Yann Leydier

Jean Duong
email: jean.duong@corenum.com

Stéphane Br Ès

Véronique Églin

Martial TOLA Frank Lebourgeois

libcrn, an Open-Source Document Image Processing Library

Keywords: document image processing, open source, library, toolbox

In this paper we introduce libcrn, a multiplatform open-source document image processing library aimed at researchers and companies. It is written in C++11 and has a non-contaminating license that makes it available for use in any project without legal constraints.

The features include low-level image processing (color format conversion, binarization, convolution, PDE. . . ), document images specific tools (connected components extraction, recursive block description, PDF export. . . ), maths (matrix arithmetics, linear algebra, GMMs, equation solvers. . . ), classification and clustering (kNN, k-means, HMMs. . . ).

The API is comprehensively documented and libcrn's architecture follows modern C++ guidelines to facilitate the handling of the library and enforce its safe usage.

A sample OCR, which is only 30 lines long, is described to illustrate libcrn's scope of possibilities.

I. INTRODUCTION

Implementing the most basic document image processing algorithm may be a good exercise for students. However, when focusing on high-level processing chains or complex methods, researchers as well as manufacturers rely on toolboxes and software libraries for low-level tasks.

Commercial software libraries are available (Intel IPP, Lead Tools. . . ) and used in industry. They generally offer the basic tools needed to perform simple image manipulation and are well suited for non-specialist engineers. Specialists often prefer Open Source libraries as it is possible to check the details of the algorithms and modify and fine tune them when needed.

The most widely used image processing library is OpenCV 1 . Whereas is contains a great amount of algorithms, it is not originally meant for document images and lacks features and services that make it inconvenient.

Qgar 2 [START_REF] Tombre | Stable, robust and off-the-shelf methods for graphics recognition[END_REF] is an Open Source document image processing library created in the early 2000s. It features the most elementary tools to create document analysis software but also lacks some crucial features such as RGB images. Although Qgar can be easily extended, its development has been stalled since 2008. 1 http://opencv.org/ 2 http://www.qgar.org/ We introduce libcrn3 , licensed in LGPL (a noncontaminating Open Source license). Its aim is to allow both researchers and engineers to implement document image processing chains and algorithms. libcrn is available for Windows (Visual C++ 2015), Linux, MacOS and Android. It is written in C++11 using the "modern" guidelines issued by the C++ committee so that users can easily and safely use it (e.g.: no memory management is required from the users and no leak can happen). We implemented many image processing algorithm but also the mathematical tools needed to process the data that can be extracted from images.

II. FEATURES

A. General points

In order to facilitate the storage of data, most of the objects in libcrn can be serialized in XML files. Multiplatform utilities are packaged so that no overwork weights on the user to make applications run on any OS (e.g.: automatic file path format conversion, file manipulation, character set conversion, dynamically loaded modules. . . ).

B. Image 1) Formats:

We provide built-in support for numerous pixel types (see tab. I). Any other type of pixel format (such as matrices!) is supported as long as it implements the basic arithmetic operators. All the color types are trivially convertible and multiple binarization methods are offered: Niblack, Sauvola [START_REF] Sauvola | Adaptive document binarization[END_REF], local min or max, Fisher, entropy and Otsu [START_REF] Otsu | A threshold selection method from gray-level histograms[END_REF].

2) Algorithms: Classical basic transformation algorithms are provided, such as rotate by shear, mathematical morphology, convolution, distance transform. . . Feature extractors can be combined to allow the comparison of shapes. Sample feature extractors are provided including profile projections and gradients histograms (see fig. 1). Shapes can also be compared directly with an FFT-based cross-correlation or gradient matching. The gradient matching is a double inclusion measure. A mask of significant gradients is automatically computed and the angles inside a shape's mask are compared to the angles on the other shape. If the location of the pixel in the other shape is out of its dilated mask, then a penalty is applied. The sum of both inclusion measures is the dissimilarity between the two shapes (see fig. 2). 3) PDE: Gradients can be obtained from both grayscale and color [START_REF] Lebourgeois | Content based image retrieval using gradient color fields[END_REF] images by different means (half derivatives, Gaussian kernel. . . ) and can be isotropically diffused. From the gradients, a large number of transforms and descriptors are computed (see fig. rectangular blocks. Each block refers to a part of the original image that is instantiated only when needed. This structure is of course serializable.

The fast connected components extraction [START_REF] He | Fast connectedcomponent labeling[END_REF] directly feeds the aforementioned block structure.

Simple line and word segmentation algorithms are implemented and more can be easily added to feed the block layout structure.

Other useful algorithms specific to document images such as stroke thickness estimation, text line height estimation and skew estimation are offered.

C. Maths

1) Algebra: libcrn's math toolbox includes matrices of integers, reals and complex numbers and provide all arithmetics. Two system solvers (Cramer and Gauss-Jordan) are included.

Fundamental linear algebra methods such as diagonalization and inversion are implemented. Effective computation procedures will be added 4 for tensor computation [START_REF] Nelson | Tensor analysis[END_REF], [START_REF] Simmonds | A brief on tensor analysis[END_REF]: Higher order structure tensors [START_REF] Schultz | A higher-order structure tensor[END_REF] are proposed as extensions of classical structure tensor used in image processing [START_REF] Zenzo | A note on the gradient of a multi-image[END_REF]. Since they are tricky to compare (e.g. for clustering, feature extraction, etc.), their glyphs are to be used for analysis [START_REF] Schultz | A maximum enhencing higherorder tensor glyph[END_REF]. Thus, Zernike moments have been implemented and should be soon added to image analysis toolbox.

2) Data analysis: Multidimensional data may be processed using Principal Component Analysis (PCA). Furthermore, Gaussian mixture models are also supported for both uni and multivariate samples. In particular, the Expectation-Maximization fitting procedure (EM) is implemented.

3) Interpolation and regression: The math module contains tools for linear interpolation, cubic splines interpolation and polynomial regression.

4) Geometry: libcrn provides angle arithmetics and tools such as circular mean and variance, but also circular histograms utilities: circular earth mover's distance [START_REF] Rabin | Circular earth mover's distance for the comparison of local features[END_REF], kurtosis, trigonometric moments. . . 5) Signal processing: FFT can be applied on complex matrices and vectors.

D. Pattern analysis 1) Clustering: libcrn provides clustering algorithms for both vectorial and metric data:

• k-means • k-medoids (PAM and fast [START_REF] Park | A simple and fast algorithm for k-medoids clustering[END_REF]) • Outliers: LOF and LoOP [START_REF] Kriegel | Loop: Local outlier probabilities[END_REF] • Spectral Clustering (all formulas from [START_REF] Ng | On spectral clustering: Analysis and an algorithm[END_REF]- [START_REF] Zelnik-Manor | Self-tuning spectral clustering[END_REF]) • Affinity Propagation [START_REF] Frey | Clustering by passing messages between data points[END_REF] 2) Classification: Many data classification problems can be addressed, using a wide variety of tools from a highly generic kNN implementation to discrete or Gaussian semicontinuous HMMs.

3) Other: Other "combinatorics oriented" algorithms are also available in libcrn, such as Kuhn and Munkres' "Hungarian" bipartition algorithm, the A* path finder or the disjoint set forest distribution.

E. GUI

Custom widgets are provided to create applications with Gtkmm -Gtk's C++ wrapper -versions 2 and 3. This include displaying and browsing the block structure of an image, image overlays and the automated generation configuration panels. A Qt widget library will be available soon.

A demonstration application is included in libcrn. It allows to test some features of the library on any image (see fig. 4). This tool is very handy to run simple algorithms over a given image without writing a new program.

III. 30 LINES FOR AN OCR

To illustrate libcrn's ease of use, we present a very simple OCR engine that is only 30 lines long. It works on a medieval manuscript excerpt written in capital letters with no spacing between words (see fig. 5). An occurrence of each letter in the alphabet was manually extracted and put in a folder named data, where each file name corresponds to the image's label. The source code is displayed in fig. 6. The first step (fig. 6.1) is to create a feature extractor that will be used to compare each character to the database. To do that, we use a FeatureSet, which can contain multiple elementary feature extractors. The FeatureSet will concatenate the feature vectors extracted by each FeatureExtractor. For simplicity, we use the four profile projections and the horizontal and vertical black pixels projections.

In step 2, we open each pre-labelled character image. As it is often impossible to know whether an image file contains RGB, grayscale or binary pixels, we directly store the image in a Block object. A Block automatically converts the input image to the format desired by the user (the default grayscaling and binarization methods can be changed programmatically at anytime). Is also contains named lists of sub-blocks that will be used later. The extracted features are stored in a list of shared pointers 5 to the base class Object.

The line segmentation is performed in 6.3. The document image is opened in the same way as the database images. We create a temporary BlockTreeExtractor that will extract the text lines and store them in a sub-block list named Lines.

Just before we actually extract the characters (fig. 6.4), we get an estimation of the mean stroke width. This will help us filter the noise. Connected components are extracted within each line. After that, each Block in the Lines subblock list will contain a sub-block list named Characters. The recursive sub-block lists can be used for more complex purposes and can even match an XML Alto's structure. Finally, connected components smaller than the mean stroke width are removed and the remaining ones are sorted from left to right.

The 5th and last step is the actual recognition. Each Block remaining in the Characters sub-sub-list represents a letter in the text. Its feature vector is extracted using the same FeatureSet as the database. We search its nearest neighbor in the database a retrieve a class number that can be used to compute the character's transcription. Now that our homemade OCR is fully described, we shall not discuss its performance: profile projections are not known to be the best features for medieval manuscript recognition! The purpose here was to illustrate the easiness of designing applications with libcrn. Variety of document analysis problems can be addressed, not restricted to ancient scripts: Printed pages or manuscripts may be considered. Historic or business documents may be processed. Even more borderline tasks are feasible (e.g.: text extraction from scenes, plate recognition, mobile applications, etc.).

IV. CONCLUSION

In this paper we introduced libcrn, a multiplatform (noncontaminating) open-source document image processing library written in C++11 and aimed at researchers and companies available for Windows (Visual C++ 2015), Linux, Ma-cOS and Android. Its API is comprehensively documented and libcrn's architecture follows modern C++ guidelines to facilitate the handling of the library and enforce its safe usage (e.g.: no memory management is required from the users and no leak can happen).

libcrn includes low-level image processing (expandable pixel formats, binarization, PDE. . . ), document images specific tools (connected components extraction, recursive block description. . . ), maths helpers (algebra, data analysis, geometry, signal processing. . . ) and pattern analysis algorithms (classification, clustering. . . ).

We described a short code example that provides OCR capacities in only 30 lines. This illustrated the use of feature extractors, segmentation providers and classification tools. Many other applications may be designed for research or industrial needs. libcrn has been useful in several projects and remains constantly improving.

Figure 1 .

 1 Figure 1. a,b) Angle histograms computed inside zones (rectangular or radial). c) 9 angle histograms computed with all pixels, weighted by their distance to automatically centered anchors.

Figure 2 .

 2 Figure 2. Two-ways inclusion measurement of two shapes.

Figure 3 .

 3 Figure 3. Sample of differential features (σ = 1.5 on the first line, σ = 4.5 on the second).

Figure 4 .

 4 Figure 4. Titus, libcrn's quick testing tool.

Figure 5 .

 5 Figure 5. Sample medieval manuscript.

https://github.com/Liris-Pleiad/libcrn

Code is already available on demand. Full integration will be done for the next release of libcrn

Shared pointers are memory management objects that deletes pointers when they are not referenced anymore.

/ / 1 . F e a t u r e e x t r a c t o r a u t o f e a t u r e E x t r a c t o r = c r n : : F e a t u r e S e t { } ; f e a t u r e E x t r a c t o r . PushBack ( s t d : : m a k e s h a r e d <c r n : : F e a t u r e E


Philippe Helluy
email: helluy@unistra.fr

Stability analysis of an implicit lattice Boltzmann scheme

We analyze the D1Q3 lattice kinetic model, which is the simplest kinetic model representing the isothermal Euler equations. We show that it is entropy unstable but that it can be made stable if the transport step is solved with an implicit numerical scheme.

Introduction

Lattice kinetic models are essential in computational fluid dynamics. They are the key ingredient of the Lattice Boltzmann Method (LBM). The idea is to construct a kinetic interpretation of a hyperbolic system of conservation laws with a minimal set of velocities. In this report we analyze the D1Q3 lattice kinetic model, which is the simplest kinetic model representing the isothermal Euler equations. We show that it is unstable but that it can be made stable if the transport step is solved with an implicit scheme. The unknown of the D1Q3 model is a three-dimensional distribution function f (x, t) ∈ R 3 , where x ∈ R and t ∈ [0, T ] are respectively the space and time variable. The distribution function satisfies transport equations with a BGK relaxation source term [START_REF] Lal Bhatnagar | A model for collision processes in gases. i. small amplitude processes in charged and neutral onecomponent systems[END_REF]

f i t + v i f i x = 1 ε (M (f ) i -f i ), i = 1 . . . 3, (1)
where we have noted partial derivatives with indices (f t = ∂ t f for instance). The kinetic velocity takes only three values

v = (-λ, 0, λ),
where λ is a positive real number. The fluid macroscopic variables are the density ρ(x, t), the momentum q(x, t) and the momentum flux z(x, t). As usual the fluid velocity is defined by u = q/ρ.

The macroscopic variables are recovered by computing discrete moments of f

  ρ q z   = P   f 1 f 2 f 3   , P =   1 1 1 -λ 0 λ λ 2 0 λ 2   . 1
The constant sound speed of the isothermal fluid is denoted by c > 0. The discrete Maxwellian state M (f ) is then given by

M (f ) = 1 λ 2   ρu(u -λ)/2 + c 2 ρ/2 ρ(λ 2 -u 2 -c 2 ) ρu(u + λ 2 )/2 + c 2 ρ/2   (2)
in such a way that

P M (f ) =   ρ q ρu 2 + c 2 ρ   .
Multiplying the kinetic equation ( 1) by P we obtain

ρ t + q x = 0, q t + z x = 0, (3)
z t + λ 2 q x = 1 ε (q 2 /ρ + c 2 ρ -z).
When ε → 0, then formally f = M (f ) and from (3) we see that ρ and u satisfy the isothermal Euler equations

∂ t ρ + ∂ x (ρu) = 0, ∂ t (ρu) + ∂ x (ρu 2 + c 2 ρ) = 0. (4)
The model ( 1), ( 2) is thus a minimalistic abstract kinetic interpretation of the isothermal Euler equation. It is also denoted as the "D1Q3" model in the lattice-Boltzmann community [START_REF] Yh Qian | Lattice bgk models for navier-stokes equation[END_REF]. It can be extended to higher dimensions. For instance, in two or three dimensions it becomes the D2Q9 or D3Q27 models.

Numerical method and asymptotic expansion

A traditional method for solving numerically (1) is the first order Lie splitting algorithm. For applying one time step of the splitting algorithm, we start from a state that is close to equilibrium: f = M (f ) + O(ε). We first apply the free transport equation for a duration of ∆t

f t + v • f x = 0.
Then in a second stage of the same duration ∆t we apply the local BGK return to equilibrium

f t = 1 ε (M (f ) -f ).
In the case of the D1Q3 model, this approach can lead to instabilities that are sometimes observed in LBM simulations [START_REF] Dubois | Stable lattice boltzmann schemes with a dual entropy approach for monodimensional nonlinear waves[END_REF]. Therefore, we replace the exact transport step by a first order implicit solver in time. Assuming high precision of the solver in the x variable the effect of the implicit solver can be modeled by

f (x, t) -f (x, t -∆t) ∆t + vf x (x, t) = 0. (5)
By a Taylor expansion, we find the equivalent equation of the implicit solver (5)

f t + vf x - ∆t 2 v 2 f xx = O(∆t 2 ). (6)
In a second step, we solve the differential equation exactly

f t = 1 ε (M (f ) -f ) = M -I ε f.
This is easy because during the relaxation step ρ, q, and thus M (f ), are constant. In the following, ε is a small parameter, but we assume that the vector field M is restricted to a manifold of f 's on which

M -I ε f = O(1). (7)
In the literature this hypothesis is often formulated by saying that f remains close to a Maxwellian state and that the initial data are "well-prepared". Hypothesis ( 7) is crucial because it will allow us to apply the Baker-Campbell-Hausdorf (BCH) formula with the good ordering for estimating the equivalent equation of the splitting algorithm. Let us also point out that we assume that (7) remains true even if ε ∼ ∆t or ε ∼ ∆t 2 for instance. For a more precise analysis of this hypothesis, we refer to [START_REF] Wanner | Solving ordinary differential equations, stiff and differential-algebraic problems[END_REF] (Section VI.3 pages 388-392).

In the Lie formalism, one time-step of the splitting scheme can be written

ϕ(τ ) = exp(τ M -I ε ) exp(τ (-v∂ x + 1 2 τ v 2 ∂ xx )) + O(τ 3
).

Now we apply the BCH formula

exp(A) exp(B) = exp(A + B + 1 2 [A, B] + 1 12 ([A, [A, B]] + [B, [B, A]]) + • • • ).
We obtain

ϕ(τ ) = exp (τ L) + O(τ 3 ), with L = -v∂ x + 1 2 τ v 2 ∂ xx + M -I ε + 1 2 τ M -I ε , -v∂ x .
Therefore at second order in time, the equivalent equation of the scheme is

f t + vf x - ∆t 2 v 2 f xx - 1 2 ∆t M -I ε , -v∂ x f = M (f ) -f ε .
For expressing the Lie bracket in a more convenient way, we introduce the matrix

V =   -λ 0 0 0 0 0 0 0 λ   .
Then the Lie bracket becomes

[M -I, -v∂ x ] f = -V ∂ x M (f ) + M (f )V ∂ x f = (M V -V M )∂ x f
Now we go back to variables (ρ, q, z). After some computations, we find that

P [M -I, -v∂ x ] P -1   ρ q z   =   0 0 0 -c 2 + u 2 -2u 1 0 c 2 -λ 2 -u 2 2u   ∂ x   ρ q z   .
We obtain the equivalent equations solved by the splitting algorithm at order 2 in ∆t

ρ t + q x - ∆t 2 z xx = 0, q t + z x - ∆t 2 λ 2 q xx = ∆t 2ε (u 2 -c 2 )ρ x -2uq x + z x , (8)
∂ t z + λ 2 ∂ x q - ∆t 2 λ 2 z xx = 1 ε (q 2 /ρ + c 2 ρ -z) + ∆t 2ε (c 2 -λ 2 -u 2 )q x + 2uz x .
On this equation we will now assume that 1 ∆t > ε. We freeze ∆t and perform a Chapman-Enskog expansion when ε → 0. The second equation implies that when ε → 0

z x = (c 2 -u 2 )ρ x + 2uq x = (q 2 /ρ + c 2 ρ) x + O(ε)
and is thus redundant with

z = q 2 /ρ + c 2 ρ + O(ε).
The third equation in (8) gives

z = q 2 /ρ + c 2 ρ -ε ∂ t z + λ 2 ∂ x q + ∆t 2 (c 2 -λ 2 -u 2 )q x + 2uz x + O(ε∆t).
We need to rewrite the factor in ε:

∂ t z + λ 2 ∂ x q,
with only spatial derivatives. At leading order we have

∂ t z = ∂ t q 2 /ρ + c 2 ρ + O(ε + ∆t) = 2q ρ q t - q 2 ρ 2 ρ t + c 2 ρ t + O(ε + ∆t).
But q t = -z x + O(ε + ∆t) and ρ t = -q x + O(ε + ∆t) thus z t = -2uz x + u 2 q x -c 2 q x + O(ε + ∆t).

Then z t = -2u 2uq x + (c 2 -u 2 )ρ x + (u 2 -c 2 )q x + O(ε + ∆t).

Finally z t + λ 2 q x = -3u 2 + λ 2 -c 2 q x -2u c 2 -u 2 ρ x + O(ε + ∆t).

We then obtain the equivalent viscous equation of the splitting method

ρ t + (ρu) x = κ ∆t 2 z xx , (ρu) t + ρu 2 + c 2 ρ x = κ ∆t 2 λ 2 q xx + D x ,
with κ = 1 (effect of the implicit solver) or κ = 0 (exact transport solver) and

D = (ε + ∆t 2 ) λ 2 -c 2 -3u 2 q x + 2u u 2 -c 2 ρ x .

Stability analysis

Now we want to analyze the entropy stability of the second order term when ε → 0. For this, we define

and thus second order equivalent equations become

An entropy of the Euler equations is

We know that with this choice there exists an entropy flux G(w) such that

Multiplying (9) on the left by S (w), integrating by part in x and neglecting boundary terms, we obtain the entropy dissipation balance

A sufficient condition for entropy dissipation is thus that E(w) = S (w)A(w) is a positive matrix. The D1Q3 model is generally used for subsonic flows. When κ = 0 (no numerical viscosity) E(w) has always a negative eigenvalue and the scheme is thus unstable. The negative eigenvalue has a minimal modulus if λ = √ 3c and is then of order O(u 6 ). It justifies the fact that the scheme can, however, be applied in practice on relatively coarse meshes for low Mach number flows. When κ = 0 Taylors expansions in u show that

If λ is large enough, the scheme is thus stable for low Mach flows.


Fatma Chabchoub
email: chabchoub.fatma@gmail.com

Yousri Kessentini
email: yousri.kessentini@gmail.com

Slim Kanoun
email: slim.kanoun@gmail.com

Veronique Eglin
email: veronique.eglin@insa-lyon.fr

SmartATID: A mobile captured Arabic Text Images Dataset for multi-purpose recognition tasks

Keywords: Smartphone Arabic document capture database, mobile OCR, capture-based distortions

Today's smartphones are able to capture documents with a good and simple way as any personal scanners. The captured document images need to be processed by specific and automated document processing systems. The systems are dedicated to textual content analysis, indexing and recognition. For instance, they may be used for font identification, writer identification and word or line segmentation. The state-of-theart works lack comprehensive database for Arabic document images which are captured by mobile phones. This paper presents the first public offline images database for both printed and handwriting Arabic mobile captured documents, named "SmartATID". The document images of the database are acquired under varying capture conditions (blur, perspective angles and light). This causes photometric and geometric distortions that influence the performance of OCR process but also the page segmentation in lines and paragraphs. Each document image of our database is provided with a ground truth file that contains the exact text transcription and all numerical capture parameters used for each image capture. The database is freely and publicly usable by the research community at the following address http:// sites.google.com/site/smartatid.

INTRODUCTION AND RELATED WORK

Document image processing is today considerably influenced by evolutions of data acquisition systems and embedded technologies for new innovative mobile applications. Especially, the smartphone revolution has impacted the way people use information and at the very beginning of the processing flow the way they digitize them. A smartphone-captured document image is at the origin of the new challenges associated to content image analysis and recognition by automated embedded document image processing systems ( [START_REF] Liang | Camera-based analysis of text and documents: a survey[END_REF], [START_REF] Shafait | Document image dewarping contest[END_REF] and [START_REF] Liang | Geometric rectification of camera-captured document images[END_REF]). These systems need to deal with the camera sensors, the resolution of the images, the light conditions, the physical distortions of the paper (folds etc.), the perspective distortion, the out-of-focus blur and/or the motion blur during capture [START_REF] Liang | Camera-based analysis of text and documents: a survey[END_REF].

In order to face this new challenge, we all need to develop reliable solutions for improving best suited image quality required to efficient embedded text recognition systems. Recognizing texts from images captured by mobile phones offers access to significant data. By using those data, we intend to perform new information retrieval system, reuse contents on demand and share them through networks and users. In that context, it becomes a necessity to make available standard databases that closely match the real world scenarios and offers new services to testify the new embedded mobile systems. Such standardized databases serve to help researchers to develop their solutions. Besides, they provide a baseline benchmark between different systems within evaluation campaigns and competitions.

The analysis of the literature shows an absolutely terrible lack of databases and baseline benchmarks for digitizing mobile captured documents. Indeed, the only available dataset comprising of documents captured in real conditions with various photometric and geometric distortions was generated for ICDAR2015 Competition on Smartphone Document Capture and OCR by Burie et al. [4]. This dataset contains 12100 document images captured from 50 different paper documents with real content from wiki-books and cooking recipes from Internet. All documents contain single column English text printed with multiple scales, fonts, fontfaces and colors. At least 240 different images are taken per document under varying capture conditions (perspective angles, light and blur). Special mention may be made for the research community working on quality assessment of smartphonecaptured-document images who created recently two datasets for evaluating and benchmarking their methods.

The first one was proposed in 2013 by Kumar et al. [5]. This dataset deals with only one type of documents, one type of capture distortions (out-of-focus blur) and the images are captured using one smartphone. The dataset of Kumar et al. [START_REF] Liang | Geometric rectification of camera-captured document images[END_REF] has 29 different documents used to capture 375 images with varying degrees of out-of-focus blur.

The second one was named Smart Doc-QA and was created in 2015 by Nayef et al. [6]. This dataset deals with different real paper document types, single and multiple capture distortions and the images are captured using two smartphones. Smart Doc-QA [START_REF] Nayef | SmartDoc-QA: A Dataset for Quality Assessment of Smartphone Captured Document Images -Single and Multiple Distortions[END_REF] contain 4260 document images captured from 30 different paper documents with representative value of different distortions (Light, perspective and blur).

A deep observation on the state-of-the-art of existing datasets for mobile captured documents shows there is no standard Arabic datasets for both printed and manuscript contents. Consequently, this paper presents a detailed description of the constitution of an Arabic multi-font and multi-size printed/handwritten text database dedicated to recognition and content based information retrieval tasks.

Our mobile printed and handwritten Arabic document images database is a collection of smartphone-captured documents. This database will be useful mainly for document analysis scenarios (page and line segmentation tasks), Arabic text recognition and also could be helpful for assessing document image quality, and improving it.

The paper is organized as follows: in section 2, we present the overall acquisition protocol and the different distortions correlated to a mobile images capture. Then in section 3 we present the ground truth representation including the text transcription encoding and different tags for the acquisition parameters required for further recognition tasks evaluations.

II. OVERVIEW OF "SMARTATID" DATABASE

In this section, we present our database of Arabic printed/handwritten texts for mobile captured documents.

A. Documents image selection

In order to create the Smartphone-captured Arabic printed text images dataset, we selected a set of document images from APTID/MF database [START_REF] Jaiem | Database for Arabic Printed Text Recognition Research[END_REF] (Arabic Printed Text Image Database /Multi-Font). This selection is composed by 32 text-blocks images which are scanned at 300 dpi resolution in grayscale format and whose content is organized in many sizes and fonts among the 10 fonts presented in Fig. 1. The set of selected images are printed with a Brother printer, in total there are 116 paper documents with multiple scales and fonts used to capture 16472 images with mobile phones for our printed text image Dataset. Fig. 2 shows examples of those paper documents.

With the same process applied to create the dataset of Smartphone-captured Arabic printed text images, we also build the Smartphone-captured Arabic handwritten text images dataset. The Arabic handwritten documents are selected from ALTID database [START_REF] Chtourou | ALTID: Arabic/Latin Text Images Database for recognition research[END_REF] (Arabic/Latin Text Images Database for recognition research). As the handwritten dataset of ALTID database is inherited from the printed document pages of APTID/MF database, we choose to use the same handwritten text images than the Smartphone-captured Arabic printed text ones. This selection is made up of four Arabic handwritten texts written by eight writers and presented according two layouts:

 a formal layout: the author writes the text in a formal style using a lined paper as a background template with 2cm for line spacing.  an informal layout: the author rewrites the same text in a freely without any layout constraints.

Consequently, there are 64 documents (32 different formal documents and 32 different informal documents) used to capture 9088 images with mobile phones of our handwritten text image Dataset. Fig. 3 shows examples of those handwriting documents presented with layout constraints and without any. For our two datasets, the documents have been chosen with plain and simple background in order to reduce the impact of the background effects on the automatic pages processing (i.e.: OCR results). Anyway the quality of the captured document images has a big influence on OCR performance.

B. Logistics for the data capture

Despite the lack of automatic material for the acquisition, we succeeded in capturing the document images for the database in a precisely repeatable and controlled environment. As a consequence, the majority of steps have been performed manually in real conditions by setting various photometric and geometric distortions to images.

In our dataset we mainly examine scene-related distortions such as light conditions, out-of-focus blur, motion blur and perspective angles distortions. To ensure real life scenarios of the possible light conditions, we performed image capture outdoors and image capture indoors with strictly controlled light conditions. In order to simulate the possible capture positions, the out-of-focus blur and the motion blur during the captures, we have employed a rolling adjustable height and swinging Camera.

For the device-related distortions, we have employed two modern smartphones, Samsung Galaxy S6 edge and I phone 6S plus, whose cameras are fitted with different sensor technologies and able to capture images at different resolutions (13MP and 8MP).

The sequence of steps dedicated to the capture is synchronized by a human so as to ensure the generation of digitized data in real situation. As examples of human interactions, we can mention: the manual change of paper document, of light conditions and the location of the focal distance from the smartphone camera and the center of the document page. Once a desired position and other capture conditions are achieved, the agent responsible to the image capture communicates with the smartphone through a Bluetooth to trigger the capture. During the creation of both datasets, the captures are taken with representative values of different types of distortions (see capture parameters below). For each image, the information about the document content and the capture conditions are stored as ground truth for upcoming evaluation.

1) Fixed Capture Parameters:

The documents capture is established through different fixed parameters setting that guarantee a controlled acquisition and similar for all images of the dataset.

 Background: simple and plain with a contrasted color with the paper documents to be captured. The background is chosen to facilitate the page border detection process.

 Document layout and dimensions: A4 paper size with a fixed orientation.

 Smartphone setting: flash is always deactivated Fig. 4 shows the fixed capture parameters related to the documents and the background. 2) Variable Capture Parameters:

The documents capture is also established through different variable parameters setting that ensure multiple acquisition conditions and offer the desired diversity of images for the dataset. The detailed capture protocol and integration of these constraints are presented in section 3 and 4.

 Smartphone camera: 2 smartphones o Distance between the camera and the document: 1 value (35cm)

In the following section, we detail our protocol for images capture and we detail the choice of specific values for the variable capture settings mentioned above. The captures are organized into two categories: single captured distortions (singly distorted document images) and multiple captured distortions (multiply distorted document images). The presence of multiple causes of distortions is undoubtedly the expression of the reality of the ground. Nevertheless, possessing singly distorted document images is useful to assess the impact of each type of distortions taken individually for the evaluation of OCR results.

For both distortion categories, we take every capture with the following two smartphones:  Samsung Galaxy S6 edge (camera: 13MP)  IPhone 6S plus (camera: 8MP)

3) Single Distortions:

Under "single distortions" constraints, we study independently light conditions, out-of-focus blur and motion blur. All images with single captured distortions are captured at the same position, called "parallel position with a minimal perspective distortion" (i.e.: a position where the perpendicular distance between the center of the document and the camera of the smartphone is 35cm and both angle of mobile rotation around Y-axis and of mobile rotation around X-axis equal zero. Under these conditions, we perform the capture process as follows:

For light distorted images, five successive images are taken for each document under five light conditions mentioned below.

 Light condition 1: Day light only (without any artificial lights) outdoors.

 Light condition 2: Day light only (without any artificial lights) indoors.

 Light condition 3: Day light + Economic lamp Led indoors.

 Light condition 4: Night + table lamp light.

 Light condition 5: Table lamp light + an object casting a shadow on a large part of the document.

For out-of-focus blur, all the images are also captured at the same parallel position presented above, with the light condition fixed to "day light + Economic lamp Led indoors". The smartphone camera is focused on the document at a distance closer than 35-cm (22-cm), then the image is captured at the position with a 35-cm distance. This leads to a controlled out-of-blur capture of the pages. The same operation is performed four times in order to capture four different images at varying out-of-focus blur degrees. Each time, the focus point is lcm shorter to the document, whereas the capture is taken at the same distance of 35cm.

For motion blur, we consider the same light condition and camera position as the out-of-focus blur captures. The motion blur is executed as follows. While the rolling support is moved at a specific speed and aimed to a certain direction, the smartphone camera is started to take the capture. The capture can occur at any moment during the motion, hence producing images with different degrees of severity of blur. Two images are captured this way for each document (image with horizontal motion blur and image with 2D motion blur). Fig. 5 shows three different images of the same document captured at the third light condition and the parallel position. The first image presents a focused (sharp) capture, the second presents a blurry image due to out-offocus blur and the third presents a blurry image due to motion blur. Finally the overall process has produced a total of 11 captures. The same operation is repeated for each smartphone camera, where we ensure that each capture shows only one type of distortion.

4) Multiple Distortions:

Here we use a combination of different capture conditions for each image. We have chosen 3 light conditions, 5 camera positions and 3 blur values. Additionally, we take a reference capture, which is not blurry focused or blurry sharp, at each combination of light condition and position. This creates 60 captures per document for each dataset. For the light conditions, we have selected conditions 3, 4 and 5 mentioned section 3. We have chosen to express the position of the camera with respect to the document in the following way (perspective 1, perspective 2, and distance), in respect to what has been mentioned above in the variable capture parameters. We have also considered five different positions: (0°, 0°, and 35cm), (_10°, _5°, 35cm), (_10°, 5°, 35cm), (_5°, 10°, 35cm) and (5°, 10°, 35cm) combined with only two of the out-of-focus blur values mentioned above and the 2D motion blur. Fig. 6 shows examples of captured images for multiple distortions.

D. Ground Truth File drscription

An essential component of any database is the presence of ground truth data [START_REF] Ahmed | A Generic Method for Automatic Ground Truth Generation of Camera-captured Documents[END_REF]. Each image of our two datasets, the smartphone-captured Arabic printed documents dataset and the smartphonecaptured Arabic handwritten documents dataset, is provided with the following ground-truth information:

 The transcription of the text in a document.

 The capture parameters (distortion types and values)

and the ID of a captured document.

 A sharp "reference" image of the document at each combination of position and light.

An example of such XML file for our smartphonecaptured Arabic handwritten text images dataset is given in figure Fig. 7.  Smartphone: in this element, we specify the type of camera and the resolution value used to capture the image document.

 Distortion: in this element, we indicate the category of the capture (singly distorted document images or multiply-distorted document images) and we present a sub-elements for each capture parameters such as: light condition, out-of-focus blur, motion blur and perspective.

Each text image in our smartphone-captured Arabic printed text images dataset is fully described using the same Xml file with a simple difference at the "Content" element where we specify the font name, the number of line of Arabic text (nbline) and sub-elements for each line where we have her identical transcription and the number of her content tokens.

III. CONCLUSION

In this paper we address the lack of Arabic databases for mobile captured documents, we have created a novel database, named Smart ATID, which contains an off-line Smartphone-captured Arabic printed text images dataset and an off-line Smartphone-captured Arabic handwritten text images dataset. This database proposes both single and multiple capture distortions for subsets of the images, allowing to address the specific issue of real conditions. Additionally, it includes a complete ground-truth with type and amount of each distortion contained in the images and reference images. We hope this database will serve the scientific community for many years to come for benchmarking mobile Arabic recognition methods.

Our future work consists in the improvement of human motion reproduction and the extension of our two datasets to more document types, more smartphones and more distortions.

Figure 1 .

 1 Figure 1. 10 Arabic fonts used in APTID/MF [5].

Figure 2 .

 2 Figure 2. Examples of Arabic printed documents used in the "SmartATID" database.

Figure 3 .

 3 Figure 3. Examples of Arabic handwritten documents used in the "SmartATID" database: (a) Formal (constrained) handwritten text, (b) Informal (unconstrained) handwritten text.

Figure 4 .

 4 Figure 4. The rolling support holding smartphone over a document on a fixed simple background.

Fig. 4

 4 Fig. 4 presents a photo of the rolling support holding the smartphone beside a document ready to be captured. C. Capture Protocol As mentioned previously, our Arabic database contains two datasets: one is composed by smartphone-captured Arabic printed text images and the second by smartphonecaptured Arabic handwritten text images. The first smartphone-captured Arabic printed document dataset has a total of 16472 document images captured from 116 different paper documents. Hundred and forty two different images are captured per document (71 captures per phone). The second smartphone-captured Arabic handwritten document dataset has a total of 9088 document images captured from 64 different paper documents. In the same way than for the printed dataset, 142 different images are captured per document (71 captures per phone).



  Light: 5 light conditions  Out-of-focus blur: 4 values  Motion blur: 2 types. We use two motions blur, a horizontal one and a 2D one.  Position of the smartphone camera with respect to the paper document (5 positions): o Perspective 1: Longitudinal incidence angle (mobile rotation around Y-axis). o Perspective 2: Lateral incidence angle (mobile rotation around X-axis).

Figure 5 .

 5 Figure 5. Examples of captured images for single distortions.

Figure 6 .

 6 Figure 6. Examples of captured images for multiple distortions.

  (a) A sharp image-Focused. (b) A blurry image-out-of-focus. (c) A blurry image-2D motion. (d) Cropped image (a). (e) Cropped image (b). (f) Cropped image (c). (a) Capture position (5°,10°,35cm) light condition #2, sharp (focused) (b) Capture position (-5°,10°,35cm) light condition #3, out-of-focus blur (c) Capture position (0°,0°,35cm) light condition #4, sharp (focused) (c) Capture position (-10°,5°,35cm) light condition #5, 2D Motion blur.

Figure 7 .

 7 Figure 7. Example of XML file present the ground truth information of smartphone-captured Arabic handwritten text images dataset.


Li Wang

Ali Mohammad-Djafari

Nicolas Gac

Mircea Dumitru

COMPUTED TOMOGRAPHY RECONSTRUCTION BASED ON A HIERARCHICAL MODEL AND VARIATIONAL BAYESIAN METHOD

Keywords: Computed Tomography (CT), Non Destructive Testing (NDT), Hierarchical Model, Bayesian Method, Student-t prior models

In order to improve the quality of X-ray Computed Tomography (CT) reconstruction for Non Destructive Testing (NDT), we propose a hierarchical prior modeling with a Bayesian approach. In this paper we present a new hierarchical structure for the inverse problem of CT by using a multivariate Studentt prior which enforces sparsity and preserves edges. This model can be adapted to the piecewise continuous image reconstruction problems. We demonstrate the feasibility of this method by comparing with some other state of the art methods. In this paper, we show simulation results in 2D where the image is the middle slice of the Shepp-Logan object but the algorithms are adapted to the big data size problem, which is one of the principal difficulties in the 3D CT reconstruction problem.

INTRODUCTION.

The concept of an inverse problem is now a familiar one to most scientists and engineers, particularly in the field of signal and image processing. In a nutshell, it involves the estimation of an unknown quantity from another observable quantity which is linked to it through a mathematical relationship known as the forward model. In this paper, we are concerned with CT. The domains of CT application range from medical imaging to NDT. X-ray computed tomography (X-ray CT) is nowadays a common technology in NDT. In X-ray tomography, the reconstructed 3D image represents the attenuation function of the volume being examined. The 3D X-ray CT image reconstruction, which is recently a popular topic in inverse problem, is usually implemented in GPU to reduce the computation time [1]. To model and solve an inverse problem, firstly a mathematical model is considered. In this paper, Cartesian coordinates is used. Different coordinates can also be applied according to the character of model. [START_REF] Thibaudeau | Fully 3d iterative ct reconstruction using polar coordinates[END_REF] has presented a 3D CT reconstruction using polar coordinates, and [START_REF] Knaup | Ct image reconstruction using hexagonal grids[END_REF] has introduced the method using hexagonal grids. Various methods have been used for doing the inversion in the forward model. The most widely used is MAP estimation method [START_REF] Mohammad-Djafari | Gauss-markov-potts priors for images in computer tomography resulting to joint optimal reconstruction and segmentation[END_REF], the Markov Chain Monte Carlo (MCMC) method [START_REF] Gamerman | Markov chain Monte Carlo: stochastic simulation for Bayesian inference[END_REF], Discrete Algebraic Reconstruction Technique (DART) reconstruction method [START_REF] Kees | Dart: a practical reconstruction algorithm for discrete tomography[END_REF] and the super-resolution method [START_REF] Van Aarle | Super-resolution for computed tomography based on discrete tomography[END_REF]. In this article we adapt a Bayesian approach. The Bayesian approach can be done via a hierarchical model, which assigns priors for the image and the noise. In this Bayes framework, by using the priors and the likelihood, we obtain the posterior distrubution, from which we estimate the unknowns of the model, together with their corresponding variances, via two techniques: the Posterior Mean (PM) estimation via Variational Bayesian Approach (VBA) and the Joint Maximum A Posterior (JMAP).

One of the estimation that we used is the PM via VBA. More details of VBA are given in [START_REF] Mohammad-Djafari | Gauss-markov-potts priors for images in computer tomography resulting to joint optimal reconstruction and segmentation[END_REF][START_REF] Michael I Jordan | An introduction to variational methods for graphical models[END_REF][START_REF] Dimitris G Tzikas | The variational approximation for bayesian inference[END_REF][START_REF] Tzikas | Variational bayesian blind image deconvolution with student-t priors[END_REF]. For the Bayesianbased methods, by assuming the prior distributions and likelihood distribution, the posterior distribution can be obtained. Different assumptions for the prior distribution will encode different prior knowledgement. Some prior models are discussed in [START_REF] Gu | Bayesian sparse estimation using double lomax priors[END_REF][START_REF] Ting | Sparse image reconstruction for molecular imaging[END_REF][START_REF] Chen | Nonlocal prior bayesian tomographic reconstruction[END_REF][START_REF] Mohammad | Hierarchical markovian models for 3d computed tomography in non destructive testing applications[END_REF][START_REF] Molina | Variational posterior distribution approximation in bayesian emission tomography reconstruction using a gamma mixture prior[END_REF]. In the mathematical model, f ∈ R N represents all the pixels or voxels of the object and g ∈ R M represents all the projection data. The projection system is denoted by the linear matrix H ∈ R M ×N , which depends on the geometry of the acquisition system. The forward model is expressed as:

g = Hf + , (1)
with ∈ R M representing the noise.

The inverse problem consists of using the real projection data to infer the original object f . Filtered Back Projection (FBP):

f = H t H -1 H t g is a common inversion methods.
Generally, the problem is ill-posed. To solve this problem, the regularization theory and Bayesian inversion are widely used.

The regularization approach reconstructs f by minimizing the criterion:

f = arg min f g -Hf 2 + λR (f ) (2)
where λ is a regularization parameter and R (f ) indicates the criterion penalty, for example

R (f ) = f 2 , R (f ) = |f |.
Our Bayesian method uses a hierarchical model with proba-bility distribution functions (pdf) for the prior on f , the likelihood function from the data, and relevant parameters. The rest of this paper is organized as follows: in Section 2, the Bayesian method is presented, Section 3 provides the new hierarchical model. Different reconstruction algorithms are introduced in Section 4. We present the experimental results in Section 5, and conclusions are drawn in Section 6.

BAYESIAN APPROACH BASIS.

Using the forward model, Equation (1), and assigning the likelihood p(g|f , H) and the prior law p(f ), the Bayesian Rule provides the posterior law:

p(f |g, H) = p(g|f , H)p(f ) p(g|H) (3)
where

p(g|H) = p(g|f , H)p(f )df (4)
We proceed to infer the unknown f by using the MAP and PM algorithms.

THE HIERARCHICAL MODEL.

Firstly, we assign a pdf for , the noise in Equation (1). We consider a non-stationary noise, so we propose a zero-mean Normal distribution, with unkown variance v . In order to estimate the variance, we assign it a pdf. We choose an Inverse Gamma (IG) with parameters α and β defining expected mean and uncertainty.

p( |v ) = N ( |0, v I) p(v |α 0 , β 0 ) = IG(v |α 0 , β 0 ) (5)
The image f can be expressed as:

f = Dz, (6)
where z is the sparse representation of f and D is the corresponding sparse transformation, for example, the wavelet transformation. Student-t probability density function is assigned to z in order to enforces the sparsity and thus preserves the edges of image. Many other distribution could also be used to enforce sparsity. Those can be divided in 3 categories: Generalized Gaussian (GG), mixture of Gaussian or heavy-tailed distribution. Student-t is one of the heavy-tailed distributions, with the convenient that it can be expressed as the marginal of a Normal-IG distribution.

In this paper, we estimate the edges z first, with the image f following.

The Student-t distribution can be expressed as the marginal of a Gaussian-IG distribution:

St(x|α, β) = ∞ 0 N (x|0, v)IG(v|α, β) dv (7)
Assuming that z is separable, i.e. p(z) = N j p(z j ), we can easily represent the prior distribution for all the elements of the vector z:

p(z j |α zj0 , β zj0 ) = St z j |α zj0 , β zj0 = ∞ 0 N (z j |0, v zj )IG(v zj |α zj0 , β zj0 )dv zj (8) α z0 , β z0 ? v z ? z ? D α 0 , β 0 ? v ? f ) ? H g
Fig. 1: The Hierarchical Model.

Equation 5, together with the Student-t prior model proposed to f which is expressed as in Equation 8, gives the proposed hierarchical model. A graphical representation of this hierarchical model is shown in Figure 1, and the corresponding equations in Equation 9.

         p(g|z, v ) = N (g|HDz, v I) p(z|vz) = N (z|0, V z ), V z = diag [vz] , vz = [vz 1 , • • • , vz N ] p(v z |α z0 , β z0 ) = N j IG(v zj |α z0 , β z0 ) p(v |α 0 , β 0 ) = IG(v |α 0 , β 0 ) (9)
The posterior distribution is obtained from Equation 9 via Bayes rule:

p(z, v z ,v |g) ∝ p(g|z, v )p(z|v z ) p(v z |α z0 , β z0 )p(v |α 0 , β 0 ). ( 10
)
From here we can apply various algorithms to solve the reconstruction problem. In this paper, we will discuss the JMAP method and the PM via VBA method.

RECONSTRUCTION ALGORITHMS.

The main Bayesian computational tools are: Maximum A Posteriori (MAP) method [START_REF] Mohammad-Djafari | Gauss-markov-potts priors for images in computer tomography resulting to joint optimal reconstruction and segmentation[END_REF], Markov Chain Monte Carlo (MCMC) method [START_REF] Gamerman | Markov chain Monte Carlo: stochastic simulation for Bayesian inference[END_REF], Expectation-Maximization (EM) method [START_REF] Šmídl | The variational Bayes method in signal processing[END_REF] and the Variational Bayesian (VB) method.

In this paper, we will mainly consider the MAP method and the VBA methods.

The Joint MAP algorithm.

In JMAP method, we estimate f and the other parameters involved in the model, i.e. the variances v and v z , iteratively by maximizing the posterior law:

( z, v z , v ) = arg max (z,vz,v ) {p (z, v z , v |g)} (11)
As we have mentioned above, the size of data in our research is so huge that we can not calculate directly the inverse of matrix in the steps of updating z. Instead, we use the gradient descent method to avoid the costly computation. The updating rules are summarized below:

         iter : z (k+1) = z (k) -γ (k) z ∇J ( z (k) ) v zj = βz j0 + 1 2 z 2 j αz j0 + 3 2 v = β 0 + 1 2 g-HD z 2 α 0 + M 2 +1 (12)
where

       J (z) = 1 2 v -1 g -HDz 2 + 1 2 Y z 2 , Y = V -1 2 z γ (k) z = ∇J ( z (k) ) 2 v -1 HD∇J ( z (k) ) 2 + Y ∇J ( z (k) ) 2
and ∇J (•) is the gradient of J (•).

The VBA algorithm.

We compute the PM via VBA method. Firstly, the posterior distribution p(z, v z , v |g) is approached by a separable distribution q(z, v z , v ):

p(z, v z , v |g) ≈ q(z, v z , v ) = q 1 (z)q 2 (v z )q 3 (v ). ( 13
)
This approaximation is done by minimizing the Kullback-Leibler divergence, defined as:

KL (q(z, v z , v ) : p(z, v z , v |g)) = q(z, v z , v ) ln q(z, v z , v ) p(z, v z , v |g) dzdv z dv (14)
The details of this minimization are discussed in [START_REF] Mohammad-Djafari | Bayesian inference with hierarchical prior models for inverse problems in imaging systems[END_REF]. Thanks to the conjugacy property of the prior model, we obtain the conjugate distributions for z and the parameters.

q(z j ) = N (z j | m zj , σ 2 zj ) where      m zj = v -1 bj v -1 hj + v -1 z j σ 2 zj = 1 v -1 hj + v -1 z j q(v zj ) = IG(v zj | α zj , β zj ) where α zj = α z0 + 1 2 β zj = β z0 + 1 2 m 2 zj + σ 2 zj q(v ) = IG(v | α , β ) where α = α 0 + M 2 β = β 0 + 1 2 g -HD m z 2 + j h j σ 2 zj ( 15
)
where

       h j = D t H t HD jj b j = D t H t (g -HD m z ) j + h j m zj v -1 = α β , v -1 zj = αz j βz j 5. NUMERICAL EXPERIMENTATION.
Several experiments in 2D image analyse have been carried out in order to demonstrate the properties of the proposed model. The original image is the middle slice of Shepp-Logan of size 128 × 128. The sparse operator D is the Haar wavelet transform. The projection matrix H is only used in the geometry projection Hf and back-projection H t g operators. The reconstructed images presented in Figure 2 show that the proposed hierarchical model can lead to results with more details and preserve better the edges of image.

The relative errors of reconstruction

δ f = f -f 2 / f
2 using different algorithms are shown in Figure 3. Figure 3(a) corresponds to a projection without noise while Figure 3(b) corresponds to a projection with 20dB noise. We compared the JMAP and VBA method using the proposed model with Least Square (LS) method, quadratic regularization (QR) method with fixed λ parameter and Gauss Markov Potts (GMP) methods. With the proposed model, both the JMAP and VBA algorthms works better than the non-hierarchical models, and their performance are as good as the GMP method which also can be considered as a hierarchical one.

The JMAP method with the three-level hierarchical model has a better convergence property than the other methods.

Figure 4 illustrate the influence of the noise and number of projections in our application. Figure 4(a) shows that the error of reconstruction is in inverse proportion to the noise value. projections, the results won't change a lot, but 45 projections is not enough for a big data CT reconstruction problem.

CONCLUSION

We presented a three-level hierarchical model for the piecewise continuous tomography reconstruction problem. We used a heavy-tailed Student-t distribution to model the sparse representation of image and the errors in order to achieve the sparsity property and model the non-stationality of the noise.

Experimental results demonstrate the advantages of using the proposed three-level hierarchical model and a Student-t distribution for the prior model. Future work is to apply this method to the 3D CT problem where the GPU processor is used to reduce the computation time.

[1] Nicolas Gac, Alexandre Vabre, Ali Mohammad-Djafari, et al., "Multi gpu parallelization of 3d bayesian ct algorithm and its application on real foam reonconstruction with incomplete data set," Proceedings FVR 2011, pp. 35-38, 2011.

Figure 4 (Fig. 2 :

 42 Figure 4 illustrate the influence of the noise and number of projections in our application. Figure4(a)shows that the error of reconstruction is in inverse proportion to the noise value. Figure4(b) indicates that by using 180 projections and 90

Fig. 3 :

 3 Fig. 3: Comparaison of the convergency of different methods using (a) a projection without noise and (b) a projection with SNR=20dB.

Fig. 4 :

 4 Fig. 4: Performance of the different methods as a function of (a) SNR with a great number of projections and (b) different number of projections. All the results are obtained with 30 iterations.