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Nonlinear image processing with α-Tension field: A geometric approach
their effects. Images (b 2 ) and (c 1 ) show the direc- the Laplacian operator (∆I), which can also de-
tional gradients retrieved from Sobel-X and Sobel- tect edges well but introduces noise amplifica-
Y, respectively, which store horizontal and verti- tion in smooth areas due to its isotropic smooth-
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cal edge details. The first term (1+ | ∇I | )∆I ing characteristics. We compute two main terms
output is shown in Image (c 2 ) that highlights for the 2-tension field, the first one describes a
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for edges while suppressing diffusion at flat ar- weighted diffusion process (1+ | ∇I | )∆I, which
eas. This is done by controlling the diffusion depends on the structure of the image, present-
process using adaptive modulation by the gradi- ing low diffusion of small magnitudes and pre-
ent magnitude, preserving, or smoothing around serving the integrity of edges. The second term,
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edges. The second term ∇I(∇ | ∇I | ) captures ∇I(∇ | ∇I | ), incorporates higher-order gradient
higher- order features of the image, detecting cor- information, which further improves the preserva-
ners, junctions, and fine details of the image. It tion of complex structures like edges or corners.
does this by capturing higher-order gradient infor- Through the addition of these two terms, one can
mation that helps to understand the local struc- arrive at the final output of the 2-tension field
tures and spatial variations present in the im- which attains a certain balance between noise re-
age. Visualizing the result of this term (Image duction and detail preservation. Moreover, the
(d 1 )) shows its power to detect and amplify these output for each step is plotted to give clarity
tiny features. Lastly, Image (d 2 ) demonstrates on how the operators affect the image across its
briefly the effect of combining all these to com- stages, from basic edge detection to complex fea-
pute the final output, which is τ 2 (I) and we can ture extraction.
see clearly that it retains the structure while suf- Comparison between tension field and 2-
ficiently filtering the noise. The extensivity of the tension field in different aspects is represented
deconstructions performed in this talk shows the in Table 1. Although the tension field is lin-
virtue of the α-tension field, and its extreme ro- ear, computationally simple, and works well for
bustness for high-level image processing tasks of simple edge detection, it is also very sensitive to
delicate detail preservation and artifact suppres- noise because of the isotropic smoothing and sec-
sion. ond derivatives because it can artificially inflate
noise in flat parts. On the other hand, the 2-
Step-by-step analysis of the Figure 2 describes tension field is non-linear and includes adaptive
different interactions of 2-tension field equation modulation that uses magnitude of gradient to
components: the tension field operator τ(I) is modulate the 2-tension field, which allows it to en-
the edge detector operator. It has high values hance edges and prevent noise amplification. The
on the edges but tends to also amplify noise due anisotropic smoothing behavior of this approach
to its isotropic smoothing nature. To overcome preserves edges in high-gradient regions while
this drawback, in the definition of τ 2 (I), the term smoothes homogeneous areas, making it more ro-
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1+ | ∇I | controls the weight of the Lapla- bust to noise. For advanced image processing
cian, with features being promoted and noise be- problems like denoising, edge-preserving, smooth-
ing suppressed through the gradient term. The ing, and feature enhancement, the 2-tension field
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second term ∇I(∇ | ∇I | ) accounts for how the with its higher-order gradient terms is more suit-
gradient magnitude changes in space, allowing for able, especially for tasks where the preservation
more adaptability and specificity when extracting of fine details is critical and the problems better
features from complex structures like corners or approximate the concept of artifacts.
junctions. Thus, by combining these two terms To sum up, this section has discussed the
together, the 2-tension field is capable of preserv- classical tension field and 2-tension field in the
ing significant details while getting rid of unde- sense of image processing and their merits and
sired artifacts, leading to better image processing drawbacks. The tension field τ(I) = ∆I, which
results. has the property of solving isotropic smoothing,
is a very good edge detector, but the linearity
Algorithm 2 presents the pseudocode that and the second gradients behavior of this oper-
outlines the detailed implementation procedure ator will made it very sensitive to noise, espe-
for both the tension field and the 2-tension field, cially in homogeneous zones. On the other hand,
as illustrated in Figure 2. It starts with comput- the 2-tension field takes into account higher-order
ing the gradient components, that is, ∇I x and gradient hints and modulates diffusion based on
∇I y using the central differences method, and fi- such local gradients adaptively. This adaptive be-
nally computing the gradient | ∇I |. This is fol- havior both minimizes amplification of deleteri-
lowed by applying the baseline tension field via ous noise as well as allowing crucial traits such
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