The total variation norm in computational fluid dynamics 
and image processing

Antonio Marquina (Universidad de Valencia)



In this talk we shall review the relevance of the total variation norm in the development of computational fluid dynamics and
lately as a useful and accurate mathematical tool in image processing and data treatment. The pursue of efficient and accurate algorithms 
for the simulation of supersonic flows where jump discontinuities 
({\sl shock waves}) propagate and interact are present makes the L2 norm 
limited because the development of the unavoidable Gibbs phenomenom around jumps, that causes the presence of spurious oscillations 
in the simulated flow. Then, the total variation norm, (and the L1 norm in general) replaced the L2 norm in the design of numerical 
schemes to preserve edges and other singular points of the flows, 
avoiding unphysical oscillations. Through the last three decades 
digital image processing has been developed since the relevant applications arising from new sensors, (digital cameras, CAT scanners, 
MRI, ....). Typical digital signals (like images) contain edges, noise and other undesirable artifacts, and therefore classical methods for
image restoration and enhancement based on Fourier transform 
and L2 norms, are very limited due to the presence of spurious 
oscillation ({\sl ringing}). Thus, the use of the total variation norm as a regularizer is a significant improvement in the numerical outputs. 
We shall describe the role of this new L1-mathematics in the development and treatment of images and digital data as well as 
the new challenges established from this new perspective.