[r ][c] | + | s[r ][c+1]− s[r+1][c] |
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Other Versions of Gradient Magnitude (2/3)
A noise-”insensitive” version of the gradient magnitude is theSobel filter:
h/xSobel = 1
4
-1 0 1-2 0 2-1 0 1
Exercise: explain why it is less sensitive to noise.
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IntroductionDistributions
Fourier and ConvolutionSampling
Convolution and Linear FilteringSome 2D Linear Filters
GradientsLaplacian
Other Versions of Gradient Magnitude (3/3)
Result of the Sobel filter:
Warning: magnitude is also video inverted here.
Exercise: explain why contours/edges look thicker here than inprevious examples. Thierry G eraud Introduction to Image Processing #5/7
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IntroductionDistributions
Fourier and ConvolutionSampling
Convolution and Linear FilteringSome 2D Linear Filters
GradientsLaplacian
Extracting Object Contours (1/2)
Extracting contours/edges can be performed thru thresholdingthe gradient magnitude:
Exercise: is it great?
Thierry G eraud Introduction to Image Processing #5/7
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IntroductionDistributions
Fourier and ConvolutionSampling
Convolution and Linear FilteringSome 2D Linear Filters
GradientsLaplacian
Extracting Object Contours (2/2)
Full size:
Exercise: is it great?Thierry G eraud Introduction to Image Processing #5/7
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IntroductionDistributions
Fourier and ConvolutionSampling
Convolution and Linear FilteringSome 2D Linear Filters
GradientsLaplacian
Outline
1 Introduction
2 DistributionsAbout the Dirac Delta FunctionSome Useful Functions and Distributions
3 Fourier and Convolution
4 Sampling
5 Convolution and Linear Filtering
6 Some 2D Linear FiltersGradientsLaplacian
Thierry G eraud Introduction to Image Processing #5/7
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IntroductionDistributions
Fourier and ConvolutionSampling
Convolution and Linear FilteringSome 2D Linear Filters
GradientsLaplacian
Definition
The Laplacian of s is:
∆s =δ2sδx2 +
δ2sδy2
Exercise: express h∆.
Thierry G eraud Introduction to Image Processing #5/7
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IntroductionDistributions
Fourier and ConvolutionSampling
Convolution and Linear FilteringSome 2D Linear Filters
GradientsLaplacian
Solution
You should end up with:
h∆ =
0 -1 0-1 4 -10 -1 0
Thierry G eraud Introduction to Image Processing #5/7
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IntroductionDistributions
Fourier and ConvolutionSampling
Convolution and Linear FilteringSome 2D Linear Filters
GradientsLaplacian
Filtering (1/2)
Result:
Thierry G eraud Introduction to Image Processing #5/7
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IntroductionDistributions
Fourier and ConvolutionSampling
Convolution and Linear FilteringSome 2D Linear Filters
GradientsLaplacian
Filtering (2/2)
Crop:
Exercise:
find how to sharpen image contours/edges,
express hsharpen in function of a strength parameter.
Thierry G eraud Introduction to Image Processing #5/7
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IntroductionDistributions
Fourier and ConvolutionSampling
Convolution and Linear FilteringSome 2D Linear Filters
GradientsLaplacian
Edge Sharpening (1/2)
Contour/edge sharpening results:
Thierry G eraud Introduction to Image Processing #5/7
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IntroductionDistributions
Fourier and ConvolutionSampling
Convolution and Linear FilteringSome 2D Linear Filters
GradientsLaplacian
Edge Sharpening (2/2)
Contour/edge sharpening results:
Thierry G eraud Introduction to Image Processing #5/7