Regularized iterative and noniterative procedures for object restoration in the presence of noise :an error analysis

An analysis is carried out, using the prolate spheroidal wave functions, of certain regularized iterative and noniterative methods previously proposed for the achievement of object restoration (or, equivalently, spectral extrapolation) from noisy image data. The ill-posedness inherent in the problem is treated by means of a regularization parameter, and the analysis shows explicitly how the deleterious effects of the noise are then contained. The error in the object estimate is also assessed, and it is shown that the optimal choice for the regularization parameter depends on the signal-to-noise ratio. Numerical examples are used to demonstrate the performance of both unregularized and regularized procedures and also to show how, in the unregularized case, artefacts can be generated from pure noise. Finally, the relative error in the estimate is calculated as a function of the degree of superresolution demanded for reconstruction problems characterized by low space–bandwidth products.

Stability Problems in Inverse Diffraction

Inverse diffraction consists in determining the field distribution on a boundary surface from the knowledge of the distribution on a surface situated within the domain where the wave propagates. This problem is a good example for illustrating the use of least-squares methods (also called regularization methods) for solving linear ill-posed inverse problem. We focus on obtaining error bounds For regularized solutions and show that the stability of the restored field far from the boundary surface is quite satisfactory: the error is proportional to ∊(ðŗ‚ ≃ 1) ,ðŗœ being the error in the data (Hölder continuity). However, the error in the restored field on the boundary surface is only proportional to an inverse power of │In∊│ (logarithmic continuity). Such a poor continuity implies some limitations on the resolution which is achievable in practice. In this case, the resolution limit is seen to be about half of the wavelength. Copyright © 1981 by The Institute of Electrical and Electronics Engineers, Inc.

Head repositioning accuracy in patients with neck pain and asymptomatic subjects: concurrent validity, influence of motion speed, motion direction and target distance

Background: Cervicocephalic kinesthetic deficiencies have been demonstrated in patients with chronic neck pain (NP). On the other hand, authors emphasized the use of different motion speeds for assessing functional impairment of the cervical spine. Purpose: The objectives of this study were (1) to investigate the head repositioning accuracy in NP patients and control subjects and (2) to assess the influence of target distance, motion speed, motion direction and pain. Materials and methods: Seventy-one subjects (36 healthy subjects and 35 NP patients; age 30–55 years) performed the head repositioning test (HRT) at two different speeds for horizontal and vertical movements and at two different distances. For each condition, six consecutive trials were sampled. Results: The study showed the validity and reproducibility of the HRT, confirming a dysfunctional threshold of 4.5°. Normative values of head repositioning error up to 3.6° and 7.1° were identified for healthy and NP subjects, respectively. A distance of 180 cm from the target and a natural motion speed increased HRT accuracy. Repositioning after extension movement showed a significantly larger error in both groups. Intensity, duration of pain as well as pain level did not significantly alter head repositioning error. Conclusions: The assessment of proprioceptive performance in healthy and NP subjects allowed the validation of the HRT. The HRT is a simple, not expensive and fast test, easily implementable in daily practice to assess and monitor treatment and evolution of proprioceptive cervical deficits.