Sampling Type Operators and their Applications to Digital Image Processing...

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Sampling Type Operators and their Applications to Digital Image Processing Gianluca Vinti [email protected]

Transcript of Sampling Type Operators and their Applications to Digital Image Processing...

  • Sampling Type Operators and theirApplications to Digital Image Processing

    Gianluca [email protected]

  • 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.

    Gianluca Vinti ([email protected]) Sampling Type Operators 2 / 51

  • 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)

    Gianluca Vinti ([email protected]) Sampling Type Operators 3 / 51

  • 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).

    Gianluca Vinti ([email protected]) Sampling Type Operators 4 / 51

  • 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

  • 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.

    Gianluca Vinti ([email protected]) Sampling Type Operators 6 / 51

  • 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)

    Gianluca Vinti ([email protected]) Sampling Type Operators 7 / 51

  • 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)

    Gianluca Vinti ([email protected]) Sampling Type Operators 8 / 51

  • 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.

    Gianluca Vinti ([email protected]) Sampling Type Operators 9 / 51

  • −∞ < 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).

    Gianluca Vinti ([email protected]) Sampling Type Operators 10 / 51

  • 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).

    Gianluca Vinti ([email protected]) Sampling Type Operators 11 / 51

  • 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.

    Gianluca Vinti ([email protected]) Sampling Type Operators 12 / 51

  • 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

  • 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.

    Gianluca Vinti ([email protected]) Sampling Type Operators 14 / 51

  • 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

  • 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).

    Gianluca Vinti ([email protected]) Sampling Type Operators 16 / 51

  • 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

  • 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.

    Gianluca Vinti ([email protected]) Sampling Type Operators 18 / 51

  • Particular Cases

    Modular Convergence in Lp-spaces

    Digital images are multivariate discontinuous signals =⇒ f ∈ Lp(Rn)

    Corollary

    Let f ∈ Lp(Rn), 1 ≤ p

  • Examples with Graphical Representations

    Consider the function

    f (x , y) =

    3, −1 ≤ x ≤ 1 and − 1 ≤ y ≤ 1,

    6

    x2 + y 2, otherwise.

    Gianluca Vinti ([email protected]) Sampling Type Operators 20 / 51

  • 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.

    Gianluca Vinti ([email protected]) Sampling Type Operators 21 / 51

  • Univariate and Bivariate Fejér’s kernel

    Gianluca Vinti ([email protected]) Sampling Type Operators 22 / 51

  • 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 ,

    ,

    Gianluca Vinti ([email protected]) Sampling Type Operators 23 / 51

  • Univariate and Bivariate B-spline kernels M3(x) and M23

    Gianluca Vinti ([email protected]) Sampling Type Operators 24 / 51

  • 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

  • 2D Kernels - JACKSON

    Gianluca Vinti ([email protected]) Sampling Type Operators 26 / 51

  • Approximations of f (blue) by means of the Sampling

    Kantorovich operators SF2w f (grey), for w = 5 and w = 10

    Gianluca Vinti ([email protected]) Sampling Type Operators 27 / 51

  • Approximations of f (blue) by means of the Sampling

    Kantorovich Operators SM23w f (grey), for w = 5 and w = 10

    Gianluca Vinti ([email protected]) Sampling Type Operators 28 / 51

  • Comparison between f (blue) and the Sampling Kantorovich

    Operators SF2w f (grey) and SM23w f (red), for w = 5 and w = 10

    Gianluca Vinti ([email protected]) Sampling Type Operators 29 / 51

  • 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.

    Gianluca Vinti ([email protected]) Sampling Type Operators 30 / 51

  • We consider the following image:

    Figura : ”Lena”. Original image. (150× 150 pixel).

    Gianluca Vinti ([email protected]) Sampling Type Operators 31 / 51

  • 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)

    Gianluca Vinti ([email protected]) Sampling Type Operators 32 / 51

  • 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)

    Gianluca Vinti ([email protected]) Sampling Type Operators 33 / 51

  • 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

  • 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

  • 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

    Gianluca Vinti ([email protected]) Sampling Type Operators 36 / 51

  • 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

  • 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.

    Gianluca Vinti ([email protected]) Sampling Type Operators 38 / 51

  • Gianluca Vinti ([email protected]) Sampling Type Operators 39 / 51

  • 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)

    Gianluca Vinti ([email protected]) Sampling Type Operators 40 / 51

  • 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)

    Gianluca Vinti ([email protected]) Sampling Type Operators 41 / 51

  • 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

  • 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)

    Gianluca Vinti ([email protected]) Sampling Type Operators 43 / 51

  • Zoom of the Reconstruction of the ROI by the SamplingKantorovich Operators with a Jackson-type kernel J 24 for

    w = 20

    Gianluca Vinti ([email protected]) Sampling Type Operators 44 / 51

  • 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

    Gianluca Vinti ([email protected]) Sampling Type Operators 45 / 51

  • 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)

    Gianluca Vinti ([email protected]) Sampling Type Operators 46 / 51

  • Application of the algorithm to the original imagetogether with an edge detection algorithm - Comparison

    with the equivalent reconstructed image

    Gianluca Vinti ([email protected]) Sampling Type Operators 47 / 51

  • Application of the algorithm to the reconstructed imagetogether with an edge detection algorithm - Comparison

    with the CT image with contrast medium

    Gianluca Vinti ([email protected]) Sampling Type Operators 48 / 51

  • 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.

    Gianluca Vinti ([email protected]) Sampling Type Operators 49 / 51

  • 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)

    Gianluca Vinti ([email protected]) Sampling Type Operators 50 / 51

  • Many

    Thanks for Your attention

    Gianluca Vinti ([email protected]) Sampling Type Operators 51 / 51