First Twenty Years of Total Variation In Image Processing and Analysis (Inventor's Perspective and Critique)

Total Variation measure for the functions of one variable, and the related properties, were established by C. Jordan in 1881. A consistent geometrical extension to the functions of two-variable was developed by A. Kronrod in 1950.

In order to address the ‘smooth solutions only’ limitations of the classical Tichonov regularization theory for ill-posed problems, particularly as it applies to the image restoration problems, L. Rudin in his 1987 work, has postulated that: “Images belong and can be modeled in the space of functions of bounded total variation BV(R2)” . In the same work a first example of a BV imbedding of the solution for a discontinuous image enhancement problem was constructed through a non-linear PDE Shock-Filter. Thus, the new field of non-linear PDE’s for image processing was initiated.

In 1989 report delivered to US Government, the author and his two collaborators demonstrated a new variational computational framework, where imaging solutions were derived via Minimization of TV functional, subject to the noise statistical constraints. This work had to wait until 1992 to appear in public.

Also, in 1989 Mumford and Shah have proposed a novel variational approach to image segmentation problem. The classical Mumford-Shah functional, while important on its own merits, is not equivalent to the Min(TV) problem, does not yield a straight-forward Euler-Lagrange minimization algorithm, imposes special geometrical constraints on the way the boundaries can intersect each other, and it is not a straight-forward generalization of Tichonov Regularization theory as the TV approach is.

The above minimizing 2-D Total Variation functional yielded a novel Euler-Lagrange equation with piece-wise smooth solutions to image denoising and deblurring. A ‘simple Min(TV) concept’ forced the solution to be in BV space, but not necessary to be in the class C8, which would happen with the classical regularization.

Ever since, the Min(TV) principle is being faithfully followed, in thousands of publications, with no or little change to the variatonal principle. The multitude of applications is an indication of an apparent success for the first twenty years of the TV-based methods. Citations show real practical applications to the fields of optical, electro-optical, acoustic image processing, 3-D image analysis, data compression, CT/MRI and other bio-medical imaging, fluid mechanics, circuits design, scattering and electromagnetic inverse problems, DNA and Chromosomes image analysis, seismic tomography, material science, etc.

As a practicing image processor, the author has spent last twenty years applying advanced algorithms (anything that works) to imaging problems arising in forensic/security and other video gathering fields. Image quality is critical here, as the identification depends on it. TV based image restoration showed to be superior to the classical Tichonov regularization.

The following practical observation however can be made: most’ bad quality’ images were not satisfactory recoverable by any of the state of the art image processing techniques. In this talk, we will examine such practical examples and consider the causes. We will question whether the blind following of the original principle of TV minimization is the best way to fight ill-posedness in image processing.

Leonid Rudin, Cognitech, Inc.

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