Page 43 - IJOCTA-15-4

Nonlinear image processing with α-Tension field: A geometric approach
on the direction of the gradient, which means in- size of time step is changed dynamically to cap-
creased smoothing takes place in homogeneous re- ture local behavior of the solution. Numerical
gions and edges are preserved. 12,21 On the other oscillations would be suppressed by regulariza-
2
hand, the second term, ∇I(∇ | ∇I | ), puts the tion techniques, acting to smooth a tight jump in
information of higher- order gradient into the evo- | ∇I |. Another important aspect is the type of
lution process. The term increases the contrast boundary conditions must be compatible with the
in the backgrounds with different gradient magni- physical interpretation of the problem. Thereby,
tudes, allowing finer features to be perceived more conservation properties, for example, Neumann
clearly. These mechanisms work in conjunction boundary conditions (zero gradient at the bound-
to preserve fine structural features while avoid- ary) are commonly used in image processing to
ing excessive smoothing and distortion of the im- avoid artificial segments of features at the bound-
age. The 2-tension field for feature enhancement aries of the image domain. 15
has numerous practical applications, including for
medical imaging. To give an example, in med- 4.5. Comparison with existing methods
ical imaging, highlighting anatomical structures Notably, the 2−tension field proposed in (16) pro-
(blood vessels, tumors, or tissue boundaries) can vides major benefits compared to classical image
improve the possibility of making an appropriate processing methods like total variation (TV), reg-
diagnosis significantly. This enables the 2-tension ularization, and anisotropic diffusion. Its adaptiv-
field to boost even weak or low-contrast features, ity is one such advantage due to the factoring of
which assist radiologists in identifying irregulari- (1+ | ∇I | ). This term interpolates the local dif-
2
ties based on selective enhancement of gradients fusion strength according to local gradient mag-
existing in such areas.
nitude | ∇I |. On the other hand, regularization
on traditional TV is frequently achieved with a
fixed or piece-wise-linear weighting scheme, which
4.4. Numerical implementation and
might not sufficiently respond to the variable
computational considerations
structure of the image. While for instance TV reg-
The code to numerically approximate the PDE ularization is good in keeping sharp edges, it has
τ 2 (I) will then involve the discretization of the the drawback of producing a staircasingeffect in
2-tension field, so as to find an approximate solu- smooth areas because it uses first-order gradients
tion on the computational grid. Common numer- only. The 2-tension field improves this by adding
2
ical discretization techniques, such as finite dif- higher-order information through ∇I(∇ | ∇I | )
ferences and finite elements, approximate contin- term to provide a more balanced approach be-
uous derivatives with discrete counterparts. For tween smoothing and edge conservation. 23 Such
example, the Laplacian term ∆I can be approx- adaptivity preserves fine details like textures and
imated by means of second-order central differ- subtle transitions without producing artifacts.
ences, and at the same time the gradient terms A novel aspect of the α−tension field comes
2
| ∇I | and ∇ | ∇I | are traditionally calculated from its capability to encode higher-order gra-
with first-order differences. Even since the 2- dient information, that is otherwise not taken
tension field is nonlinear because of terms includ- into consideration by most simple models, such
2
2
ing (1+ | ∇I | ) and ∇I(∇ | ∇I | ), one often uses as anisotropic diffusion or TV-based methods.
2
iterative methods such as explicit or semi- implicit ∇I(∇ | ∇I | ) contains the spatial variation of
schemes. 15 Explicit methods update the image in- the gradient-magnitude explicitly, which can lead
tensity I directly at each time step, but their sta- the model to respond to second order changes of
bility may depend on very small time steps. The the image intensity. 14,24 This is especially use-
semi-implicit schemes will treat some terms im- ful in cases where preserving or enhancing fine-
plicitly, thus allowing for larger time steps which scale structures (like ridges or corners) for im-
gives better computational efficiency while main- proved segmentation is ideal. For example, in
taining stability. medical imaging, where delicate anatomical struc-
Due to the fact that the α−tension field tures are of utmost diagnostic significance, includ-
is nonlinear, a critical challenge when approxi- ing higher-order terms ensures that such details
mating this field numerically is ensuring stable are not smoothed out during denoising. In con-
and accurate results. The diffusion coefficient trast, many popular existing methods use only
2
(1+ | ∇I | ) is space- and time-varying, which if first-order derivatives, causing them to produce a
not treated well, can introduce instabilities. 13,22 less realistic model of complex structures. The
This can be mitigated by adaptive time-stepping interpretation in terms of the 2-tension field illus-
techniques as follows: the procedure used where trates a more rigorous theoretical background of
585