Introduction to Medical Imaging Week 6: Introduction to Medical Imaging Week 6: Denoising (part II) – Variational Methods and Evolutions Guy Gilboa Course

Motivation Mathematical formulations for edge preserving processes and functionals. The models allow discontinuities and can follow edge directions.

MRI Denoising video oY4jRQ

Denoising of ultrasound image 3D volume visualization of a 7 week old human fetus in the belly. Left: original data. Right: 3D nonlinear diffusion.

Denoising of CT data 1D cross-section of different nonlinear diffusion denoising methods (EED-Weickert, RPM – regularized Perona-Malik, compared to GS – Gaussian smoothing and simulated low dose). Taken from Mendrik et al, IEEE Trans. Med. Imaging, 2009.

Evolutions Nonlinear processes that start with the noisy image as initial condition and smooth it gradually. Formalized as partial-differential- equations (PDE’s) Perona-Malik Weickert anisotropic diffusion.

Variational methods

Total Variation Denoising The simplest nonlinear convex variational method. Accounts very well for edges. Does not handle texture that well. Very useful, produces piece-wise smooth approximation of the signal without oscillations.

TV – based on smoothness and fidelity terms Variational methods – optimize with respect to some energy E ◦ Spatial smoothness, total variation: ◦ Fidelity term (distance to input image):

TV Denoising Taken from

TV-L1 – removing outliers Mila Nikolova. "A variational approach to remove outliers and impulse noise."Journal of Mathematical Imaging and Vision (2004):

TV deconvolution Model – energy to be minimized Euler-Lagrange Numerical implementation Matlab files and results

TV – numerics and discretization

Total variation example g n f, PSNR=28.1dB u, PSNR=39.8dB f-u g-u

Spectral Total Variation (extra) Creates a transform based on the TV functional. In the case of TV eigenfunctions (like disks) behaves as an ideal filter, without loss of shape or contrast. Designed for image analysis and synthesis, can work quite well for non-textured images. Gilboa, Guy. "A spectral approach to total variation." Scale Space and Variational Methods in Computer Vision. Springer Berlin Heidelberg, pp

Spectral TV example g n f, PSNR=28.1dB u, PSNR=42.1dB f-u g-u

Filter comparison: Gaussian filter u, PSNR=23.8dB f-u

Bilateral filter u, PSNR=37.8dB f-u

Total variation u, PSNR=39.8dB f-u

Nonlocal means u, PSNR=40.6dB f-u

Spectral TV u, PSNR=42.1dB f-u

Clean image and noise g, PSNR=∞ n