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4.4 Destriping via gradient field integration
This far, we introduced variational methods based on PDE’s and energy functional minimization,
with image denoising applications in mind. These techniques are actually used
extensively in computer vision/graphics, to solve over-constrained geometric problems.
It is typically the case of Photometric Stereo Methods (PSM) and Shape From Shading
(SFS) applications where the goal is to recover a depth map (or an image) by integrating
a gradient field with discrete values.
Let us denote by G =(G x ,G y ) a bidimensional gradient vector defined in a subspace
Ω of R 2 . The problem of gradient field integration consists in determining a function u
whose gradient ∇u is close to G. Two classes of techniques can be used to tackle this
ill-posed problem and estimate the true image u. Local integration techniques [Coleman
et al., 1982], [Healey and Jain, 1984] and [Wu and Li, 1988] rely on a curve integration :
∫
u(x, y) =u(x 0 ,y 0 )+ G x dx + G y dy (4.69)
where γ is the integration path from a starting point (x 0 ,y 0 ) to pixel (x, y) ∈ Ω. Local integration
techniques recover an image u starting with an initial height and then propagating
height values according to the neighborhood gradient values. Local integration techniques
are dependent on the data accuracy and can propagate error values when dealing with
noisy gradient fields. Global integration methods [Horn and Brooks, 1986], [], [Frankot
and Chellappa, 1988] and [Horn, 1990] can be formulated in a variational framework as
the minimization of an energy functional :
∫
E(u) =
Ω
γ
( ) ∂u 2 ( ) ∂u 2
∂x − G x +
∂y − G y dx dy (4.70)
Unlike denoising and decomposition variational models described this far, both terms in the
energy functional (4.70) are fidelity terms that measure the L 2 norm difference between the
gradient field components of u and the observed gradient field G. For instance, the energy
functional does not include any lagrange multiplier λ. The Euler-Lagrange equations of
the previous functionals leads to :
∂
∂x
( ∂u
∂x − G x
which can be simplified into the following Poisson equation :
)
+ ∂ ( ) ∂u
∂x ∂y − G y = 0 (4.71)
∇ 2 u = ∇.G (4.72)
The previous equation can be solved using a fast marching method [Ho et al., 2006] or
the streaming multigrid method proposed in [Kazhdan and Hoppe, 2008]. Among several
other techniques, we draw a particular attention to the well-known Frankot-Chellapa