We present a computational procedure to evaluate the integral ∫ y x s p -1 e -μs ds for 0 ≤ x < y ≤ +∞,μ = ±1, p > 0, which generalizes the lower ( x =0) and upper ( y =+∞) incomplete gamma functions. To allow for large values of x , y , and p while avoiding under/overflow issues in the standard double precision floating point arithmetic, we use an explicit normalization that is much more efficient than the classical ratio with the complete gamma function. The generalized incomplete gamma function is estimated with continued fractions, with integrations by parts, or, when x ≈ y , with the Romberg numerical integration algorithm. We show that the accuracy reached by our algorithm improves a recent state-of-the-art method by two orders of magnitude, and it is essentially optimal considering the limitations imposed by floating point arithmetic. Moreover, the admissible parameter range of our algorithm (0 ≤ p,x,y ≤ 10 15 ) is much larger than competing algorithms, and its robustness is assessed through massive usage in an image processing application.

Detecting changes between two satellite images of the same scene generally requires an accurate (and thus often uneasy to obtain) model discriminating relevant changes from irrelevant ones. We here present a generic method, based on the definition of four different a-contrario detection models (associated to arbitrary features), whose aggregation is then trained from specific examples with gradient boosting. The results we present are encouraging, and in particular the low false positive rate is noticeable.

Abstract Designing realistic noise patterns from scratch is hard. To solve this problem, recent contributions have proposed involved spectral analysis algorithms that enable procedural noise models to faithfully reproduce some class of textures. The aim of this paper is to propose the simplest and most efficient noise model that allows for the reproduction of any Gaussian texture. Texton noise is a simple sparse convolution noise that sums randomly scattered copies of a small bilinear texture called texton . We introduce an automatic algorithm to compute the texton associated with an input texture image that concentrates the input frequency content into the desired texton support. One of the main features of texton noise is that its evaluation only consists to sum 30 texture fetches on average. Consequently, texton noise generates Gaussian textures with an unprecedented evaluation speed for noise by example. A second main feature of texton noise is that it allows for high‐quality on‐the‐fly anisotropic filtering by simply invoking existing GPU hardware solutions for texture fetches. In addition, we demonstrate that texton noise can be applied on any surface using parameterization‐free surface noise and that it allows for noise mixing.

Image inpainting consists in filling missing regions of an image by inferring from the surrounding content. In the case of texture images, inpainting can be formulated in terms of conditional simulation of a stochastic texture model. Many texture synthesis methods thus have been adapted to texture inpainting, but these methods do not offer theoretical guarantees since the conditional sampling is in general only approximate. Here we show that in the case of Gaussian textures, inpainting can be addressed with perfect conditional simulation relying on kriging estimation. We thus obtain a microtexture inpainting algorithm that is able to fill holes of any shape and size in an efficient manner while respecting exactly a stochastic model.

Aggregates of misfolded proteins are a hallmark of many age-related diseases. Recently, they have been linked to aging of Escherichia coli (E. coli) where protein aggregates accumulate at the old pole region of the aging bacterium. Because of the potential of E. coli as a model organism, elucidating aging and protein aggregation in this bacterium may pave the way to significant advances in our global understanding of aging. A first obstacle along this path is to decipher the mechanisms by which protein aggregates are targeted to specific intercellular locations. Here, using an integrated approach based on individual-based modeling, time-lapse fluorescence microscopy and automated image analysis, we show that the movement of aging-related protein aggregates in E. coli is purely diffusive (Brownian). Using single-particle tracking of protein aggregates in live E. coli cells, we estimated the average size and diffusion constant of the aggregates. Our results provide evidence that the aggregates passively diffuse within the cell, with diffusion constants that depend on their size in agreement with the Stokes-Einstein law. However, the aggregate displacements along the cell long axis are confined to a region that roughly corresponds to the nucleoid-free space in the cell pole, thus confirming the importance of increased macromolecular crowding in the nucleoids. We thus used 3D individual-based modeling to show that these three ingredients (diffusion, aggregation and diffusion hindrance in the nucleoids) are sufficient and necessary to reproduce the available experimental data on aggregate localization in the cells. Taken together, our results strongly support the hypothesis that the localization of aging-related protein aggregates in the poles of E. coli results from the coupling of passive diffusion-aggregation with spatially non-homogeneous macromolecular crowding. They further support the importance of "soft" intracellular structuring (based on macromolecular crowding) in diffusion-based protein localization in E. coli.

The total variation image (or signal) denoising model is a variational approach that can be interpreted, in a Bayesian framework, as a search for the maximum point of the posterior density (maximum a posteriori estimator). This maximization aspect is partly responsible for a restoration bias called the “staircasing effect,” that is, the outbreak of quasi-constant regions separated by sharp edges in the intensity map. In this paper we study a variant of this model that considers the expectation of the posterior distribution instead of its maximum point. Apart from the least square error optimality, this variant seems to better account for the global properties of the posterior distribution. We present theoretical and numerical results that demonstrate in particular that images denoised with this model do not suffer from the staircasing effect.

The restoration of images corrupted by blur and Poisson noise is a key issue in medical and biological image processing. While most existing methods are based on variational models, generally derived from a maximum a posteriori (MAP) formulation, recently sparse representations of images have shown to be efficient approaches for image recovery. Following this idea, we propose in this paper a model containing three terms: a patch-based sparse representation prior over a learned dictionary, the pixel-based total variation regularization term and a data-fidelity term capturing the statistics of Poisson noise. The resulting optimization problem can be solved by an alternating minimization technique combined with variable splitting. Extensive experimental results suggest that in terms of visual quality, peak signal-to-noise ratio value and the method noise, the proposed algorithm outperforms state-of-the-art methods.

We propose a definition of a Sharpness Index that is closely related to the notion of Global Phase Coherence recently introduced for automatic image restoration and image quality assessment. Using Gaussian random fields instead of random phase images, we can estimate the probability that a random image has a given Total Variation, which leads us to an explicit formula and a fast algorithm. Theoretical arguments and numerical experiments are given to assess the similarity between the Sharpness Index and the Global Phase Coherence, and an application to non-parametric blind deconvolution is presented, that illustrates the possibilities offered by this new approach.

In this paper, we are interested in the mathematical analysis of the micro-textures that have the property to be perceptually invariant under the randomization of the phases of their Fourier Transform. We propose a compact representation of these textures by considering a special instance of them: the one that has identically null phases, and we call it “texton”. We show that this texton has many interesting properties, and in particular it is concentrated around the spatial origin. It appears to be a simple and useful tool for texture analysis and texture synthesis, and its definition can be extended to the case of color micro-textures.

Multiplicative noise removal is a challenging image processing problem, and most existing methods are based on the maximum a posteriori formulation and the logarithmic transformation of multiplicative denoising problems into additive denoising problems. Sparse representations of images have shown to be efficient approaches for image recovery. Following this idea, in this paper, we propose to learn a dictionary from the logarithmic transformed image, and then to use it in a variational model built for noise removal. Extensive experimental results suggest that in terms of visual quality, peak signal-to-noise ratio, and mean absolute deviation error, the proposed algorithm outperforms state-of-the-art methods.

In the Rudin–Osher–Fatemi (ROF) image denoising model, total variation (TV) is used as a global regularization term. However, as we observe, the local interactions induced by TV do not propagate much at long distances in practice, so that the ROF model is not far from being a local filter. In this paper, we propose building a purely local filter by considering the ROF model in a given neighborhood of each pixel. We show that appropriate weights are required to avoid aliasing-like effects, and we provide an explicit convergence criterion for an associated dual minimization algorithm based on Chambolle's work. We study theoretical properties of the obtained local filter and show that this localization of the ROF model brings an interesting optimization of the bias-variance trade-off, and a strong reduction of an ROF drawback called the “staircasing effect.” Finally, we present a new denoising algorithm, TV-means, that efficiently combines the idea of local TV-filtering with the nonlocal means patch-based method.

According to Shannon Sampling Theory, Fourier interpolation is the optimal way to reach subpixel accuracy from a properly-sampled digital image. However, for most images this interpolation tends to produce an artifact called ringing, that consists in undesirable oscillations near objects contours. In this work, we propose a way to detect this ringing artifact. Using Euler zigzag numbers, we compute the probability that neighboring gray-levels form an alternating sequence by chance, and characterize these undesirable ringing blocks as structures that would be very unlikely in a random image. We then show two applications where the associated algorithm is used to test or enforce the compliance of an image with Fourier interpolation.

We propose a new algorithm able to detect the presence and the localization of aliasing in a single digital image. Considering the image in Fourier domain, the fact that two frequencies in aliasing relation contribute to similar parts of the image domain is a suspicious coincidence, that we detect with an a-contrario model. This leads to a localization of the aliasing phenomenon in both spatial and spectral domains, with a detection algorithm that keeps control of the number of false alarms. Experiments on several images show that this new method favorably compares to the state of the art, and opens interesting perspectives in terms of image enhancement.

This paper presents a new method for unsupervised subpixel change detection using image series. The method is based on the definition of a probabilistic criterion capable of assessing the level of coherence of an image series relative to a reference classification with a finer resolution. In opposition to approaches based on an a priori model of the data, the model developed here is based on the rejection of a nonstructured model-called a-contrario model-by the observation of structured data. This coherence measure is the core of a stochastic algorithm which automatically selects the image subdomain representing the most likely changes. A theoretical analysis of this model is led to predict its performances, in particular regarding the contrast level of the image as well as the number of change pixels in the image. Numerical simulations are also presented that confirm the high robustness of the method and its capacity to detect changes impacting more than 25 percent of a considered pixel under average conditions. An application to land-cover change detection is then provided using time series of satellite images.

The Fourier phase spectrum of an image is well known to contain crucial information about the image geometry, in particular its contours. In this paper, we show that it is also strongly related to the image quality, in particular its sharpness. We propose a way to define the Global Phase Coherence (GPC) of an image, by comparing the likelihood of the image to the likelihood of all possible images sharing the same Fourier power spectrum. The likelihood is measured with the total variation (Rudin-Osher-Fatemi implicit prior), and the numerical estimation is realized by a Monte-Carlo simulation. We show that the obtained GPC measure decreases with blur, noise, and ringing, and thus provides a new interesting sharpness indicator, that can be used for parametric blind deconvolution, as demonstrated by experiments.

In this paper, a new method is presented for a subpixelic land cover classification using both high-resolution structural information and coarse-resolution (CR) temporal information. To that aim, the linear mixture model is used for pixel disaggregation. It enables us to describe a CR time series in terms of the mixture of classes that are represented within each pixel. Then, the Bayes' rule and the maximuma posterioricriterion lead to the definition of an energy function whose minimum corresponds to the researched optimal classification. A theoretical analysis of the labeling errors that may be obtained using this energy function is provided, raising the main parameters for labeling performance. The optimal classification is computed by combining linear regressions and simulated annealing, leading to an unsupervised algorithm. The method is validated with numerical results obtained on two different agricultural scenes (i.e., the Danubian plain and the Coet Dan watershed).

When sampling a continuous image or subsampling a discrete image, aliasing artifacts can be controlled by filtering the data prior to sampling. Bandlimiting filters completely avoid aliasing artifacts, but have to find a compromise between blur and ringing artifacts. In this paper, we propose a new joint definition of blur/ringing artifacts, that associates to a given bandlimited prefilter its so-called spread-ringing curve. We then build a set of filters yielding the optimal blur/ringing compromise according to the previous definition. We show on experiments that such filters yield sharper images for a given level of ringing artifact.

Wertheimer's theory suggests a general perception law according to which objects having a quality in common get perceptually grouped. The Helmholtz principle is a quantitative version of this general grouping law. It states that a grouping is perceptually "meaningful" if its number of occurrences would be very small in a random situation: geometric structures are then characterized as large deviations from randomness. In two previous works, we have applied this principle to the detection of orientation alignments and boundaries in a digital image. In this paper, we show that the method is fully general and can be extended to a grouping by any quality. We treat as an illustration the alignments of objects, their grouping by color and by size, and the vicinity gestalt (clusters). Collaboration of the gestalt grouping laws and their pyramidal structure are illustrated in a case study.

We address the problem of computing a local orientation map in a digital image. We show that standard image gray level quantization causes a strong bias in the repartition of orientations, hindering any accurate geometric analysis of the image. In continuation, a simple dequantization algorithm is proposed, which maintains all of the image information and transforms the quantization noise in a nearby Gaussian white noise (we actually prove that only Gaussian noise can maintain isotropy of orientations). Mathematical arguments are used to show that this results in the restoration of a high quality image isotropy. In contrast with other classical methods, it turns out that this property can be obtained without smoothing the image or increasing the signal-to-noise ratio (SNR). As an application, it is shown in the experimental section that, thanks to this dequantization of orientations, such geometric algorithms as the detection of nonlocal alignments can be performed efficiently. We also point out similar improvements of orientation quality when our dequantization method is applied to aliased images.

We present an accurate numerical scheme for the affine plane curve evolution and its morphological extension to grey-level images. This scheme is based on the iteration of a nonlocal, fully affine invariant and numerically stable operator, which can be exactly computed on polygons. The properties of this operator ensure that a few iterations are sufficient to achieve a very good accuracy, unlike classical finite difference schemes that generally require a lot of iterations. Convergence results are provided, as well as theoretical examples and experiments.

We reformulate the "shape from motion" problem in terms of movie processing: how to focus the redundant depth information of a noisy image sequence into a perfect depth-coherent movie? We give an answer in the simple case of a straight movement of the camera parallel to the focal plane. From an axiomatic approach, we deduce a unique multiscale analysis of movies compatible with the depth recovery, and study its properties both in the image and the scene space. In particular, we show that this movie filtering preserves the geometrical constraints satisfied by ideal movies of rigid 3D scenes. We also study a numerical scheme in agreement with the theoretical axioms: and produce some experiments on synthetic noisy movies.