Sampling Type Operators and their Applications to Digital Image Processing...
Transcript of Sampling Type Operators and their Applications to Digital Image Processing...
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Sampling Type Operators and theirApplications to Digital Image Processing
Gianluca [email protected]
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The Classical Signal Theory:
WKS - SAMPLING THEOREM
E.T. Whittaker - V.A. Kotelnikov - C.E. Shannon (’30 - ’50)
Theorem
Let be f ∈ L2(R) (f with finite energy) such that:
supp f̂ ⊆ [−πW , πW ], W > 0; (f is band limited)
Then ∑k∈Z
f
(k
W
)· sinc(Wt − k) = f (t), for every t ∈ R.
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Univariate Generalized Sampling Operators
Let ϕ ∈ Cc(R) and f : R→ R be a bounded function. We define
(SϕW f )(x) :=∑k∈Z
f
(k
W
)ϕ(Wx − k) (x ∈ R, W > 0).
ϕ is a kernel function, such that
∑k∈Z
ϕ(x − k) = 1, for every x ∈ R.
(P.L.Butzer, S.Ries, R.L.Stens, 1987), (C.Bardaro, P.L.Butzer, R.L.Stens, G.Vinti,2010)
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Univariate Generalized Sampling Operators
SϕW are bounded linear operators mapping C (R) into itself, havingoperator norm
‖SϕW ‖[C(R),C(R)] = supu∈R
∑k∈Z|ϕ(u − k)|
For functions not necessarily continuous on R but integrable there, theabove operators are not suitable, since they depend on single functionvalues f (k/W ).
Let f ∈ Lp(R) be a function such that f (k) = 1 for all k ∈ Z.Then for W = 1 and x ∈ R, (Sϕ1 f )(x) = 1 and S
ϕ1 f 6∈ Lp(R).
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Univariate Generalized Sampling Operators in Lp setting
Λp := {f ∈ M(R) : ‖f ‖lp(Σ) < +∞, for each admissible sequence Σ},
where ‖f ‖lp(Σ) := {∑j∈Z|f (xj)|p∆j}
1p .
Σ := (xj)j∈Z is an admissible partition of R and ∆j = xj − xj−1. In case of
uniform sampling, Σw := (j
W)j∈Z
Proposition
For 1 ≤ p < +∞, there holds
‖SϕW f ‖Lp ≤ m0(ϕ)1−1/p‖ϕ‖1/p
L1‖f ‖lp(w) (f ∈ Λp; W > 0)
where m0(ϕ) := supu∈R∑
k∈Z |ϕ(u − k)|.Gianluca Vinti ([email protected]) Sampling Type Operators 5 / 51
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Univariate Generalized Sampling Operators in Lp setting
SϕW are bounded linear operators from Λp endowed with the
lp(W )-seminorm into Lp;
the operator norms are bounded by m0(ϕ)1−1/p‖ϕ‖1/p
L1, uniformly
with respect to W > 0.
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Unidimensional Sampling Kantorovich Operators
We replace the samples values f (k/w) by an average of f on a smallinterval around k/w , namely the mean
w
∫ (k+1)/wk/w
f (u)du
Practically, more information is usually known around a point thanprecisely at that point; this procedure simultaneously reduces jitter errors.
This will lead to a study of the series
(Sχw f )(x) :=∞∑
k=−∞
{w
∫ (k+1)/wk/w
f (u) du}χ(wx − k) (x ∈ R,w > 0)
(1)where f : R→ R is a locally integrable function such that the above seriesis convergent for every x ∈ R. (C.Bardaro, P.L.Butzer, R.Stens, G.Vinti, 2007)
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Multivariate Sampling Kantorovich Operators
Let (Sχw f )w>0 be the family of operators defined by
(Sχw f )(x) :=∑k∈Zn
χ(wx−tk)·
[wn
Ak·∫Rwk
f (u) du
](x ∈ Rn,w > 0),
where f : Rn → R is locally integrable functions, such that the above series
is convergent for every x ∈ Rn.
(D. Costarelli, G. Vinti, 2011), (G.Vinti, L.Zampogni, 2013), (F.Ventriglia,G.Vinti, 2013)
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Assumptions on the kernel functions
χ : Rn → R is a kernel function if satisfies the following conditions:
χ ∈ L1(Rn) is locally bounded in 0 ∈ Rn;∑k∈Zn
χ(u − tk) = 1 for every u ∈ Rn;
∃ β > 0: supu∈Rn
∑k∈Zn
∣∣χ(u − tk)∣∣ · ∥∥u − tk∥∥β2 < +∞,where Πn = (tk)k∈Zn is a sequence defined by tk = (tk1 , ..., tkn), whereeach (tki )ki∈Z, i = 1, ..., n is a sequence of real numbers with:
limki→±∞
tki = ±∞;
there exist ∆, δ > 0 for which δ ≤ ∆ki = tki+1 − tki ≤ ∆, forevery i = 1, ..., n.
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−∞ < tki < tki+1 < +∞.
Denote by ∆ki := tki+1 − tki > 0 for every i = 1, 2, ..., n,
Ak := ∆k1 ·∆k2 · ... ·∆kn > 0
Rwk are defined by
Rwk :=[ tk1
w,
tk1+1w
]×[ tk2
w,
tk2+1w
]× ...×
[ tknw,
tkn+1w
](w > 0).
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Multivariate Sampling Kantorovich Operators: Properties
There holds:
m0,Πn(χ) := supu∈Rn
∑k∈Zn
∣∣χ(u − tk)∣∣ < +∞;if f ∈ L∞(Rn), the operators Sχw f are well-defined. Indeed,
|(Sχw f )(x)| ≤ m0,Πn(χ) ‖f ‖∞ < +∞,
for every x ∈ Rn, i.e. Sχw : L∞(Rn)→ L∞(Rn).
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Main Results
Theorem
Let f : Rn → R be a bounded and continuous function. For every x ∈ Rnwe have
limw→∞
(Sχw f )(x) = f (x).
Moreover, if f is uniformly continuous and bounded, we have
limw→∞
‖Sχw f − f ‖∞ = 0.
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Orlicz Spaces
W. Orlicz (’30)
ϕ : R+0 → R+0 is a ϕ-function if satisfies the following conditions
ϕ is non decreasing and continuous for every u ≥ 0;
ϕ(0) = 0, ϕ(u) > 0 for every u > 0;
ϕ(u)→ +∞ as u →∞.
Let (Ω,Σ, µ) be a measure space.
M(Ω) := {f : Ω→ R : Σ−measurable, finite µ− a.e.}
(M.A.Krasnosel’skĭı,Ja.B.Rutickĭı, 1961), (J.Musielak, 1983), (M.M.Rao,Z.D.Ren,
1991, 2002), (C.Bardaro,J.Musielak,G.Vinti, 2003).Gianluca Vinti ([email protected]) Sampling Type Operators 13 / 51
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Let ϕ be a convex ϕ-function.
We define the functional
Iϕ[f ] :=
∫Ωϕ(|f (x)|) dµ(x) (f ∈ M(Ω)).
The Orlicz space generated by ϕ is defined by
Lϕ(Ω) := {f ∈ M(Ω) : ∃ λ > 0 s.t. Iϕ[λf ] < +∞} .
Modular Convergence
(fw )w>0 ⊂ Lϕ(Ω) is modularly convergent to f ∈ Lϕ(Ω) if
∃ λ > 0 :lim
w→∞Iϕ[λ(fw − f )] = 0.
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Examples of Orlicz Spaces
1) ϕ(u) = up (Iϕ[f ] = ‖f ‖p) ⇒ Lϕ(Rn) = Lp(Rn)
(1 ≤ p 0) Zygmund spaces
3) ϕγ(u) = euγ − 1 ⇒ Lϕγ (Rn)
(γ > 0) Exponential spaces
(P.L.Butzer,R.J.Nessel, 1971), (J.Stein, 1969), (A.Fiorenza, 1992),
(D.E.Edmunds,M.Krbec, 1995), (S.Hencl, 2003)Gianluca Vinti ([email protected]) Sampling Type Operators 15 / 51
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Multivariate Sampling Kantorovich Operators in Lϕ setting
In order to obtain a modular convergence result in Lϕ(Rn), we need amodular continuity property for the operators Sχw .
Theorem
Let ϕ be a convex ϕ-function. For every f ∈ Lϕ (Rn) there holds
Iϕ[λSχw f ] ≤‖χ‖1
δn ·m0,Πn(χ)Iϕ[λm0,Πn(χ)f ], for some λ > 0
In particular, Sχw maps Lϕ (Rn) in Lϕ (Rn).
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Multivariate Sampling Kantorovich Operators in Lp setting
In the important particular case when ϕ(u) = up for u ∈ R+0 , we haveLϕ(Rn) = Lp(Rn), 1 ≤ p
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Sampling Kantorovich Operators in Orlicz Spaces:Modular Convergence Theorem
Theorem
Let f ∈ Lϕ(Rn). Then there exists λ > 0 such that
limw→∞
Iϕ[λ(Sχw f − f )] = 0.
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Particular Cases
Modular Convergence in Lp-spaces
Digital images are multivariate discontinuous signals =⇒ f ∈ Lp(Rn)
Corollary
Let f ∈ Lp(Rn), 1 ≤ p
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Examples with Graphical Representations
Consider the function
f (x , y) =
3, −1 ≤ x ≤ 1 and − 1 ≤ y ≤ 1,
6
x2 + y 2, otherwise.
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Bivariate Fejér’s kernel
F2(x , y) := F (x) · F (y) ((x , y) ∈ R2),
where F is ”classical” (univariate) Fejér’s kernel , defined by
F (x) :=1
2sinc2
(x2
),
where the sinc-function is given by
sinc (x) :=
sinπxπx , x ∈ R \ {0} ,
1, x = 0.
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Univariate and Bivariate Fejér’s kernel
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Bivariate B-spline kernel
M23(x , y) := M3(x) ·M3(y) ((x , y) ∈ R2),
where M3 is the (univariate) B-spline of order 3, defined by
M3(x) :=
34 − x
2, |x | ≤ 12 ,
1
2
(3
2− |x |
)2,
1
2< |x | ≤ 3
2,
0, |x | > 32 ,
,
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Univariate and Bivariate B-spline kernels M3(x) and M23
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Other important examples of kernels: Jackson-type kernels
J nk (x) :=n∏
i=1
Jk(x1), x ∈ Rn,
where the univariate Jackson-type kernes are defined by:
Jk(x) = ck sinc2k( x
2kπα
), x ∈ R,
with k ∈ N, α ≥ 1, where the normalization coefficients ck are given by
ck :=
[∫R
sinc2k( u
2kπα
)du
]−1Gianluca Vinti ([email protected]) Sampling Type Operators 25 / 51
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2D Kernels - JACKSON
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Approximations of f (blue) by means of the Sampling
Kantorovich operators SF2w f (grey), for w = 5 and w = 10
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Approximations of f (blue) by means of the Sampling
Kantorovich Operators SM23w f (grey), for w = 5 and w = 10
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Comparison between f (blue) and the Sampling Kantorovich
Operators SF2w f (grey) and SM23w f (red), for w = 5 and w = 10
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Applications to Image Processing
digital image = matrix
The image A = (aij)ij ∈ Rm×m can be modeled by
I(x , y) :=m∑i=1
m∑j=1
aij · 1ij(x , y) (image function)
where
1ij(x , y) :=
1, (x , y) ∈ (i − 1, i ]× (j − 1, j ],
0, otherwise.
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We consider the following image:
Figura : ”Lena”. Original image. (150× 150 pixel).
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Approximations of ”Lena” by the bivariate SamplingKantorovich operators SF2w I, where F2 is the bivariate
Fejer’s kernel, for w = 5 and w = 10 (150× 150 pixel)
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Approximations of ”Lena” by the Sampling Kantorovichoperators S
M23w I, where M23 is the bivariate B-spline kernel
of order 3, for w = 5 and w = 10 (150× 150 pixel)
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Approximations of ”Lena” by the bivariate SamplingKantorovich operators SF2w I, where F2 is the bivariate
Fejer’s kernel, for w = 5 and w = 10 (300× 300 pixel)
Gianluca Vinti ([email protected]) Sampling Type Operators 34 / 51
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Approximations of ”Lena” by the Sampling Kantorovichoperators S
M23w I, where M23 is the bivariate B-spline kernel
of order 3, for w = 5 and w = 10 (300× 300 pixel)
Gianluca Vinti ([email protected]) Sampling Type Operators 35 / 51
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Biomedical Applications
The advent of digital has given a fundamental importance to imageprocessing and has pioneered the study of reconstruction algorithms,analysis and improvement
For example 3D static and dynamic images allow to visualize in anever more accurate the object of study allowing a diagnosisincreasingly precise
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Biomedical Applications
Digital Biomedical images play an important rule in:
clinical diagnosis;
surgery (EVAR) ;
patients follow up.
Is therefore fundamental that the contours of the images are
clearly visible, together with some others details.
Becomes essential to have at disposal algorithms for image
reconstruction and image enhancement
=⇒ Multivariate Sampling Kantorovich OperatorsGianluca Vinti ([email protected]) Sampling Type Operators 37 / 51
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Biomedical Applications
The images of our interest are in the field of Vascular Surgery and take
account of our collaboration with a group of radiologists and vascular
surgeons at the University of Perugia. In particular, the images are related
with infrarenal abdominal aortic aneurysms.
Moreover, a concrete clinical case involving a pathology of the aorta
artery will be analyzed, in order to show the importance of Digital Image
Processing (D.I.P.) techniques for improving diagnosis.
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Gianluca Vinti ([email protected]) Sampling Type Operators 39 / 51
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Reconstruction of the original image (left, 100 × 100pixel) by the Sampling Kantorovich Opertors with theB-spline kernel M23 for w = 10 (right, 400 × 400 pixel)
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A concrete clinical case
CT (computer tomography) image depicting the aorta artery delimited bya red square (Region of Interest - ROI)
(images courtesy of Santa Maria della Misericordia Hospital - Department ofSurgical and Biomedical Sciences - Section of Vascular Surgery - Perugia)
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Contrast Mediums
Pro:
becomes radiopaque the arteries
Versus:
no patient-friendly
critical in presence of kidney issuesGianluca Vinti ([email protected]) Sampling Type Operators 42 / 51
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Reconstruction of the ROI (left, 240 × 240 pixel) by theSampling Kantorovich Opertors with a Jackson-type
kernel J 24 for w = 20 (right, 480 × 480 pixel)
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Zoom of the Reconstruction of the ROI by the SamplingKantorovich Operators with a Jackson-type kernel J 24 for
w = 20
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Wavelet Decomposition of the ROI (480 × 480 pixel) afterthe application of the Sampling Kantorovich Operators
with a Jackson-type kernel J 24 for w = 20
sc=20 sc=21 sc=22
sc=23 sc=24 LP
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Enhancement of the ROI (left, 240 x 240 pixel) by theWavelet Algorithm, Equalization and Normalization
procedure (right, 480 x 480 pixel)
Original image (left) – Reconstructed image (right)
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Application of the algorithm to the original imagetogether with an edge detection algorithm - Comparison
with the equivalent reconstructed image
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Application of the algorithm to the reconstructed imagetogether with an edge detection algorithm - Comparison
with the CT image with contrast medium
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Conclusions
Multivariate Sampling Kantorovich Operators together with suitableDIP algorithms in Diagnostics of Medical Images:
improve the possibility of medical diagnose without contrast medium;
enhance the difference in granularity of ROI’s structures;
highlight the general morphology of the occlusion.
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Researchers involved in the subject
Mathematical aspects
L. Angeloni, C. Bardaro, I. Mantellini, F. Ventriglia, G. Vinti
and L. Zampogni, D. Costarelli (Perugia), P.L. Butzer,
R.L. Stens (Aachen), J. Musielak (Poznan).
Engineering aspects
F. Cluni, A.M. Minotti , M. Seracini (Perugia).
Medical aspects
E. Cieri, G. Isernia, G. Simonte (Perugia)
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Many
Thanks for Your attention
Gianluca Vinti ([email protected]) Sampling Type Operators 51 / 51