Exploiting Redundancy for Aerial Image Fusion using Convex ...

4 Stefan Kluckner, Thomas Pock and Horst Bischof
sought solution u and reflects the regularization in terms of a smooth solution.
The second term accounts for the summed errors between u and the (noisy) input
data f. The scalar λ controls the fidelity between data fitting and regularization.
In following we derive our model for the task of image fusion from multiple
observations.
As a first modification of the TV-L 1 model defined in (1), we extend the convex
minimization problem to handle a set of K scene observations (f 1 , . . . , f K ).
Introducing multiple input images can be accomplished by summing the deviations
between the sought solution u and available observations f k , k = 1 . . . K
according to
⎧
⎫
⎨
K∑
min
u∈X ⎩ ‖∇u‖ ∑
⎬
1 + λ |u i,j − fi,j|
k ⎭ . (2)
k=1 i,j∈Ω
Since orthographic image generation from gray or color information with
sampling distances of approximately 10 cm requires an accurate recovery of fine
details and complex textures, we replace the TV-based regularization with a
dual-tree complex wavelet transform (DTCWT) [9, 16]. The DTCWT is nearly
invariant to rotation, which is important for regularization, but also to translations
and can be efficiently computed by using separable filter banks. The
transform is based on analyzing the signal with two separate wavelet decompositions,
where one provides the real-valued part and the other one yields the
complex part. Due to the redundancy in the proposed decomposition, the directionality
can be improved, compared to standard discrete wavelets [9]. In order
to include the linear wavelet-based regularization into our generic formulation we
replace the gradient operator ∇ by the linear transform Ψ : X → C. The space
C ⊆ C D denotes the real- and complex-valued transform coefficients c ∈ C. The
dimensionality of C D directly depends on parameters like the image dimensions,
the number of levels and orientations. The adjoint operator of the transform Ψ,
required for the signal reconstruction, is denoted as Ψ ∗ and is defined through
the identity 〈Ψu, c〉 C
= 〈u, Ψ ∗ c〉 X
.
As the L 1 norm in the data term is known to be sensitive to Gaussian noise
(we expect a small amount), we use the robust Huber norm [19] to estimate
the error between sought solution and observations instead. The Huber norm
is quadratic for small values, which is appropriate for handling Gaussian noise,
and linear for larger errors, which amounts to median like behavior. The Huber
norm is defined as
{ t
2
|t| ɛ = 2ɛ
: 0 ≤ t ≤ ɛ
t − ɛ 2 : ɛ < t . (3)
Because of the height field driven alignment of the appearance information,
undefined areas can be simply determined in advance for a geometrically transformed
image f k . Therefore, we support our formulation with a spatially varying
term w k i,j ∈ {0, 1}W H , which encodes the inpainting domain. The choice w k i,j = 0
corresponds to pure inpainting at a pixel location (i, j).
Considering the wavelet-based regularization, the encoded inpainting domain
and the Huber norm, our extended energy minimization problem for redundant