Faculty of Information Technology and Bionics Institute de Recherche en Informatique de Toulouse 50/a Pr´ater utca, H-1083 Budapest 118 r. de Narbonne, F-31062 Toulouse
ABSTRACT
A volumetric non-blind single image super-resolution tech- nique using tensor factorization has been recently introduced by our group. That method allowed a 2-order-of-magnitude faster high-resolution image reconstruction with equivalent image quality compared to state-of-the-art algorithms. In this work a joint alternating recovery of the high-resolution im- age and of the unknown **point** **spread** **function** parameters is proposed. The method is evaluated on dental computed to- mography images. The algorithm was compared to an ex- isting 3D super-resolution method using low-rank and total variation regularization, combined with the same alternating PSF-optimization. The two algorithms have shown similar improvement in PSNR, but our method converged roughly 40 times faster, under 6 minutes both in simulation and on exper- imental dental computed tomography data.

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In most digital cameras, and even in high-end digital single lens re- flex cameras, the acquired images are sampled at rates below the Nyquist critical rate, causing aliasing effects. This work introduces an algorithm for the subpixel estimation of the **point** **spread** **function** of a digital cam- era from aliased photographs. The numerical procedure simply uses two fronto-parallel photographs of any planar textured scene at different dis- tances. The mathematical theory developed herein proves that the camera psf can be derived from these two images, under reasonable conditions. Mathematical proofs supplemented by experimental evidence show the well-posedness of the problem and the convergence of the proposed al- gorithm to the camera in-focus psf. An experimental comparison of the resulting psf estimates shows that the proposed algorithm reaches the accuracy levels of the best non-blind state-of-the-art methods.

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3 T´eSA Laboratory, 14-16 Port Saint-Etienne, 31000 Toulouse, France
{nzhao, jean-yves.tourneret}@enseeiht.fr, {adrian.basarab, denis.kouame}@irit.fr
ABSTRACT
This paper addresses the problem of blind deconvolution for ultra- sound images within a Bayesian framework. The prior of the un- known ultrasound image to be estimated is assumed to be a product of generalized Gaussian distributions. The **point** **spread** **function** of the system is also assumed to be unknown and is assigned a Gaus- sian prior distribution. These priors are combined with the likeli- hood **function** to build the joint posterior distribution of the image and PSF. However, it is difficult to derive closed-form expressions of the Bayesian estimators associated with this posterior. Thus, this paper proposes to build estimators of the unknown model parameters from samples generated according to the model posterior using a hy- brid Gibbs sampler. Simulation results performed on synthetic data allow the performance of the proposed algorithm to be appreciated.

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Index Terms— Ultrasound imaging, image deconvolution, Bayesian inference, Gibbs sampler.
1. INTRODUCTION
Ultrasound (US) imaging is widely used due to its advantages such as being portable, cost effective and noninvasive. However, the US images are contaminated by an intrinsic noise called speckle and have low contrast and relatively low spatial resolution at a given fre- quency. A 2D convolution model between the tissue reflectivity im- age and the system **point** **spread** **function** (PSF) is commonly used to model US images. As a consequence, deconvolution methods are widely used to improve the quality of US images.

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noise (AWGN) and H ∈ R N ×N is the system impulse re-
sponse/**point** **spread** **function** (PSF) assumed to be a circulant matrix [3, 4]. In US imaging systems, the PSF is usually un- known. Existing methods to address this problem include ei- ther the estimation of the PSF in a pre-processing step [3,5] or the estimation of the PSF and the TRF simultaneously [6, 7]. In this paper, we follow the second strategy to estimate the US TRF and PSF jointly. In particular, a parametric model for the PSF of the form of a modulated 2D Gaussian **function** is proposed. This parametric model allows us to reduce the estimation of the PSF during the blind deconvolution process to the estimation of a few parameters of the PSF model. In addition, a generalized Gaussian distribution is proposed for

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Received 8 December 2004 / Accepted 5 September 2006
ABSTRACT
A new method is presented for determining the **point** **spread** **function** (PSF) of images that lack bright and isolated stars. It is based on the same principles as the MCS image deconvolution algorithm. It uses the information contained in all stellar images to achieve the double task of reconstructing the PSFs for single or multiple exposures of the same field and to extract the photometry of all **point** sources in the field of view. The use of the full information available allows us to construct an accurate PSF. The possibility to simultaneously consider several exposures makes it well suited to the measurement of the light curves of blended **point** sources from data that would be very di ﬃcult or even impossible to analyse with traditional PSF fitting techniques. The potential of the method for the analysis of ground-based and space-based data is tested on artificial images and illustrated by several examples, including HST /NICMOS images of a lensed quasar and VLT/ISAAC images of a faint blended Mira star in the halo of the giant elliptical galaxy NGC 5128 (Cen A).

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We apply our method to a dataset of Euclid-like simulated PSFs (**Point** **Spread** **Function**). ESA’s Euclid mission will cover a large area of the sky in order to accurately measure the shape of billions of galaxies. PSF estimation and correction is one of the main sources of systematic errors on those galaxy shape measurements. PSF variations across the field of view and with the incoming light’s wavelength can be highly non-linear, while still retaining strong geometrical information, making the use of Optimal Transport distances an attractive prospect. We show that our representation does indeed succeed at capturing the PSF’s variations.

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noise (AWGN) and H ∈ R N ×N is the system impulse re-
sponse/**point** **spread** **function** (PSF) assumed to be a circulant matrix [3, 4]. In US imaging systems, the PSF is usually un- known. Existing methods to address this problem include ei- ther the estimation of the PSF in a pre-processing step [3,5] or the estimation of the PSF and the TRF simultaneously [6, 7]. In this paper, we follow the second strategy to estimate the US TRF and PSF jointly. In particular, a parametric model for the PSF of the form of a modulated 2D Gaussian **function** is proposed. This parametric model allows us to reduce the estimation of the PSF during the blind deconvolution process to the estimation of a few parameters of the PSF model. In addition, a generalized Gaussian distribution is proposed for

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acts as a dispersive optical element for ultrashort pulses of light. In this regime, the optical transformation of the field induced by the medium is very complex but still remains linear and deterministic, hence controllable. Owing to the availability of spatial light modulators (SLMs), several techniques based on wavefront shaping were developed to experimentally characterize this process. A recurrent application is to find the incident wavefront that counterbalances the effects of scattering and thus re-compress the pulse to its initial duration and focus it to a diffraction limited spot. For instance it can be achieved by iteratively optimizing the incident wavefront [ 19 , 20 ] but also by using digital phase conjugation [ 21 ], spectral pulse shaping [ 22 ], or time-gating techniques [ 23 ]. Another and more global approach to describe and manipulate the outgoing broadband light consists of measuring the multi-spectral transmission matrix (MSTM) [ 24 ]. The MSTM is a set of N λ ≈ ∆λ laser /δλ m monochromatic transmission matrices (TMs); each TM linearly relates the input field to the output field of the medium [ 25 ] for a given spectral component of the pulse. The full set of matrices provides both spatial and spectral/temporal information for controlling the transmitted pulse; in particular enhancing a single spectral component of the output pulse or focusing it at a given time can be performed [ 26 – 28 ]. The key **point** here is that these techniques manipulate both spatial and spectral degrees of freedom of the pulse by only using a single SLM. This is possible thanks to the spatio-spectral coupling resulting from the propagation through the medium. This is what we exploit here, now implemented as 3D spatio-spectral control, relying on a single SLM. Although pulse control in complex media has already been studied in the last years, to our knowledge, the spatio-temporal degrees of freedom of a scattering medium have never been used to spectrally engineer the **point**-**spread** **function** (PSF) of an ultrashort pulse.

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50/a Pr´ater utca, H-1083 Budapest 118 r. de Narbonne, F-31062 Toulouse
ABSTRACT
A volumetric non-blind single image super-resolution tech- nique using tensor factorization has been recently introduced by our group. That method allowed a 2-order-of-magnitude faster high-resolution image reconstruction with equivalent image quality compared to state-of-the-art algorithms. In this work a joint alternating recovery of the high-resolution im- age and of the unknown **point** **spread** **function** parameters is proposed. The method is evaluated on dental computed to- mography images. The algorithm was compared to an ex- isting 3D super-resolution method using low-rank and total variation regularization, combined with the same alternating PSF-optimization. The two algorithms have shown similar improvement in PSNR, but our method converged roughly 40 times faster, under 6 minutes both in simulation and on exper- imental dental computed tomography data.

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In this paper, we model the **point**-**spread** **function** (PSF) of a fluo- rescence MACROscope with a field aberration. The MACROscope is an imaging arrangement that is designed to directly study small and large specimen preparations without physically sectioning them. However, due to the different optical components of the MACRO- scope, it cannot achieve the condition of lateral spatial invariance for all magnifications. For example, under low zoom settings, this field aberration becomes prominent, the PSF varies in the lateral field, and is proportional to the distance from the center of the field. On the other hand, for larger zooms, these aberrations become gradually ab- sent. A computational approach to correct this aberration often relies on an accurate knowledge of the PSF. The PSF can be defined either theoretically using a scalar diffraction model or empirically by ac- quiring a three-dimensional image of a fluorescent bead that approx- imates a **point** source. The experimental PSF is difficult to obtain and can change with slight deviations from the physical conditions. In this paper, we model the PSF using the scalar diffraction approach, and the pupil **function** is modeled by chopping it. By comparing our modeled PSF with an experimentally obtained PSF, we validate our hypothesis that the spatial variance is caused by two limiting optical apertures brought together on different conjugate planes.

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In this paper we propose a method for retrieving the **Point**
**Spread** **Function** (PSF) of an imaging system given the ob-
served image sections of a fluorescent microsphere. Theoret- ically calculated PSFs often lack the experimental or micro- scope specific signatures while empirically obtained data are either over sized or (and) too noisy. The effect of noise and the influence of the microsphere size can be mitigated from the experimental data by using a Maximum Likelihood Ex-

method [23], which is an extension of the step-edge tech- nique to achieve sub pixel resolution on the estimation. By aligning the step-edge slightly off the orthogonal scan direction the effective sampling rate is increased. Also, scan-line averaging successfully suppresses noise and increases signal-to-noise ratio making the estima- tion more stable. In [29] the authors propose a slanted- edge non-parametric sub-pixel psf estimation method that admits geometrical distortions. A parametric and non-parametric edge **spread** **function** estimation proce- dure is proposed in [9]. Non-uniform illumination is also taken into account. However, the differentiation step that gives back the psf requires regularization and therefore loses accuracy. Since the previous methods are based on estimating several one-dimensional responses, several images or symmetry assumptions are needed to reconstruct a full bi-dimensional psf.

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shifted in depth but not in the radial plane. If the sphere is assumed to be pla
ed at a relative
position (x o , y o , z o ) in a given volume, then the observed image will have the
entroid in the volume as (x i , y i , z i ) , with x i ≈ x o and y i ≈ y o . Another **point** to be noted is that due to photon loss, although uniformly distributed, the true intensity of the observed sphere s is unknown. In Subse
tion 1.1.3 , we saw how a bandlimited obje
t
ould be simulated for the

For this generation of instruments and the next, understand- ing the **point** **spread** **function** (PSF) of AO instruments on giant telescopes will be important for the development of algorithms optimized in the search for planets. 5 , 6 The analysis in this paper expands on our previous work, 7 which demonstrated the origin of azimuthal asymmetry in the PSF as a consequence of the time lag error, to explore asymmetry along the preferential axis intro- duced by scintillation. This effect has been demonstrated previ- ously by Cantalloube et al. 8 We will expand on their discussion using a more general method of analyzing the structure of the AO-corrected PSF analytically, as well as validating our conclu- sions with observations and atmospheric datasets. More specifi- cally, our formalism demonstrates that the asymmetry grows linearly only for small spatial frequencies, and at higher spatial frequencies becomes nonlinear. We include solutions for the zeros of the log of the asymmetry metric, which are image loca- tions with an observable return to symmetry.

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The usual route to conformal blocks is through the short- distance expansion for ’ 1 ðt 1 Þ’ 2 ðt 2 Þ. In our construction ’ 1 ðt 1 Þ is replaced by O y ðt 1 Þ, which does not have an evident short distance expansion with ’ 2 ðt 2 Þ. Nevertheless, within our approach we are able to derive a block representation for the four-**point** **function**. This puts into evidence once again that our method, with its cancellation of defects, preserves conformal covariance.

Submitted to Proceedings of the National Academy of Sciences of the United States of America
Enhanced phosphorus export from land into streams and lakes is a primary factor driving the expansion of deep-water hypoxia in lakes during the Anthropocene. However, the interplay of regional scale environmental stressors and the lack of long-term instrumen- tal data often impede analyses attempting to associate changes in land cover with downstream aquatic responses. Herein we performed a synthesis of data that link paleolimnological recon- structions of lake bottom-water oxygenation to changes in land cover/use and climate over the last 300 years in order to evaluate whether the **spread** of hypoxia in European lakes was primarily associated with enhanced phosphorus exports from either grow- ing urbanization, intensiﬁed agriculture or climatic change. We showed that hypoxia started spreading in European lakes around CE 1850 and was greatly accelerated after CE 1900. Socio-economic changes in Europe beginning in CE 1850 resulted in widespread urbanization as well as a larger and more intensively cultivated surface area. However, our analysis of temporal trends demon- strated that the onset and intensiﬁcation of lacustrine hypoxia were more strongly related to the growth of urban areas than to changes in agricultural areas and the application of fertilizers. These results suggest that anthropogenically-triggered hypoxia in European lakes were primarily caused by enhanced phosphorus discharges from urban **point** sources. To date, there have been no signs of sustained recovery of bottom water oxygenation in lakes following the enactment of European water legislation in the 1970s to 1980s, and the subsequent decrease in domestic phosphorus consumption.

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As many earlier methods, ours analyses model disease **spread** on networks from a node centric summary statistics, by considering the number of contacts and transmission events per time. Therefore, it inherently neglects correlation between nodes. In other words there is no consideration of assortativity between individuals based on their number of contacts or transmission events per time. At the same time, individuals share their activity randomly among all their contacts (weights are homogeneously, or multinomially, distributed among edges that leave a node), which can enforce correlations among nodes in certain networks. Also clustering is observed in many contact networks [38] and this issue should be addressed in an extended version of our model.

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Finally we would like to underline the fact that the combination of the two-**point** **function** technique with the assumption of positive second variation is quite natural. In fact, on a general stationary **point** the Jacobi operator does not satisfy the maximum principle. It is precisely the stability condition which guarantees its validity. We hope that this simple observation can be useful also in other contexts.