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Accurate image segmentation via a high-order scheme for level-set equations Silvia Tozza joint work with Maurizio Falcone and Giulio Paolucci INdAM/Dept. of Mathematics, SAPIENZA Workshop CMIPI - Computational Methods for Inverse Problems in
Silvia Tozza
joint work with Maurizio Falcone and Giulio Paolucci
INdAM/Dept. of Mathematics, SAPIENZA
Workshop CMIPI - Computational Methods for Inverse Problems in Imaging Como, July 18, 2018
Silvia Tozza (INdAM/Dept. of Mathematics, SAPIENZA) Accurate image segmentation via a high-order scheme for level-set equations 1 / 34
1
Introduction
2
Image Segmentation via Level-Set equation
3
Adaptive filtered scheme
4
Numerical experiments
Silvia Tozza (INdAM/Dept. of Mathematics, SAPIENZA) Accurate image segmentation via a high-order scheme for level-set equations 2 / 34
Introduction
Time dependent HJ equation
We are interested in computing the approximation of viscosity solution of Hamilton-Jacobi (HJ) equation:
(t, x) ∈ (0, T) × Rd, v(0, x) = v0(x), x ∈ Rd. (1) (H1) H(x, p) is Lipschitz continuous w.r.t. all variables (H2) v0(x) is Lipschitz continuous.
viscosity solution for (1). GOAL: Construct convergent schemes to the viscosity solution v of (1) with the property to be of high-order in the region of regularity.
Silvia Tozza (INdAM/Dept. of Mathematics, SAPIENZA) Accurate image segmentation via a high-order scheme for level-set equations 3 / 34
Introduction
Time dependent HJ equation
We are interested in computing the approximation of viscosity solution of Hamilton-Jacobi (HJ) equation:
(t, x) ∈ (0, T) × Rd, v(0, x) = v0(x), x ∈ Rd. (1) (H1) H(x, p) is Lipschitz continuous w.r.t. all variables (H2) v0(x) is Lipschitz continuous.
viscosity solution for (1). GOAL: Construct convergent schemes to the viscosity solution v of (1) with the property to be of high-order in the region of regularity.
Silvia Tozza (INdAM/Dept. of Mathematics, SAPIENZA) Accurate image segmentation via a high-order scheme for level-set equations 3 / 34
Introduction Level-Set Method
The model equation corresponding to the LS method is
(t, x, y) ∈ [0, T] × R2, v(0, x, y) = v0(x, y), (x, y) ∈ R2. The unknown is a "representation" function v : [0, T] × R2 → R of the interface The position of the interface Γt at time t is given by the 0-level set of v(t, .), i.e. Γt = {(x, y) : v(t, x, y) = 0} v0 must be a representation function for the initial front ∂Ω0 where Ω0 ⊂ R2 is an open and bounded set, i.e. v0(x, y) > 0 in Ω0, v0(x, y) = 0
∂Ω0 := Γ0, v0(x, y) < 0 in R2 \ Ω0. c(x, y) is the velocity of the front in the normal direction η(t, x, y) =
∇v(t,x,y) |∇v(t,x,y)|.
Silvia Tozza (INdAM/Dept. of Mathematics, SAPIENZA) Accurate image segmentation via a high-order scheme for level-set equations 4 / 34
Introduction Level-Set Method
Figure: Illustration of the LS idea.
Silvia Tozza (INdAM/Dept. of Mathematics, SAPIENZA) Accurate image segmentation via a high-order scheme for level-set equations 5 / 34
Image Segmentation via Level-Set equation
Goal
Detect the boundaries of objects represented in a picture. A very popular method for segmentation is based on the level set method, this application is often called “Active contour" since the segmentation is obtained following the evolution of a simple curve (a circle for example) in its normal direction.
Key idea behind LS
The boundaries of a specific object inside a given image, described by the intensity function I(x, y), are characterized by an abrupt change of the values
In order to make use of this intuition, we have to define the velocity c(x, y) accordingly.
Silvia Tozza (INdAM/Dept. of Mathematics, SAPIENZA) Accurate image segmentation via a high-order scheme for level-set equations 6 / 34
Image Segmentation via Level-Set equation
c1(x, y) = 1 (1 + |∇(G ∗ I)|µ), µ ≥ 1 has been proposed in Malladi-Sethian-Vemuri (1993) for µ = 1 in Caselles-Catte-Coll-Dibos (1993) for µ = 2.
Properties of c1
takes values in [0, 1] is close to 0 if there is a rapid change in the values of I
Silvia Tozza (INdAM/Dept. of Mathematics, SAPIENZA) Accurate image segmentation via a high-order scheme for level-set equations 7 / 34
Image Segmentation via Level-Set equation
Another velocity proposed in Malladi-Sethian-Vemuri (1993) c2(x, y) = 1 − |∇(G ∗ I(x, y))| − M2 M1 − M2 , where M1 and M2 are the maximum and minimum values of |∇(G ∗ I(x, y))|.
Properties of c2
Takes values in [0, 1] is close to 0 if the magnitude of the image gradient is close to its maximal value, close to 1 otherwise.
Silvia Tozza (INdAM/Dept. of Mathematics, SAPIENZA) Accurate image segmentation via a high-order scheme for level-set equations 8 / 34
Image Segmentation via Level-Set equation
The two velocities share common properties, but present different features:
Features of c1
c1 depends more heavily on the changes in the magnitude of the gradient. ⇒ Easier detection of the edges but can produce false edges inside the object (e.g. in presence of specularities).
Features of c2
c2 is smoother inside the objects, being less dependent on the relative changes in the gradient. ⇒ Possible problems in the detection of all the edges if at least one is “more marked".
Silvia Tozza (INdAM/Dept. of Mathematics, SAPIENZA) Accurate image segmentation via a high-order scheme for level-set equations 9 / 34
Image Segmentation via Level-Set equation
Considering v0(x, y) = dist{(x, y), Γ0} then by construction all the C-level set are at a distance C from the 0-level set. If we consider a generic point (xc, yc) on a C-level set, then it is reasonable to assume that the closest point on Γ0 should be (x0, y0) = (xc, yc) − v(t, xc, yc) ∇v(t, xc, yc) |∇v(t, xc, yc)|.
Silvia Tozza (INdAM/Dept. of Mathematics, SAPIENZA) Accurate image segmentation via a high-order scheme for level-set equations 10 / 34
Image Segmentation via Level-Set equation
Therefore, it seems natural to define the extended velocity ˜ c(x, y) as ˜ c(x, y, v, vx, vy) = c
|∇v|, y − v vy |∇v|
(2) which coincides with c(x, y) on the 0-level set, as it is needed. Since the idea behind the modification of the velocity c(x, y) into ˜ c is to follow the evolution of the 0-level set and then to define accordingly the evolution on the other level sets, we can see the new definition, in some sense, as a characteristic based velocity.
Silvia Tozza (INdAM/Dept. of Mathematics, SAPIENZA) Accurate image segmentation via a high-order scheme for level-set equations 11 / 34
Image Segmentation via Level-Set equation
Figure: Initial fronts for the two cases tested.
Silvia Tozza (INdAM/Dept. of Mathematics, SAPIENZA) Accurate image segmentation via a high-order scheme for level-set equations 12 / 34
Adaptive filtered scheme
We look for non-monotone schemes since we want to get a high-order scheme We want to find a convergent scheme that approximates the viscosity solution of (1) We start from the results in Bokanowski, Falcone and Sahu (2016) and by Oberman and Salvador (2015) and we extend them introducing an adaptive choice of the parameter controlling the filter.
Silvia Tozza (INdAM/Dept. of Mathematics, SAPIENZA) Accurate image segmentation via a high-order scheme for level-set equations 13 / 34
Adaptive filtered scheme
The proposed adaptive scheme is un+1
i,j
= SAF (un)i,j := SM(un)i,j + φn
i,jεn∆tF
SA(un)i,j − SM(un)i,j εn∆t
starting from the initial condition u0
i,j.
εn = εn(∆t, ∆x, ∆y) > 0 is the switching parameter that will satisfy lim
(∆t,∆x,∆y)→0 εn = 0
F : R → R is the filter function φn
i,j is the smoothness indicator function at the node (xj, yi) and time tn, based
[4]
Silvia Tozza (INdAM/Dept. of Mathematics, SAPIENZA) Accurate image segmentation via a high-order scheme for level-set equations 14 / 34
Adaptive filtered scheme
The proposed adaptive scheme is un+1
i,j
= SAF (un)i,j := SM(un)i,j + φn
i,jεn∆tF
SA(un)i,j − SM(un)i,j εn∆t
starting from the initial condition u0
i,j.
εn = εn(∆t, ∆x, ∆y) > 0 is the switching parameter that will satisfy lim
(∆t,∆x,∆y)→0 εn = 0
F : R → R is the filter function φn
i,j is the smoothness indicator function at the node (xj, yi) and time tn, based
[4]
Silvia Tozza (INdAM/Dept. of Mathematics, SAPIENZA) Accurate image segmentation via a high-order scheme for level-set equations 14 / 34
Adaptive filtered scheme
Assumptions on SM
The scheme is consistent, monotone and can be written in differenced form un+1
i,j
= SM(un)i,j := un
i,j − ∆t hM
xj, yi, D−
x un i,j, D+ x un i,j; D− y un i,j, D+ y un i,j
D±
x un i,j := ± un
i,j±1−un i,j
∆x
and D±
y un i,j := ± un
i±1,j−un i,j
∆y
.
Assumptions on SA
SA has a high-order consistency and can be written in differenced form un+1
i,j
= SA(un)i,j:= un
i,j−∆thA
xj, yi, D−
k,xui,j, . . . , D− x un i,j, D+ x un i,j, . . . , D+ k,xun i,j,
D−
k,yui,j, . . . , D− y un i,j, D+ y un i,j, . . . , D+ k,yun i,j
for a Lipschitz continuous function hA(x, y, p−, p+, q−, q+) (in short), with D±
k,xun i,j := ± un
i,j±k−un i,j
k∆x
and D±
k,yun i,j := ± un
i±k,j−un i,j
k∆y
. No assumptions on the stability of the high-order scheme are made.
Silvia Tozza (INdAM/Dept. of Mathematics, SAPIENZA) Accurate image segmentation via a high-order scheme for level-set equations 15 / 34
Adaptive filtered scheme
In our approach the filter function F must satisfy F(r) ≈ r for |r| ≤ 1 so that if |SA − SM| ≤ ∆tεn and φn
i,j = 1 ⇒ SAF ≈ SA
F(r) = 0 for |r| > 1 so that if |SA − SM| > ∆tεn or φn
i,j = 0 ⇒ SAF = SM
⇒ Several choices for F, different for regularity properties. In the numerical experiments we will use the discontinuous filter F(r) = r if |r| ≤ 1
Silvia Tozza (INdAM/Dept. of Mathematics, SAPIENZA) Accurate image segmentation via a high-order scheme for level-set equations 16 / 34
Adaptive filtered scheme
If we want our scheme un+1
i,j
= SAF (un)i,j := SM(un)i,j + φn
i,jεn∆tF
SA(un)i,j − SM(un)i,j εn∆t
to switch to the high-order scheme when some regularity is detected, we have to choose εn such that
εn∆t
εn
(4) for (∆t, ∆x, ∆y) → 0, in the region of regularity at time tn Rn :=
i,j = 1
Silvia Tozza (INdAM/Dept. of Mathematics, SAPIENZA) Accurate image segmentation via a high-order scheme for level-set equations 17 / 34
Adaptive filtered scheme
For the definition of a function φ, needed to detect the region Rn, we require φn
i,j :=
1 if the solution un is regular in Ii,j, if Ii,j contains a point of singularity, where Ii,j = [xj−1, xj+1] × [yi−1, yi+1],
Remark
For εn ≡ ε∆x, with ε > 0 and φn
i,j ≡ 1, we get the standard Filtered Schemes
exploit more carefully the local regularity of the solution at every time tn and cell Ii,j. ⇒ Under all these assumptions, the AF Scheme is convergent (See [3] for details on the convergence theorem). ⇒ Falcone’s talk for more details!
Silvia Tozza (INdAM/Dept. of Mathematics, SAPIENZA) Accurate image segmentation via a high-order scheme for level-set equations 18 / 34
Adaptive filtered scheme
For the definition of a function φ, needed to detect the region Rn, we require φn
i,j :=
1 if the solution un is regular in Ii,j, if Ii,j contains a point of singularity, where Ii,j = [xj−1, xj+1] × [yi−1, yi+1],
Remark
For εn ≡ ε∆x, with ε > 0 and φn
i,j ≡ 1, we get the standard Filtered Schemes
exploit more carefully the local regularity of the solution at every time tn and cell Ii,j. ⇒ Under all these assumptions, the AF Scheme is convergent (See [3] for details on the convergence theorem). ⇒ Falcone’s talk for more details!
Silvia Tozza (INdAM/Dept. of Mathematics, SAPIENZA) Accurate image segmentation via a high-order scheme for level-set equations 18 / 34
Adaptive filtered scheme
For the definition of a function φ, needed to detect the region Rn, we require φn
i,j :=
1 if the solution un is regular in Ii,j, if Ii,j contains a point of singularity, where Ii,j = [xj−1, xj+1] × [yi−1, yi+1],
Remark
For εn ≡ ε∆x, with ε > 0 and φn
i,j ≡ 1, we get the standard Filtered Schemes
exploit more carefully the local regularity of the solution at every time tn and cell Ii,j. ⇒ Under all these assumptions, the AF Scheme is convergent (See [3] for details on the convergence theorem). ⇒ Falcone’s talk for more details!
Silvia Tozza (INdAM/Dept. of Mathematics, SAPIENZA) Accurate image segmentation via a high-order scheme for level-set equations 18 / 34
Adaptive filtered scheme
We use the scheme SAF defined in (3), composed by Lax-Wendroff as SA and Local Lax-Friedrichs as SM. To compute εn we use the following
Formula for εn
εn = max
(xj,yi)∈RnK
2
xun
+ Hq
yun
+ 2HpHqvxy)
x un, Dyun, Dyun) − hM(x, y, Dxun, D− x un, Dyun, Dyun)
x un, Dxun, Dyun, Dyun) − hM(x, y, D− x un, Dxun, Dyun, Dyun)
y un) − hM(x, y, Dxun, Dxun, Dyun, D− y un)
y un, Dyun) − hM(x, y, Dxun, Dxun, D− y un, Dyun)
finite difference approximations around the point (i, j), with K > 1
2.
In this way the tuning is automatic.
Silvia Tozza (INdAM/Dept. of Mathematics, SAPIENZA) Accurate image segmentation via a high-order scheme for level-set equations 19 / 34
Adaptive filtered scheme
How detect the front Γt?
Figure: Neighborhood θδ of Γt.
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Adaptive filtered scheme
Stopping Criterion: The iterations stop when the 0-level, or more precisely a neighborhood θδ of Γt of radius δ = max{∆x, ∆y}, ceases to move. E∞ := ||un+1 − un||L∞(θδ) = max
i,j |F n i,j − F n−1 i,j
| < tol where tol > 0 is the prescribed tolerance. Another possible choice: E1 := ||un+1 − un||L1(θδ) = ∆x∆y
|F n
i,j − F n−1 i,j
| < tol(∆x, ∆y) where now the tolerance tol > 0 depends also on the discretization parameters. Error Formulas: P-Errrel = |Ne − Na| Ne , P-Err1 = |Ne − Na|∆x∆y where Ne = # pixels of the exact object Na = # pixels of the approximated object via the chosen scheme.
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Adaptive filtered scheme
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Numerical experiments Synthetic Tests
Figure: Plots of the final front varying ∆x with error computed in L∞ norm, using the monotone scheme and the AF-LW scheme with µ = 3, Kreg = 2, velocity ˜ c.
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Numerical experiments Synthetic Tests
Table: Errors and number of iterations varying ∆x.
c Norm || ||∞ Monotone AF-LW ∆x tol µ Kreg Ni P-Errrel P-Err1 Ni P-Errrel P-Err1 0.02 0.001 3 2 505 0.0657 0.4088 653 0.0502 0.3128 0.01 0.0005 3 2 1240 0.0433 0.2692 1507 0.0399 0.2482
Table: Errors and number of iterations varying ∆x .
Norm || ||∞ Monotone AF-LW ∆x tol µ Kreg Ni P-Errrel P-Err1 Ni P-Errrel P-Err1 0.02 0.0002 3 2 375 0.0923 0.5748 507 0.0669 0.4164 0.01 0.00004 3 2 1162 0.0487 0.3024 1276 0.0439 0.2726
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Numerical experiments Real Tests
Silvia Tozza (INdAM/Dept. of Mathematics, SAPIENZA) Accurate image segmentation via a high-order scheme for level-set equations 25 / 34
Numerical experiments Real Tests
Figure: Plots of the final front using the monotone scheme (left) and the AF-LW scheme (right) with velocity c, with L∞ norm and tol = 0.0005, ∆x = 0.02, µ = 3 and Kreg = 5.
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Numerical experiments Real Tests
Table: Errors and number of iterations using c varying the stopping rule.
c ∆x = 0.02 Monotone AF-LW Norm tol µ Kreg Ni P-Errrel P-Err1 Ni P-Errrel P-Err1 || ||∞ 0.002 5 329 0.0807 0.2628 319 0.0507 0.1652 || ||1 0.0001 5 315 0.0861 0.2804 308 0.0511 0.1664
Table: Errors and number of iterations using c varying the stopping rule
∆x = 0.02 Monotone AF-LW Norm tol µ Kreg Ni P-Errrel P-Err1 Ni P-Errrel P-Err1 || ||∞ 0.0005 3 5 322 0.1827 0.5948 330 0.1515 0.4932 || ||1 0.0001 2 2 312 0.0097 0.0316 299 0.0074 0.0240
Silvia Tozza (INdAM/Dept. of Mathematics, SAPIENZA) Accurate image segmentation via a high-order scheme for level-set equations 27 / 34
Numerical experiments Real Tests
Figure: Plots of the final front using the monotone scheme, Ni = 376 (left), and the AF-LW scheme, Ni = 403 (right), all using L1 norm in the stopping criterion and tol = 0.00001, µ = 4 and Kreg = 5, ∆x = 0.01, and velocity c.
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Numerical experiments Real Tests
Figure: Plots of the final front using the monotone scheme (left) and the AF-LW scheme (right) with velocity c.
∆x = 0.02 Monotone AF-LW tol µ Kreg Ni P-Errrel P-Err1 Ni P-Errrel P-Err1 0.00005 5 3 228 0.0118 0.2476 262 0.0078 0.1628
Silvia Tozza (INdAM/Dept. of Mathematics, SAPIENZA) Accurate image segmentation via a high-order scheme for level-set equations 29 / 34
Numerical experiments Real Tests
Figure: Plots of the final front using the monotone scheme (Ni = 156), and the AF-LW scheme (Ni = 243), with L1 norm and tol = 0.00005, µ = 4, Kreg = 5, ∆x = 0.02 and velocity c.
Silvia Tozza (INdAM/Dept. of Mathematics, SAPIENZA) Accurate image segmentation via a high-order scheme for level-set equations 30 / 34
Numerical experiments Real Tests
Figure: Plots of the final front using the monotone scheme (Ni = 185), and the AF-LW scheme (Ni = 223), with L1 norm and tol = 0.00001, µ = 4, Kreg = 5, ∆x = 0.02 and velocity c.
Silvia Tozza (INdAM/Dept. of Mathematics, SAPIENZA) Accurate image segmentation via a high-order scheme for level-set equations 31 / 34
Numerical experiments Real Tests
We presented a rather simple way to construct convergent schemes, which are of high-order in regions of regularity. Our procedure is able to stabilize an otherwise unstable (high-order) scheme, still preserving its accuracy. The adaptive filtered scheme can be used efficacily and in a easily way to solve the image segmentation problem. We noticed that the new velocity function ˜ c introduced in the LS model can improve a lot the results, especially for biomedical images.
Work in progress/Future works
Explore more the different filter fuctions Using different smoothness indicator functions φn
i,j
Silvia Tozza (INdAM/Dept. of Mathematics, SAPIENZA) Accurate image segmentation via a high-order scheme for level-set equations 32 / 34
Numerical experiments Real Tests
We presented a rather simple way to construct convergent schemes, which are of high-order in regions of regularity. Our procedure is able to stabilize an otherwise unstable (high-order) scheme, still preserving its accuracy. The adaptive filtered scheme can be used efficacily and in a easily way to solve the image segmentation problem. We noticed that the new velocity function ˜ c introduced in the LS model can improve a lot the results, especially for biomedical images.
Work in progress/Future works
Explore more the different filter fuctions Using different smoothness indicator functions φn
i,j
Silvia Tozza (INdAM/Dept. of Mathematics, SAPIENZA) Accurate image segmentation via a high-order scheme for level-set equations 32 / 34
38(1):A171–A195, 2016.
Hamilton-Jacobi equations, Lecture Notes in Computational Science and Engineering, Proc. ENUMATH 2017, to appear.
first order evolutive Hamilton-Jacobi equations, in preparation.
Segmentation via a modified Level-Set method, in preparation.
level set approach, Center for Pure and Applied Mathematics, Report PAM-589,
A.M. Oberman and T. Salvador, Filtered schemes for Hamilton-Jacobi equations: a simple construction of convergent accurate difference schemes, Journal of Computational Physics, 284:367–388, 2015
Silvia Tozza (INdAM/Dept. of Mathematics, SAPIENZA) Accurate image segmentation via a high-order scheme for level-set equations 33 / 34
THANK YOU FOR YOUR ATTENTION!
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