Regularized interpolation for noisy images.
Interpolation is the means by which a continuously defined model is fit to discrete data samples. When the data samples are exempt of noise, it seems desirable to build the model by fitting them exactly. In medical imaging, where quality is of paramount importance, this ideal situation unfortunately does not occur. In this paper, we propose a scheme that improves on the quality by specifying a tradeoff between fidelity to the data and robustness to the noise. We resort to variational principles, which allow us to impose smoothness constraints on the model for tackling noisy data. Based on shift-, rotation-, and scale-invariant requirements on the model, we show that the L(p)-norm of an appropriate vector derivative is the most suitable choice of regularization for this purpose. In addition to Tikhonov-like quadratic regularization, this includes edge-preserving total-variation-like (TV) regularization. We give algorithms to recover the continuously defined model from noisy samples and also provide a data-driven scheme to determine the optimal amount of regularization. We validate our method with numerical examples where we demonstrate its superiority over an exact fit as well as the benefit of TV-like nonquadratic regularization over Tikhonov-like quadratic regularization.
Ramani
Sathish
S
Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109 USA. sramani@umich.edu
Thevenaz
Philippe
P
Unser
Michael
M
D000465:Algorithms; D006257:Head / Q000033:anatomy & histology; D006801:Humans; D007091:Image Processing, Computer-Assisted / Q000379:methods*; D008279:Magnetic Resonance Imaging / Q000379:methods; D015233:Models, Statistical; D016012:Poisson Distribution; D015203:Reproducibility of Results; D014057:Tomography, X-Ray Computed / Q000379:methods