Spatial and Spatiotemporal Variation Problems

In image processing it is required to formulate the variation problem for spatially and temporally varying variables. The path of the mass point x(t), a scalar function, has to be replaced by a spatial function or spatiotemporal f ( x ), i. e., by a scalar vector function of a vector variable. For image sequences, one of the components of x is the time t.

Consequently, the Lagrange function now depends on the vector vari- able x. Furthermore, it will not only be a function of f ( x ) and x explic- itly. There will be additional terms depending on the spatial (and possi- bly temporal) partial derivatives of f. They are required as soon as we demand that f at a point should be dependent on f in the neighborhood. In conclusion, the general formulation of the error functional ε (f ) as a variation integral for g reads

∈

ε (f ) = ∫ L.f, f_xp, x Σ  dxW → minimum.                               (17.6)

The area integral is calculated over a certain image domain Ω RW. Equation (17.6) already contains the knowledge that the extreme is a minimum. This results from the fact that f should show a minimum deviation from the given functions at certain points with additional con- straints.

The corresponding Euler-Lagrange equation is:

 

.

W

L_f −    ∂ _xp L_fxp = 0.                                         (17.7)

p=1

The variational approach can also be extended to vectorial features such as the velocity in image sequences. Then, the Lagrange function depends on the vectorial feature f = [f₁, f₂,..., f_W ]^T, the partial deriv- atives of each component f_i of the feature in all directions (f_i)_xp, and explicitly on the coordinate x:

ε ( f ) = ∫ L. f , (f_i)_xp, x Σ  dxW → minimum.                              (17.8)

From this equation, we obtain an Euler–Lagrange equation for each component f_i of the vectorial feature:

.

W

L_fi −    ∂ _xp L_(fi)xp = 0.                                        (17.9)

p=1

17.3 Variational Image Modeling†                                                                    447

 

Similarity Constraints

The similarity term is used to make the modeled feature similar to the measured feature. For a simple segmentation problem, in which the objects can be distinguished by their gray value, the measured feature is the gray value itself and the similarity term S is given by

S(f, x ) = ⊗ f ( x ) − g( x )⊗ _n.                                     (17.10)

This simply means that the deviation between the modeled feature and the image measured with the L_n norm should be minimal. The most commonly used norm is the L₂ norm, leading to the well known least squares (LS) approach.

For a linear restoration problem, the original image f ( x ) is degraded by a convolution operation with the point spread function of the degra- dation h( x ) (for further details, see Section 17.8). Thus the measured image g( x ) is given by

g( x ) = h( x ) ∗ f ( x ).                                       (17.11)

In order to obtain a minimum deviation between the measured and re- constructed images, the similarity term is

 

S(f, x ) = ⊗ h( x ) ∗ f ( x ) − g( x )⊗ _n.                               (17.12)

As a last example, we discuss the similarity constraint for motion de- termination. In Section 14.3.2 we discussed that the optical fl ow should meet the brightness constraint equation (14.9):

 

f ( x, t) ∇ g( x, t) + g_t( x, t) = 0                                     (17.13)

and used an approach that minimized the deviation from the optical fl ow in a least squares sense (Eq. (14.15)). With the L_n norm, we obtain the following similarity term:

 

S( f , x, t) = ⊗ f ∇ g + g_t⊗ _n.                                      (17.14)

This equation simply expresses that the continuity equation for the op- tical fl ow (Eq. (14.9)) should be satisfi ed as well as possible in a least squares sense. Note that the similarity now also depends explicitly on time, because the minimization problem is extended from images to space-time images.

From the following example, we will learn that similarity constraints alone are not of much use with the variational approach. We use the motion determination problem with the L₂-norm (least squares). With Eq. (17.14), the Lagrange function depends only on the optical fl ow f . To compute the Euler-Lagrange equations, we only need to consider the

448                                                                     17 Regularization and Modeling

 

partial derivatives of the similarity term Eq. (17.14) with respect to the components of the optical fl ow, ∂ L/∂ f_i:

L_fi = 2. f ∇ g + g_tΣ  g_xi.                                        (17.15) Inserting Eq. (17.15) into the Euler-Lagrange equation (Eq. (17.9)) yields

. f ∇ g + g_tΣ  g_x = 0,       . f ∇ g + g_tΣ g_y = 0,                    (17.16)

or, written as a vector equation,

. f ∇ g + g_tΣ ∇ g = 0.                                         (17.17)

∇

These equations tell us that the optical fl ow cannot be determined when the spatial gradient of g is a zero vector. Otherwise, they yield no more constraints than the continuity of the optical fl ow. This example nicely demonstrates the limitation of local similarity constraints. They only yield isolated local solutions without any constraints for the spatial variation of the optical fl ow. This results from the fact that the formula- tion of the problem does not include any terms connecting neighboring points. Thus, real progress requires inclusion of global constraints.

 

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