Aspheres in optical design are frequently used to achieve high optical performance in imaging applications. However, manufacturing aspheres involves serious precision and cost issues that must be considered. We show that, by applying a hybrid digital-optical design approach, the amount of the asphericity cost in a single surface may be significantly alleviated without compromising the performance. First, we compute the amount of spherical aberration depending on the optical parameters (conic constant and shape factor), and compute its corresponding point spread function (PSF). From the PSF and the spectral distribution of clean images and noise, we set a statistical observation model for estimating the expected image quality (in mean square error terms) in the image sensor and also in the digitally restored image. In addition, we use a previously proposed metric for quantifying the asphere fabrication cost, to set different cost scenarios. For each of these scenarios, we study how image quality is optimized, before and after digital restoration. Reversely, we find the optical configurations of minimal asphericity cost amongst those providing a very low aberration level. Although here we have limited our study to just on-axis, monochromatic imaging, we show in simulations how our digital-optical combined approach has a high potential for boosting the cost-effectiveness trade-off.
The geometric point spread function (PSF) is an appropriate tool for modeling image degradation of an optical
system, whenever the effect of diffraction is small compared to that of aberrations. The PSF is conventionally
estimated by computing the density of ray intersections with the image plane (ray-counting method). We studied
the effect of two factors on the estimation: the number of rays, using an error model, and the influence of the
ray sampling pattern. We measured the accuracy of the PSFs estimation in three ideal cases, where we could
derive an analytical expression for the irradiance. Additionally we estimated the PSFs generated by a single rear
landscape lens. We have observed a consistent improvement of 4.5 dB (Signal-to-Noise Ratio) when doubling
the number of rays. This ensures that an arbitrarily high accuracy on the estimation of geometric PSFs is
theoretically attainable. However, the method is not very effcient because of the slow convergence rate. As an
alternative, to avoid tracing a large number of rays, we investigated the benefits of interpolating rays intersections
(using cubic splines) of the ray mapping. For instance, by interpolating only 100 traced rays we may obtain a
similar quality in the estimation as when using 100 million of real traced rays. Among different uniform ray pupil
sampling patterns (square, jittered square, hexapolar and hexagonal) we found that the hexagonal outperformed
the other ones.
Wavefront coding (WFC) is a powerful hybrid optical-numerical technique for increasing the depth of focus of imaging
systems. It is based on two components: (1) an optical phase element that codifies the wavefront, and (2) a numerical
deconvolution algorithm that reconstructs the image. Traditionally, some sophisticated optical WFC designs have been
used to obtain approximate focus-invariant point spread functions (PSFs). Instead, we present a simple and low cost
solution, implemented on infrared (IR) cameras, which uses a decentred lens inducing coma as an adjustable and
removable phase element. We have used an advanced deconvolution algorithm for the image reconstruction, which is
very robust against high noise levels. These features allow its application to low cost imaging systems. We show
encouraging preliminary results based on realistic simulations using optical PSFs and noise power spectral density (PSD)
laboratory models of two IR imaging systems. Without induced optical phase, the reconstruction algorithm improves the
image quality in all cases, but it performs poorly when there are both in and out-of-focus objects in the scene. When
using our coding/decoding scheme with low-noise detectors, the proposed solution provides high quality and robust
recovery even for severe defocus. As sensor noise increases, the image suffers a graceful degradation, its quality being
still acceptable even when using highly noisy sensors, such as microbolometers. We have experienced that the amount of
induced coma is a key design parameter: while it only slightly affects the in-focus image quality, it is determinant for the
final depth of focus.
KEYWORDS: Signal to noise ratio, Point spread functions, Statistical analysis, Visualization, Matrices, Image processing, Denoising, Image restoration, Image filtering, Global system for mobile communications
We describe here two ways to improve on recent results in image restoration using Bayes least squares estimation
with local Gaussian scale mixtures (BLS-GSM) in overcomplete oriented pyramids. First one consists of allowing
for a spatial adaptation of the covariance matrix defining the GSM model at each pyramid subband. This can
be implemented in practice by dividing the subbands into spatial blocks. The other, more powerful, method is
to generalize the GSM model to include more than one covariance matrices for each subband. The advantage of
the latter method is its flexibility, as it allows for mixing Gaussian densities with different covariance matrices
at every spatial location in every subband. It also allows for non-local selective processing, taking advantage
of the repetition in the scene of image features that are not necessarily spatially grouped. We also describe
an empirical method to adapt denoising algorithms for doing image restoration, with the only constraint on
the denoising method of being applicable to non-white noise sources. Here we present mature results of the
spatially adaptive method applied to denoising and deblurring, plus some estimation techniques and encouraging
preliminary results of the multi-GSM concept.
We propose two methods for sparse approximation of images under l2 error metric. First one performs an approximation
error minimization given a lp-norm of the representation through alternated orthogonal projections
onto two sets. We study the cases p = 0 (sub-optimal) and p = 1 (optimal), and find that the l0-AP method is
neatly superior, for typical images and overcomplete oriented pyramids. Given that l1-AP is optimal, this shows
that it is not equivalent in practical image processing conditions to minimize one or the other norm, contrarily
to what is often assumed. The second method is more powerful, and it performs gradient descent onto decreasingly
smoothed versions of the sparse approximation cost function, yielding a method previously proposed as a
heuristic. We adapt these techniques for being applied to image restoration, with very positive results.
In this work we demonstrate the relationship existing between two important issues in vision: multi-scale local spectrum analysis, and log-polar foveatization. We show that, when applying a continuous set of self-similar (rotated and scaled) band-pass filters to estimate the local spectrum at a given point of attention of the image, the inverse Fourier transform of this local spectrum is a log- polar foveated version of the original image at that position. Both the local spectrum and its associated foveated image can be obtained through log-polar warping of the spectral/spatial domain followed by a conventional invariant low-pass filtering and the corresponding inverse warping. Furthermore, the low-pass filters in the warped space and frequency domains are mirror versions of each other. Thus, filters with mirror symmetry under the log- polar warping are self-dual, and make the foveatization process commute with the Fourier transform. Nevertheless, in order to implement a fovea that can be easily moved across the image, it is preferable to use a fixed bank of steerable filters, instead of applying log-polar warpings with different centers. Using low-pass scalable filters we have implemented a real-time moving fovea. We believe that a dual finite spatial/spectral local representation of images could be a very powerful tool for many visual tasks, which could benefit from a dual explicit representation in space and spatial frequency, as well as from the rotation and scale invariance naturally achieved in both domains.
We propose a new texture synthesis-by-analysis method inspired by current models of biological early vision and based on a multiscale Gabor scheme. The analysis stage starts with a log-polar sampling of the estimated power spectral density of the texture by a set of 4 by 4 Gabor filters, plus a low-pass residual (LPR). Then, for each channel, we compute its energy and its two (X,Y) bandwidths. The LPR is coded by five parameters. In addition, the density function of the original texture is also estimated and compressed to sixteen values. Therefore, texture is coded by only 69 parameters. The synthesis method consists of generating a set of 4 by 4 synthetic channels (Gabor filtered noise signals). Their energies and bandwidths are corrected to match the original features. These bandpass filtered noise signals are mixed into a single image. Finally, the histogram and LPR frequencies of the resulting texture are modified to fit the original values. We have obtained very satisfactory results both with highly random textures and with some quasi-periodic textures. Compared to previous methods, ours has other important advantages: high robustness (stable, non iterative and fully automatic), high compactness of the coding, and computational efficiency.