Walther Nagler

Martin Ebner

Nick Sherbakov

Flexible teaching with structured micro-content How to structure content for sustainable multiple usage with recombinable character

Keywords: webbased learning, micro-content, authoring tool, content standards, Web2.0, e-Learning

This paper reflects the possibilities of reasonable structured content production and management as it is exercised by the Work Group Social Learning (WG Social Learning) at Graz University of Technology (TU Graz). By splitting content into standardised pieces adjusted to lecture-units they turn highly qualified for multiple usages and commutability purpose. Three different examples of such micro-content and a way of implementing it into online and offline learning scenarios fitting the requirements of eLearning2.0 are presented.

Introduction

What is elearning2.0?

Elearning2.0 as described by Wikipedia "refers to a second phase of e-Learning based on Web 2.0 and emerging trends in eLearning" [6]. A more human centred and therefore better definition of what eLearning2.0 can be is given by Bret Schlenker in one of his "Corporate Elearning Strategies and Development"-blog posts from September the 11 th of 2006 reminding: "So stop thinking about Learning2.0 as a new toolset…It's so not about that! Learning2.0 is about people. The tools simply allow us to do what we do best…and that's connecting with other people to support, share, and learn with each other" [7]. In the course of the second revolution of how we use the internet by speaking about the phenomenon Web2.0 (a term by Tim O´Reilly [8]) elearning2.0 was a logical progression of it. The new possibilities of user-editing online content on demand and the high grade of connectivity are perfect ingredients to be adapted for new didactical opportunities [START_REF] Maurer | Wikis and other e-Communities are changing the Web[END_REF]. The ongoing and growing success of Web2.0 has been made possibly by a wide coverage of broadband technology and the breakthrough of the RSS-technology. Elearning2.0 uses the efforts of Web2.0 within educational settings. In this context informal learning becomes more important.

Elearning2.0 at TU Graz

Electronically supported learning, briefly elearning, in the form of very good single initiatives has been practiced for years at the TU Graz. First steps in the eighties for instance go back to several programs for load flow-and short-circuit calculations as well as for inductive and ohmic interference [START_REF] Fickert | New Development of Teaching Concepts in Multimedia Learning for Electrical Power Systems Introducing Sonification[END_REF]. These activities to bundle ("to network") was the basic idea for the Vice-Rectorate for Teachings, the LifeLongLearning office, the Institut for Information systems and computers media, as well as for the Center of Computer and Information Services (CIS). In September 2006 the Working Group "Vernetztes Lernen" (WG Social Learning) became a new service of the Vice-Rectorate for Teachings. By embedding of the Working Group in the CIS established synergies can be used and the service achievements of CIS are being extended with a further component. Main focus of the Working Group is to implement (where necessary) the possibilities of network-based, communication-oriented teaching and learning methods in a didactically meaningfully way and establish them lastingly at the TU Graz [START_REF] Nagler | LifeLongLearning and Doctoral Studies -Facilitation with Podcasting Techniques[END_REF].

A great step forward was done by the WG Social Learning by implementing the so called TU Graz TeachCenter within the digital structure of the TU Graz study-services landscape (see Figure 1-1). The TU Graz TeachCenter is a free electronic platform for the promotion and management of digital teaching activities for all teachers of the TU Graz. It is an adjusted system of the so called "WBT Master"-system that has been developed at the Institute for Information Systems and Computer Media (IICM) at Graz University of Technology. The range of possibilities for the teachers nearly exceeds the state of the art in elearning2.0. The outstanding difference to other comparable platforms is the modular design of the TU Graz TeachCenter, which means that the platform can be used by each teacher very individually. This modular principle reaches the smallest unities of content within the TU Graz TeachCenter. Anything structural unit within the system can be seen as a single module -a separate PDF-file as well as a whole forum or a logged in user. There are no default applications or basic tools coming with a ready made new online course. It depends on the teacher which functionalities to choose and to be added to his/her course. Once an application is selected it can be removed by one click again. Amongst others the high operability and speed of the system results from the fact that there is no database storing the content at all. Furthermore in October 2006 the TU Graz LearnLand was launched. By using the open source software ELGG [9] a blogosphere is offered to each member (lecturers and students) of the university [START_REF] Ebner | Web 2.0 Technology: Future Interfaces for Technolgoy Enhanced Learning?[END_REF]. After the first logon a personal weblog is created where contribution can be published very easily. Users can establish digital identities and connect with other members, collaborate with them and discover new resources through their connections [START_REF] Fickert | New Development of Teaching Concepts in Multimedia Learning for Electrical Power Systems Introducing Sonification[END_REF]. Users can create own communities with own member-management. A bookmark benchmarking function and a prospective presentation tool collecting selected blog posts are further features of this blogosphere. The high grade of networking is established by tagging of any contribution which turns searching into an effective interaction. Both the TU Graz TeachCenter and the TU Graz LearnLand offer RSS functionality for their content including media files. By using RSS the administration of a course is being very simplified and the just-in-time communication is being enhanced as well as building up communities easilier.

Didactics on Micro Content

General didactical aspects of micro content

"Microcontent is information published in short form, with its length dictated by the constraint of a single main topic and by the physical and technical limitations of the software and devices that we use to view digital content today" as described by Anil Dash in 2002 [10]. From a didactical point of view it must be pointed out the term "single main topic". With other words microcontent can be described as information chunk, a small piece of a big knowledge cake. The idea is to separate the content of a lecture in didactical reasonable parts. This is not very new because it is standard for each teacher in a traditional setting. The new thing is to do the same with digital content and enhance the teaching and learning process with internet technologies. If content is very granular, following advantages can be pointed out:

• Exchangeability: Content can be replaced easily by exchanging. For example a special learning object can be used in one year but is not needed in the following year.

Removing this object will be easily. • Multiplicity: Microcontent can not only used nearly endless, but also in many different environments. Multiple usage for special teaching and learning settings -just when needed -is a very important strength. • Stand alone: Each part represents a stand alone content (html file, Flashmovie, podcast ..) and this means also direct retrievable. • Servicability: If the content is granular changes can be done just in time. Editing online files just on the fly, comparable with a wiki or a Weblog are possible. The main concept behind using microcontent is that an amount of such files can be put into a structured environment, simply by bringing it into a hierarchical order. The lecturer is preparing his lecture by connecting and bringing content together. With this possibility real new didactical scenarios are possible, because not only each kind of media files can be presented also content from anyone. With other words contributions of students or learners can also become an active part in an electronically environment. Bearing these thoughts in mind the first experiments and trials have been done on Graz University of Technology.

Micro-content used at TU Graz

The Blog-presentation tool within TU Graz LearnLand

A new application running at TU Graz LearnLand offers the user to create a presentation filled with former blog posts. By selecting different posts from any user a new stand alone presentation can be easily created displaying the individual posts and their authors in an optional sequence. The range of usage is to be explored. An example that already has been exercised can be that the teacher uses this functionality to sum up students blog post-activities that have been part of a seminar or workshop lesson. Besides that blog posts of students may so become more important to others when selected by the teacher for a presentation. So podacsting is becoming a part of teaching methods at TU Graz. The didactical scenarios are to be worked out yet. Besides screening a whole lesson of about 90 minutes one usage of podcasts is to create short videos presenting small pieces of information such as for explaining purposes within help desks.

An introduction to the ABC-eBook

A third example for practicing the ideas of micro-content at the TU Graz is the "ABC-eBook Project" by merging advantages of the so called "ABC-system" and the eBook functionality of the TU Graz TeachCenter. With the ABC-system content is worked into small standardised units embedded within a well structured ABC-course system. The eBook environment allows mashing up these units independently to their ABC-course structure. Therefore each unit can be handled alone. Motivated by the initiative "New Media in teaching at universities and polytechnics in Austria" (NML) of the Austrian Federal Ministry for Education, Science and Culture (bm:bwk) in spring 2000 the TU Graz started the project "MultiMediaLearning" which lasted till the foundation of the WG Social Learning in September 2006. The Institute for Electrical Power Systems (IFEA) at the TU Graz was one of the first partners of MML. Since spring 2001 IFEA undertakes its own elearning strategy. For this aim a brand new course-system called "ABC-course system" with a specially designed layout for electronic scripts has been developed and evaluated by independent psychologists. The uniform structure and the high usability of the system allow a wide ranged application of the courses as for the lesson in classroom, for online variant, for offline CD-ROM version and for the hardcopy paper script (see Figure 3-1) [START_REF] Ebner | Web 2.0 Technology: Future Interfaces for Technolgoy Enhanced Learning?[END_REF].

The ABC-eBook Project

The ABC concept

A course consists of separate standardised "ABC-screens" written in HTML and containing the content of the course. The several screens are placed in a predefined course structure of directories. Once the content is didactically adapted and realized as ABC-screens within the ABC-course system, this new course automatically is compiled to a standalone CD ROMversion by an application called "IFEA-Viewer". Another application called "wbt2paper" generates a PDF-file of the course for hardcopy usage.

The main benefit of the ABC concept is its modular character of multimedia interactive content. It allows the course designer to update already existing courses or to create new ones very easily by mixing, changing and creating new single ABC-screens in general. But it must be said that this process is not standardised yet. The task for the future is to make it that simple so that a teacher is able to manage ABC-courses offline without any programming knowledge. The eBook concept (see next chapter) is the first step realising this aim for an online variant.

The eBook concept

TU Graz TeachCenter (see chapter 1.2) offers a variety of possibilities to optimise a teachers´ digital teaching activities including functionalities for managing meta-data about training courses. One of basic applications supported by the system is called eBook. A particular eBook can be seen as a number of Multi-Media elements such as (HTML documents, PowerPoint-slides, PDF-files, movies, etc …) combined into reusable, navigable collections. eBook provides a convenient document accessing facilities (table of content, map, search, bookmark, etc,). Additionally, an eBook provides such communication features as annotation and discussion plus conventional user-control features -statistics, map of visited pages, results of answering embedded questions, etc.. eBooks can be presented in three different formats: internal S-Collections, SCORM packages [11] and LRN [12] packages.

TU Graz TeachCenter provides an automatic conversion between the internal formats; hence, an actual authoring may be carried out using internal TU Graz TeachCenter editors or any third-party authoring component compliant with the SCORM or LRM formats. Since all the TU Graz TeachCenter authoring tools operate with S-collections we provide a simplified description of the model below. Educational content, according to this concept, consists of addressable logical composites called Structured Collections (S-collections or just collections, for short). An S-collection does nothing but encapsulates members together with some internal structure (i.e. navigational topology). This structure is in fact a link structure expressing the relationships or associations between members. A member is either a document or another S-collection. One member in an S-collection is chosen and designated by the author as its head.

6(8)

Additionally, an S-collection may have an associated document called its label to provide contents synopsis. If a label is missing then the head is used instead. We may think of an S-collection as an opaque container: if we are outside the container, its members will not be visible to us. To see what's inside it, we must enter it. Of course, we can only be inside one container -the current container -at any given time. But once inside, we will be able to visit its members by navigating its link topology. A member we visit in the current container may be a document or another S-collection. If a document, it will be visualised in some appropriate way -typically involving the presentation of the document's media objects on the computer display and/or sound system. If an S-collection, its label (which is a document, by definition) will be visualised. The most recently visited member of the current container is the current member. Access to other members from the current member is determined by the container's link structure: only links emanating from the current member can be selected. All such links are typically visualised as anchors on the current display (these may be icons, hot-words, push-buttons, etc). Navigation within the current container therefore is from member to member as allowed by its link topology.

From a teacher perspective, eBook can be perceived as an environment for changeable multimedia content letting a user to make annotations online to any part of the content. In the context of this paper, the modularity of an eBook is to be especially pointed out. Thus, potentially any piece of content available at a TU Graz TeachCenter server can be placed into one or more S-collections and thus reused in different contexts.

To link ABC-system and eBook concept

The ABC-course system fits very well into the concept of eBook. The ABC concept structures content into small pieces of teaching units that can be perfectly displayed and managed with the eBook application. Therefore the eBook functionality is the online realisation of an ABC course fulfilling the demand of mashing up ABC screens like PPT-slides. The manager of a new eBook just chooses online different ABC-screens from different ABC-courses and may add some new screens as well. This new eBook may then be linked to any lecture within the TU Graz TeachCenter (see Figure 3-2).

The ABC-Manager

The use of the ABC-system has been limited to the purpose of IFEA but is exceeding now. First attempts to apply the system for other institutes have shown that it needs a long introduction and a much disciplined method of operating to fulfil the standards of ABC system which is necessary according to the production of the different outputs. Therefore there is a need to make the creation and the management of ABC-Screens much simpler. The future application called ABC-Manager will be useable for teachers without knowledge of HTML or time to create ABC-Screen. The ABC-Manager is a project just started. Aim of the project is to combine the comfortable mash up functionality for ABC-Screens as provided by the online WBT-Master with the multiple output possibilities of the ABC-system. The prospective application will allow doing the mash up offline. Furthermore it will include a tool for creating ABC-Screens guided by a wizard so no further programming is necessary.

Discussion & Summary

Evaluations on Microcontent

In the past the usage of ABC-courses was highly appreciated by students. Evaluations during the first three years of the implementation of the ABC-system into the teaching of IFEA pointed out that a slowly shifting of the students learning customs was noticeable. The acceptance of using a CD ROM-version for learning efforts increased during these years account of a printed paper-version. Although the evaluations were worked out and exercised by independent psychologists towards significance the rather small number of polled students (150) must be taken into account [START_REF] Fickert | Experiences and Adaptation of Teaching Concepts in the Field of Multimedia Learning for Electrical Power Systems at the University of Technology Graz[END_REF].

First results of an ongoing online evaluation of the podcasting activities according to the course "Basics of Power Systems" show that fourth of the students listened to the complete podcasting offer for this course, more than a third listened some parts of the offer but more than one time and actually fourth of the students did not use any podcast. A huge majority emphasizes an enlargement of the podcast offer on other courses even to the point of a podcast-standard for any course held at the TU Graz.

Summary

In times of rapidly growing information input there is a rising call for methods to manage information. Flexible and reasonable management of content for learning purposes becomes more important. The efforts of Web2.0 make it easier for teachers to structure and redesign their teaching processes intentions. The reuse of content in different meanings and the multiple adaptability of content will be better to be manage the smaller these pieces of contents are. The rearrangement therefore should be as simple and user-friendly as possible. The TU Graz TeachCenter bases on a system taking these requirements into account. The systems´ eBook in combination with the ABC-concept is a very good example for flexible content management. A standardised usage at the university depends on a simplification of the processes. The ABC-Manager will do so in future.

Figure 1 - 1 :

 11 Figure 1-1: The different platforms of the TU Graz and their impact on elearning

2

 2 Using short podcasts (BTZ / Help-Site) During the last year podcasts have gained popularity. Since October 2006 the TU Graz is offering podcasts from a small number of lectures to their students in an experimental setting. First evaluations encourage optimizing and enlarging Podcast activities. In July 2007 the first Austrian symposium on Podcasting took place very successfully at the University of Graz organized by iUNIg (Interuniversitäre Initiative für Neue Medien Graz) an new initiative for collaboration in the subject of new media between the four universities of Graz [13]und [14].

Figure 3 - 1 :

 31 Figure 3-1: The IFEA-ABC concept: multiple input, structured modular multiple output

Figure 3 - 2 :

 32 Figure 3-2: TU Graz TeachCenter content and applications on base of WBT Master functionality with an example of mashing eBook content from different ABC-Screens for different online courses.

Author(s):

1 Walther, Nagler


Qiyu Jin

Osamu Miyashita

Florence Tama

Jie Yang

Slavica Jonic
email: slavica.jonic@upmc.fr

Poisson image denoising by piecewise principal component analysis and its application in single-particle X-ray diffraction imaging

teaching and research institutions in France or abroad, or from public or private research centers.

Poisson image denoising by piecewise principal component analysis and its application in single-particle X-ray diffraction imaging

Introduction

Poisson noise (also called photon noise or shot noise [START_REF] Schottky | Über spontane stromschwankungen in verschiedenen elektrizitätsleitern[END_REF]) is due to a discrete ("quantized") nature of the signal measured by a detector. For instance, light can be considered as a discrete flow of individual "packets" (photons) that arrive to the detector randomly and independently from each other. As a result, photon counting obeys a classical Poisson statistics, and the number of photons y measured by a given detector element over a time interval t is described by the following probability distribution:

P(y|f ) = (f ) y y! e -f , ( 1
)
where f is the expected number of photons in time interval t. This is the standard Poisson distribution with rate f . For large numbers, the Poisson distribution approaches a normal distribution, meaning that shot noise is practically indistinguishable from a true Gaussian noise, except in the case of very few elementary events (photons, electrons, etc.) that are individually observed. Since the standard deviation of the shot noise is equal to the square root of the expected number of incident photons f , the signal-to-noise ratio (SNR) is given by:

SNR = f √ f = √ f . ( 2
)
This equation indicates that the SNR is large when f is large. However, in many applications (e.g., night vision, medical imaging, weather forecasting, astronomy, and spectral imaging), very few photons are collected by the detector, due to various constraints (weak light source, short exposure time, phototoxicity). This is referred to as photon-limited imaging. Thus, the limited number of available photons limits the SNR, due to the signal-dependent Poisson statistics. An approach to improve the SNR is to first reduce the dependence of the noise on the signal by an image transformation (e.g., Variable-Stabilizing Transformation (VST), multiscale Variable-Stabilizing Transformation (MVST), conditional variance stabilization (CVS), or Fisz transform [START_REF] Danielyan | Denoising of multispectral images via nonlocal groupwise spectrum-pca[END_REF][START_REF] Boulanger | Patch-based nonlocal functional for denoising fluorescence microscopy image sequences[END_REF][START_REF] Deledalle | Poisson nl means: Unsupervised non local means for poisson noise[END_REF][START_REF] Makitalo | On the inversion of the Anscombe transformation in lowcount Poisson image denoising[END_REF][START_REF] Makitalo | Optimal inversion of the anscombe transformation in low-count poisson image denoising[END_REF][START_REF] Boulanger | Non-parametric regression for patch-based fluorescence microscopy image sequence denoising[END_REF][START_REF] Zhang | Wavelets, ridgelets, and curvelets for poisson noise removal[END_REF][START_REF] Lefkimmiatis | Bayesian inference on multiscale models for poisson intensity estimation: Applications to photon-limited image denoising[END_REF][START_REF] Luisier | Fast interscale wavelet denoising of poisson-corrupted images[END_REF][START_REF] Anscombe | The transformation of poisson, binomial and negative-binomial data[END_REF][START_REF] Borovkov | Estimates for the distribution of sums and maxima of sums of random variables without the cramer condition[END_REF][START_REF] Fisz | The limiting distribution of a function of two independent random variables and its statistical application[END_REF][START_REF] Fryzlewicz | A haar-fisz algorithm for poisson intensity estimation[END_REF][START_REF] Fryzlewicz | Goes-8 x-ray sensor variance stabilization using the multiscale data-driven haar-fisz transform[END_REF][START_REF] Jansen | Multiscale poisson data smoothing[END_REF][START_REF] Prucnal | Transformation of image-signal-dependent noise into image-signal-independent noise[END_REF][START_REF] Foi | Practical poissoniangaussian noise modeling and fitting for single-image raw-data[END_REF][START_REF] Azzari | Variance stabilization for noisy+ estimate combination in iterative poisson denoising[END_REF]) and, then, to use methods that assume signal-independent noise [START_REF] Buades | A review of image denoising algorithms, with a new one[END_REF][START_REF] Kervrann | Optimal spatial adaptation for patch-based image denoising[END_REF][START_REF] Hammond | Image modeling and denoising with orientation-adapted gaussian scale mixtures[END_REF][START_REF] Katkovnik | From local kernel to nonlocal multiple-model image denoising[END_REF]. Other approaches to the SNR improvement have also been proposed, such as maximum likelihood estimation [START_REF] Chan | Multilevel algorithm for a poisson noise removal model with total-variation regularization[END_REF][START_REF] Srivastava | Restoration of poisson noise corrupted digital images with nonlinear pde based filters along with the choice of regularization parameter estimation[END_REF][START_REF] Shepp | Maximum likelihood reconstruction for emission tomography[END_REF][START_REF] Van Kempen | Comparing maximum likelihood estimation and constrained tikhonov-miller restoration[END_REF][START_REF] Moon | Three dimensional imaging and recognition using truncated photon counting model and parametric maximum likelihood estimator[END_REF][START_REF] Bardsley | Tikhonov regularized poisson likelihood estimation: theoretical justification and a computational method[END_REF][START_REF] Johnathan | Regularization parameter selection methods for illposed Poisson maximum likelihood estimation[END_REF][START_REF] Le | A variational approach to reconstructing images corrupted by poisson noise[END_REF][START_REF] Giryes | Sparsity-based poisson denoising with dictionary learning[END_REF],Plug-and-Play scheme [START_REF] Rond | Poisson inverse problems by the plug-and-play scheme[END_REF], Deep convolutional denoising [START_REF] Remez | Deep convolutional denoising of low-light images[END_REF] Kullback-Leibler divergence [START_REF] Brune | Forwardbackward em-tv methods for inverse problems with poisson noise[END_REF][START_REF] Wen | Primal-dual algorithms for total variation based image restoration under poisson noise[END_REF][START_REF] Li | A reweighted l2 method for image restoration with poisson and mixed poisson-gaussian noise[END_REF], optimal weights filtering [START_REF] Jin | A new poisson noise filter based on weights optimization[END_REF] based on Non-Local Means approach [START_REF] Buades | A review of image denoising algorithms, with a new one[END_REF], or Non-Local Principal Component Analysis (NLPCA) [START_REF] Salmon | Poisson noise reduction with non-local pca[END_REF].

NLPCA combines Principal Component Analysis (PCA) and sparse Poisson intensity estimation methods in a nonlocal estimation framework and takes into account the heteroscedastic nature of observations. It clusters (K-Means clustering) a set of patches extracted from the entire image into 14 groups, each group containing similar patches. Then, it separates noise and signal in each group by reducing the dimension of the data in the group using PCA. More precisely, most of the low-frequency information will be contained in the first few principal components and the other principal components, corresponding to the high-frequency information, will not be taken into account in the signal reconstruction. Therefore, NLPCA has lower efficiency with images rich in details [START_REF] Salmon | Poisson noise reduction with non-local pca[END_REF].

Theoretical analyses and simulation results of Jin et al. [START_REF] Jin | Nonlocal means and optimal weights for noise removal[END_REF][START_REF] Jin | Convergence theorems for the non-local means filter[END_REF] suggest that larger sizes of the window from which the patches are extracted do not necessarily produce better denoising results. More precisely, there is an optimal value of the window size for the best denoising results [START_REF] Jin | Nonlocal means and optimal weights for noise removal[END_REF][START_REF] Jin | Convergence theorems for the non-local means filter[END_REF]. The probability of similarity of patches is higher and the similarity estimation is more accurate for the patches extracted from the same area than for the patches extracted from different areas of the same image. Building on the results of that work [START_REF] Jin | Nonlocal means and optimal weights for noise removal[END_REF][START_REF] Jin | Convergence theorems for the non-local means filter[END_REF], we here propose a denoising method that is based on combining the NLPCA results obtained on different areas of the same image. The method first splits the given image into large pieces. Then, the pieces are individually denoised using NLPCA. Finally, the denoised pieces are combined into a full-size image. The information dimension (complexity) in a piece (local area) of the image is much lower than the complexity of information in the entire image. Therefore, it is easier to separate noise from signal in the local areas than in the entire image.

The approach that we have developed for Poisson denoising using the principle of piecewise PCA is referred to as PieceWise PCA (PWPCA) and it is described in this article. We show that it provides better results than several state-of-the-art approaches, including NLPCA. PWPCA was evaluated using standard real-space test images affected by Poisson noise as well as using single-particle X-ray diffraction images, such as those that can be collected with an X-ray free-electron laser (XFEL). XFEL experiments are currently under rapid development to allow high-resolution determination of biomolecular structures under near-physiological hydration conditions [START_REF] Hosseinizadeh | Conformational landscape of a virus by single-particle x-ray scattering[END_REF][START_REF] Miao | An approach to three-dimensional structures of biomolecules by using single-molecule diffraction images[END_REF][START_REF] Seibert | Single mimivirus particles intercepted and imaged with an x-ray laser[END_REF][START_REF] Yefanov | Orientation determination in single particle x-ray coherent diffraction imaging experiments[END_REF]. The diffraction image collected by XFEL is, in a first approximation, the intensity (the square of the amplitude) of the 2D Fourier transform of the object's 2D real-space projection (or a central slice of the object's 3D Fourier-transform intensity, according to the central slice theorem), whereas the phase information cannot be collected experimentally. One diffraction image contains the diffraction data collected from one biomolecular particle at a random and unknown orientation in 3D space. Therefore, to compute a 3D reconstruction from the 2D diffraction data collected from many copies of the same particle, their orientations (the orientation of different particles hit by the XFEL beam) and phases should be determined computationally. The amplitude volume can be obtained from the square root of the volume assembled from the oriented diffraction images. The phase images can be retrieved from the amplitude images and assembled in 3D at the determined orientations or the 3D phase volume can be retrieved from the 3D amplitude volume, and the phase retrieval is usually done with the standard Fienup method or its variants [START_REF] Miao | An approach to three-dimensional structures of biomolecules by using single-molecule diffraction images[END_REF][START_REF] Fienup | Reconstruction of an object from the modulus of its fourier transform[END_REF][START_REF] Tegze | Atomic structure of a single large biomolecule from diffraction patterns of random orientations[END_REF]. The real-space 3D object can then be computed by inverse Fourier transform of the combined amplitude and phase volumes. Both orientation and phase determination are difficult tasks because the majority of nonzero diffraction-image pixels are concentrated in a very small central region of the diffraction image and the pixel intensities obey Poisson statistics. As very few XFEL experimental data sets had been obtained so far, we have performed experiments using simulated data, as commonly done when validating new methods for XFEL. To the best of our knowledge, diffraction data restoration using Poisson denoising methods has not been previously addressed.

The remaining part of the article is organized as follows. Basic concepts of NLPCA are provided in Section 2. PWPCA is described in Section 3. Results obtained with PWPCA using simulated noisy real-space images and using simulated XFEL data are presented and discussed in Sections 4 and 5, respectively. Conclusions are provided in Section 6.

Background: Non-Local Principal Component Analysis

In this section, we briefly remind the NLPCA approach [START_REF] Salmon | Poisson noise reduction with non-local pca[END_REF] that we here propose to be used in a piecewise fashion over an image (Section 3). The set of image pixels is denoted by I = {x = (x (1) , x (2) )|x (1) , x (2)

= 1, 2, • • • , √ M }
, where both M > 0 and √ M are integers. Let y = {y(x)|x = (x (1) , x (2) ) ∈ I} be the observed image obtained by an image acquisition device and each y(x) is an independent random Poisson variable given by (Eq. 1), whose mean f (x) > 0 is the underlying intensity value to be estimated. Let Y be the M × N matrix of all denoising patches of size √ N × √ N pixels reordered in N -long vectors referred to as denoising vectors (one patch per image pixel for the image size of √ M × √ M pixels). Y i,j is the jth pixel in the ith patch of the noisy image. Let F be defined similarly to Y for the underlying intensity f = {f (x)|x = (x (1) , x (2) ) ∈ I}.

The estimate of F is given by

F i,j = exp ( [U * V * ] i,j ) , i = 1, 2, • • • , M, j = 1, 2, • • • , N, (3)
where

(U * , V * ) = arg min (U,V) L(U, V). (4)
In Eq. 4, the matrices U/U * and V/V * have dimensions M × l and l × N and are referred to as matrix of coefficients and matrix of dictionary components (axes), respectively. L is the loss function that accounts for the Bregman divergence D Φ to measure the proximity between two matrices X and Y of size M × N :

D Φ (X||Y ) = Φ(Y ) -Φ(X) -T r((∇Φ(X)) ⊤ (X -Y )), (5)
where

Φ(θ) =< exp(θ)|1n >= ∑ n i=1 e θi , < a|b > is inner prod- uct of a and b, θ = (θ 1 , • • • , θn) ⊤ , exp is the component-wise exponential function: exp : θ → (e θ1 , • • • , e θn ) ⊤ , 1n is the vector (1, • • • , 1) ⊤ ∈ R n and ∇Φ(θ) is the gradient of Φ(θ). Moreover ∇Φ(θ) = exp(θ).
The use of Eq. 5 amounts to searching for U and V that minimize the following loss function:

L(U, V) = M ∑ i=1 N ∑ j=1 exp ( [UV] i,j ) -Y i,j [UV] i,j , (6)
the rows of V are dictionary components and l (l ≪ M ) is the number of components. The purpose of the rank l is to regularize the solution.

More details on the NLPCA algorithm are provided in Appendices A and B, in the context of denoising image pieces (Section 3.2).

New Approach: Poisson Piecewise Principal Component Analysis

The PWPCA approach proposed here consists of three steps. The given image is first divided into a set of overlapping image pieces. Then, Poisson denoising of each individual piece is performed. Finally, all denoised pieces are combined, with some weights, into a full (original size) image. In the remaining part of this section, we describe in detail each of the three steps. The pseudocode of the algorithm is provided in Algorithm 1.

Extraction of image pieces

The first step is to split the given image y i.e. its pixels I into a set of pieces of size (1) , x (2) ) ∈ I|x (1)

√ M ′ × √ M ′ pixels. The set of pixels in each image piece yq, q = 1, 2, • • • , Q is denoted by Bq = {x = (x
∈ [x (1) q -m, x (1) q + m], x (2) ∈ [x (2) q -m, x (2)
q + m]}, where m is the number of pixels covered by the image piece from its central pixel and xq = (x

(1) q , x (2) q ) are the coordinates of the central pixel, meaning that √ M ′ = 2 × m + 1. This produces the total number of √ Q × √ Q pieces, where √ Q = ⌈( √ M - √ M ′ )/( 2 3 √ M ′
)⌉ and ⌈t⌉ = min{n ∈ Z|t ≤ n} is the top integral function whose value is the smallest integer greater than the independent variable or equal to it. Let us define T as

T = √ M - √ M ′ .
Then, the central pixel of each piece, xq, is located in the set {(x (1) , x (2) )|x (1) , x (1)

∈ {1 + m, ⌈T / √ Q⌉ + 1 + m, ⌈2T / √ Q⌉ + 1 + m, T + 1 + m}} and the sets of pixels Bq, q = 1, 2, • • • , Q cover I completely.
As shown in the Fig. 1, the pieces overlap to avoid block type of artifacts and each pixel is covered at least once but not more than 4 times. Note here that each piece of the image (Fig. 1) will be further split into smaller pieces (as it will explained in Section 3.2) and that these sub-pieces will here be referred to as "denoising patches" (Fig. 2), as was the case in the original publication ( [START_REF] Salmon | Poisson noise reduction with non-local pca[END_REF]).

Denoising of each image piece

In this step, Poisson denoising of each image piece yq = {y(x)|x = (x (1) , x (2) ) ∈ Bq} is preformed using the NLPCA approach (Section 2) that was adapted to process pieces of an image.

Let Yq be the M ′ × N matrix of all (M ′ ) patches of size √ N × √ N pixels extracted from yq (one patch per pixel of yq) and reordered in N -long vectors (referred to as denoising vectors).

, q = 1, 2, • • • , Q, each containing M ′ pixels. 3: for each q = 1, 2, • • • , Q do 4:
Patchization: create M ′ × N matrix Yq of all denoising vectors (one patch per pixel of piece yq).

5:

Clustering: create K clusters of patches using K-means algorithm (see Algorithm 2 in Appendix B). The kth cluster (represented by a matrix Y k q ) has M k q elements. 6:

for each cluster k do 7:

Initialize U 0 q = randn(M k q , l) and V 0 q = randn(l, N ) 8:

while t ≤ N iter and test > ε stop do 9:

for all i ≤ M k q do 10:
Update the ith row of U t q using Eq. 20 (Appendix A) 11:

end for 12:
for all j ≤ l do 13:

Update the jth column of V t q using Eq. 21 (Appendix A)

F k q = exp(U t q V t q )
18:

end for Calculate f (x) using Eq. ( 10) 24: end for Let Fq be defined similarly to Yq for the underlying intensity to be estimated fq = {f (x)|x = (x (1) , x (2) ) ∈ Bq}. We approximate each Fq, q = 1, 2, • • • , Q as follows:

Fq ≈ exp ( [UqVq] i,j ) i ∈ {1, 2, • • • , M ′ }, j ∈ {1, 2, • • • , N }, ( 7
)
where Uq is the M ′ × l matrix of coefficients, Vq is the l × N matrix of dictionary components or axes, and l ≪ M ′ .

Given Yq ∈ R M ′ ×N , with Y q,i,j being the jth pixel in the ith patch, the underlying intensity Fq is estimated by minimizing the following loss function: [START_REF] Zhang | Wavelets, ridgelets, and curvelets for poisson noise removal[END_REF] with respect to the matrices Uq and Vq. The minimization of L is done using Newton's method (for more details, see Appendix A).

L(Uq, Vq) = M ′ ∑ i=1 N ∑ j=1 exp ( [UqVq] i,j ) -Y i,j [UqVq] i,j
For each image piece yq, we separate its patches into different groups (clusters) according to their similarity and apply matrix factorization on each cluster, which allows representing the data within each cluster with a lower dimensional dictionary and results in more efficient denoising and higher SNR in the denoised image. The clustering of patches of each image piece is here done using K-means clustering algorithm (for more details, see Appendix B). The K-means algorithm is used in the original NLPCA approach as well, where it serves to cluster patches extracted from the entire image [START_REF] Salmon | Poisson noise reduction with non-local pca[END_REF]. However, denoising results may be poor when clustering patches of the entire noisy image because noise lowers the clustering accuracy and large cluster dimensions make the matrix factorization difficult. As a remedy to this problem, we here propose to perform a separate patch clustering for each piece extracted from the image.

The denoising of each image piece is vizualized in Fig. 2. Note here that the reprojection of patches in Fig. 2 means a reconstitution of the original, full-size piece of the image from the denoised patches. During the reprojection, the overlapping pixels are averaged.

Full-size image reconstitution

The final entire estimate f is computed by a weighted average of the piecewise estimates fq obtained as explained in Section 3.2. Let pixel xq from the set Bq be the center pixel of the denoised image piece fq (whose size is

√ M ′ × √ M ′ pixels, √ M ′ = 2 × m + 1
). In the merging procedure, each pixel x ∈ I is first assigned the following weight [START_REF] Jin | A new poisson noise filter based on weights optimization[END_REF]:

κq,m (x) =            m ∑ s=∥x-xq∥ ∞ 1 m(2s+1) 2 if x ̸ = xq and x ∈ Bq, m ∑ s=1 1 m(2s+1) 2 if x = xq, 0 if x ∈ I \ Bq, (9)
where ∥ • ∥∞ denotes supremum norm. Then, the full-size estimate f is obtained as follows ( [START_REF] Dabov | Image denoising by sparse 3-d transform-domain collaborative filtering[END_REF]):

f (x) = Q ∑ q=1 κq,m(x) fq(x) Q ∑ q=1 χq,m(x)κq,m(x) , ∀x ∈ I, (10)
where

χq,m(x) = { 1, x ∈ Bq, 0, x ∈ I \ Bq.
For the purpose of illustration, we here give an example of the weighting kernel if the entire image is processed as a single piece (i.e. Q = 1) and the image size is 5 × 5 pixels (m = 2). The weighting kernel in this case (κ 1,2 ) is as follows:

           1 2 +       0 0 0 0 0 0 1 18 1 18 1 18 0 0 1 18 1 18 1 18 0 0 1 18 1 18 1 18 0 0 0 0 0 0       1 2 =      
      1 .
However, note that the number of image pieces is usually larger than 1 and m is usually much larger than 2 (see Eq 11).

which takes into account the size of the images used in our experiments (256 × 256 or 257 × 257 pixels). All other parameters are the same as in [START_REF] Salmon | Poisson noise reduction with non-local pca[END_REF] (Table 1).

Fig. 3:

No-noise images used in this article and synthetic Poissonnoise images with Peak = 2 ("Spots", "Galaxy", "Cells", "Barbara", "Peppers", "House", "Camera", "Bridge").

Computation complexity

The complexity of NLPCA is O(M lN 32 ) (cf. [START_REF] Giryes | Sparsity-based poisson denoising with dictionary learning[END_REF]). As explained in section 3.1, the method proposed here splits the given image into pieces so that each pixel is covered not more than 4 times. Thus, the complexity of the proposed method is not more than O(4M lN 3 2 ), meaning that the proposed method has the same order of computational cost as NLPCA.

Experiments with standard test images

In this section, we show results of experiments conducted with the new method (PWPCA) using standard test images with Poisson noise. Also, we compare the results of PWPCA with those of several state-of-the-art denoising methods, namely PURE-LET [START_REF] Luisier | Fast interscale wavelet denoising of poisson-corrupted images[END_REF], BM3D [START_REF] Makitalo | Optimal inversion of the anscombe transformation in low-count poisson image denoising[END_REF][START_REF] Makitalo | A closed-form approximation of the exact unbiased inverse of the anscombe variance-stabilizing transformation[END_REF], NLPCA [START_REF] Salmon | Poisson noise reduction with non-local pca[END_REF][START_REF] Salmon | Poisson noise reduction with non-local PCA[END_REF], and SPDA [START_REF] Giryes | Sparsity-based poisson denoising with dictionary learning[END_REF][START_REF] Giryes | Sparsity based poisson denoising[END_REF].

The noise-free standard test images (image size: 256 × 256 pixels) were graciously obtained from Salmon et al. [START_REF] Salmon | Poisson noise reduction with non-local pca[END_REF] and Giryes and Elad [START_REF] Giryes | Sparsity-based poisson denoising with dictionary learning[END_REF]. A noise-free image f with the Peak value P eak was obtained as follows:

f (x) = P eak × S(x)/ max{S(x)|x ∈ I},
where S is a standard test image. The image with Poisson noise (y) corresponding to image f was generated using MATLAB. The standard test images with P eak = 2 and Poisson noise are presented in Fig. 3. When the image intensity is high, the Poisson distribution approaches a normal distribution. In this section, we show the results obtained with images of low intensities, characterized by Peak values of 0.5, 1, 2 and 4.

The denoising results obtained with different methods were evaluated using the so-called Peak Signal-to-Noise Ratio (PSNR) [START_REF] Makitalo | Optimal inversion of the anscombe transformation in low-count poisson image denoising[END_REF], expressed in decibels (dB) as follows:

PSNR = 10 log 10 P eak 2 MSE , ( 12
)
MSE = 1 card(I) ∑ x∈I (f (x) -f (x)) 2 , ( 13
)
where card(I) is the number of elements of the set I, f is the noisefree image and f is the denoised image.

The PWPCA and NLPCA denoising results for images with these Peak values, in terms of PSNR, and the difference between the PSNR values obtained with these two methods are shown in Table 2. The PSNR results of all tested methods are listed in Table 3. The proposed PWPCA method achieves 0.25db -1.08db improvement over the NLPCA method, on average, for all four noise levels and the PSNR difference increases significantly as the Peak value increases (Table 2). Also, the proposed PWPCA method achieves the highest average PSNR value in all four noise cases (Table 3). In Fig. 4 and Fig. 5, we compare the visual quality of images denoised by different methods. Fig. 4 shows that PWPCA outperforms the other methods regarding the restoration of image details from the noisy observation such as in the tripod leg area of ąřCamera ąś image, as SPDA and NLPCA result in oversmoothing and PURE-LET and BM3D produce more artifacts (the area indicated by arrows in Fig. 4). Furthermore, Fig. 5 shows that PWPCA produces fewer artifacts and preserves the texture better than other methods (e.g., in the area indicated by arrows in Fig. 5). In summary, PWPCA has strong denoising capability and produces better quantitative (PSNR) and visual results than other tested methods.

Experiments with simulated XFEL diffraction data

In this section, we show an application of Poisson denoising methods in 3D reconstruction of biomolecular structures from single-particle XFEL diffraction data. As very few single-particle XFEL experimental data were obtained so far, we here used simulated data, which is common in evaluating new methods for XFEL. Fig. 8: Fourier shell correlation between the density map of the structure used to simulate XFEL diffraction data (Fig. 7b) and 3D reconstructions from four data sets for the beam intensity of 10 14 photons/pulse/µm 2 (original, processed by PWPCA, processed by BM3D, and ground-truth sets, with ground-truth phases and orientations) (Fig. 10a-d). XFEL diffraction images were simulated using an atomicresolution structure of yeast elongation factor 2 obtained by Xray crystallography, which is available in the Protein Data Bank (PDB entry: 1N0U [START_REF] Jørgensen | Two crystal structures demonstrate large conformational changes in the eukaryotic ribosomal translocase[END_REF]) (Fig. 6a). The simulation was done using SPSIM [START_REF] Maia | Single particle diffraction simulator[END_REF], for two different XFEL beam intensities i.e. 10 13 photons/pulse/µm 2 and 10 14 photons/pulse/µm 2 . We found that 10 13 photons/pulse/µm 2 was amongst the lowest beam intensities that could be used with this structure and that the Peak value of the simulated diffraction images was larger than 3 in both cases of beam intensity. Though Table 3 was obtained with a different type of data, it shows that, for P eak > 3, BM3D is the next best performing method after PWPCA. Thus, the results of processing these data using PWPCA are here compared with those obtained using BM3D.

(i) (j) (k) (l) (q) (r) (s) (t) (m) (n) (o) (p) (u) (v) (w) (x) (a) (b) (c) (d) (e) (f ) (g) (h)
The simulated data contain 642 XFEL diffraction images (257 × 257 pixels) for each beam intensity, corresponding to 642 different XFEL beam directions (i.e. to 642 projection directions) uniformly distributed over a sphere with an angular sampling rate of 8 • . In addition, we computed the amplitudes and the phases of the 2D Fourier transforms of the object's 2D projections (or, according to the central slice theorem, central slices of the object's 3D Fourier transform at the same orientations as those of the simulated diffraction images). Two examples of diffraction images (corresponding to two different beam orientations) for the beam intensity of 10 14 photons/pulse/µm 2 are provided in Fig. 7a,e and the same two examples for the beam intensity of 10 13 photons/pulse/µm 2 are provided in Fig. 10a,e. One can notice a concentration of nonzero pixels in a small central region of the image (50 × 50 pixels) and a smaller number of non-zero pixels in the case of lower beam intensity (Figs. 7i,m and10i,m). These diffraction images can be compared with the Fourier intensity data (Figs. 7d,h and 10d,h), whose central regions are shown in Figs. 7l,p and 10l,p. The Fourier intensity data represent expected mean value (the diffraction data as they would be obtained in ideal conditions that are unachievable in practice) and will be here referred to as ground-truth diffraction data. Instead of showing the phase that corresponds to a simulated diffraction image (here referred to as ground-truth phase), we here show the corresponding real-space image. The real-space images obtained by combining the square root of diffraction images and the phase images for the two example beam orientations and intensities are shown in Figs. 7q,u and 10q,u while those obtained by combining the Fourier amplitude and phase images (here referred to as ground-truth real-space images) are shown in Figs. 7t,x and10t,x.

The PWPCA data processing was performed using the parameters provided in Table 1 (with √ M ′ = 101, which corresponds to P eak > 3 according to Eq. 11) while the BM3D data processing was preformed using the same parameters as in [START_REF] Makitalo | Optimal inversion of the anscombe transformation in low-count poisson image denoising[END_REF] ,o and10k,o). One can note that both methods enhance the given diffraction patterns and that PWPCA results in more non-zero pixels recovered than BM3D, which is particularly visible in the case of lower beam intensity (Fig. 10). The real-space images obtained by combining the results of PWPCA (or BM3D) and the ground-truth phases are more similar to the ground-truth real-space images in the case of higher beam intensity (Figs. 7r,v and 7t,x) than in the case of lower beam intensity (Figs. 10r,v and10t,x). In both intensity cases, the 3D structure is globally well recovered, though its resolution is higher in the case of higher beam intensity (Figs. 89) than in the case of lower beam intensity . Indeed, the Fourier shell correlation (FSC) between the recovered density map of the ground-truth structure (Fig. 6b; the density map was obtained using the method introduced in [START_REF] Sorzano | Fast and accurate conversion of atomic models into electron density maps[END_REF]) and the 3D reconstructions (reconstruction in Fourier space using Kaiser-Bessel interpolation) from four data sets for each of the two beam intensities (original, processed by PWPCA, processed by BM3D, and ground-truth sets, with ground-truth phases and groundtruth orientations; Figs. 9 and 12) shows that data processing using any of the two methods (PWPCA or BM3D) improves the resolution of the 3D reconstruction (the spatial frequencies at which the FSC falls below 0.5 are higher for the data processed by PWPCA and BM3D than for the original data) and that this resolution improvement is slightly worse in the case of lower beam intensity (Fig. 11) than in the case of higher beam intensity (Fig. 8). Interestingly enough, PWPCA performs slightly better than BM3D regarding the resolution improvement in the case of lower beam intensity (the FSC falls below 0.5 at a higher spatial frequency when using PWPCA than when using BM3D; Fig. 11).

The 3D reconstruction step requires that phases and orientations of all diffraction images are known (determined by some means prior to 3D reconstruction). As the tasks of orientation determination and phase retrieval are not trivial and prone to errors, we used the groundtruth phases and orientations for the 3D reconstructions. Also, we used the same 3D reconstruction algorithm for all 3D reconstructions. Thus, differences between 3D reconstructions obtained from data processed with PWPCA and BM3D can fully be explained by differences in denoising results of the two methods.

(q) (r) (s) (t) (m) (n) (o) (p) (u) (v) (w) (x) (i) (j) (k) (l) (e) (f ) (g) (h) (a) (b) (c) (d)

Conclusions

In this article, we presented a new method for reducing Poisson noise from images and compared it with several state-of-the-art methods. We showed that the proposed method, referred to as Piecewise Principal Component Analysis (PWPCA), has the best overall performance with respect to other methods when denoising images with Peak = 0.5, 1, 2 and 4. The overall performance of a method for a particular Peak value was evaluated by averaging the Peak Signalto-Noise Ratio (PSNR) values that the given method achieved on different images with the same, given value of Peak. The visual quality of recovered images was consistent with the obtained PSNR values.

Additionally, we showed that PWPCA can be used to preprocess single-particle XFEL diffraction data of biological macromolecules in order to improve the resolution of 3D reconstruction of the macromolecular structure from such data. In this context, PWPCA was compared with BM3D. The two methods provided similar results in the case of high XFEL beam intensities, such as 10 14 photons/pulse/µm 2 . Furthermore, PWPCA provided slightly better results than BM3D in the case of low beam intensities, more difficult to work with, such as 10 13 photons/pulse/µm 2 . These experiments were performed using simulated data. For our future work, we plan to extend the method to applications with experimental XFEL data, which should allow gaining new biological insights.

MATLAB implementation of PWPCA is available from the authors upon request. (The Estimation step: Center updates) 10:

for each k = 1, 2, • • • , K do 11:

µ k := 1 ♯C k ∑ Yi,:∈C k Y i,:
12:

end for 13: until convergence

Fig. 1 :

 1 Fig. 1: Piecewise image denoising in PWPCA. Image pieces (right) overlap (left) to eliminate block type of artifacts.

Fig. 2 :

 2 Fig. 2: Denoising of each image piece in PWPCA.

Algorithm 1 1 :

 11 Pseudocode of the Poisson Piecewise PCA (PWPCA) method proposed here Require: Noisy image y Parameters: Number of pixels in square-size image pieces, M ′ (piece size: √ M ′ × √ M ′ pixels); length of denoising vectors containing vectorized denoising patches of square size, N (denoising patch size: √ N × √ N pixels); number of patch clusters for each image piece, K; number of dictionary components, l; maximum number of iterations, N iter ; stopping tolerance, ε stop . Ensure: Estimated image f 2: Partitioning: divide image y into square-size pieces yq

19 : 20 :

 1920 Concatenation: fuse the denoising clusters in patches F k q Reprojection: To get the estimate of the qth image piece ( fq), average the pixel estimates in overlapping areas 21: end for 22: for each x ∈ I do 23:

Fig. 4 :

 4 Fig. 4: Denoising of Camera (Peak=0.5) using different methods, with the resulting PSNR indicated in decibels.

Fig. 5 :

 5 Fig. 5: Denoising of Barbara (Peak=1) using different methods, with the resulting PSNR indicated in decibels.

Fig. 6 :

 6 Fig. 6: Structure used to simulate XFEL diffraction data and the corresponding density map. (a) Atomic-resolution structure of yeast elongation factor 2 obtained by X-ray crystallography (from Protein Data Bank; PDB: 1N0U). (b) Density map computed from the structure shown in (a), used for computing the Fourier shell correlation curves shown in Fig. 9 and Fig.12.

Fig. 7 :

 7 Fig. 7: Filtering of diffraction images simulated for the beam intensity of 10 14 photons/pulse/µm 2 and the influence of filtering of diffraction images on real-space projection images. (a,e) Two examples of simulated diffraction images corresponding to two different beam orientations (257 × 257 pixels). (b,f) Filtering of images shown in (a,e) using PWPCA, respectively. (c,g) Filtering of images shown in (a,e) using BM3D, respectively.(d,h) Ground-truth diffraction images related to the simulated diffraction images shown in (a,e), respectively (Fourier intensity data representing the diffraction data as they would be obtained in ideal conditions that are unachievable in practice). (i-p) Central region (50 × 50 pixels) of the diffraction images shown in (a-h), respectively. (q-x) Real-space images obtained by combining the square root of the data shown in (a-h) and the ground-truth phases (not shown), respectively.

Fig. 9 :

 9 Fig. 9: 3D reconstructions in the case of beam intensity of 10 14 photons/pulse/µm 2 . (a-d) 3D reconstructions from the following four data sets: the original data set (a), the set processed by PWPCA (b), the set processed by BM3D (c), and the ground-truth data set (d) (ground-truth phases and orientations were used in each set). (e-f) Ground-truth atomic structure (f) and the corresponding density map (e), which are also shown in Fig. 7b,a, respectively.

  . The results of PWPCA for the two example beam orientations and intensities are shown in Figs. 7b,f and 10b,f (the central regions are shown in Figs. 7j,n and 10j,n) while those of BM3D are shown in Figs. 7c,g and 10c,g (the central regions are shown in Figs. 7k

Fig. 10 :

 10 Fig. 10: Filtering of diffraction images simulated for the beam intensity of 10 13 photons/pulse/µm 2 and the influence of filtering of diffraction images on real-space projection images. (a,e) Two examples of simulated diffraction images corresponding to two different beam orientations (257 × 257 pixels). (b,f) Filtering of images shown in (a,e) using PWPCA, respectively. (c,g) Filtering of images shown in (a,e) using BM3D, respectively.(d,h) Ground-truth diffraction images related to the simulated diffraction images shown in (a,e), respectively (Fourier intensity data representing the diffraction data as they would be obtained in ideal conditions that are unachievable in practice). (i-p) Central region (50 × 50 pixels) of the diffraction images shown in (a-h), respectively. (q-x) Real-space images obtained by combining the square root of the data shown in (a-h) and the ground-truth phases (not shown), respectively.

Fig. 11 :

 11 Fig.11: Fourier shell correlation between the density map of the structure used to simulate XFEL diffraction data (Fig.7b) and 3D reconstructions from four data sets for the beam intensity of 10 13 photons/pulse/µm 2 (original, processed by PWPCA, processed by BM3D, and ground-truth sets, with ground-truth phases and orientations) (Fig.13a-d).

Fig. 12 :

 12 Fig. 12: 3D reconstructions in the case of beam intensity of 10 13 photons/pulse/µm 2 . (a-d) 3D reconstructions from the following four data sets: the original data set (a), the set processed by PWPCA (b), the set processed by BM3D (c), and the ground-truth data set (d) (ground-truth phases and orientations were used in each set). (e-f) Ground-truth atomic structure (f) and the corresponding density map (e), which are also shown in Fig. 7b,a, respectively.

Algorithm 2 1 : 2 : repeat 3 : 5 :

 21235 Bregman clusteringRequire: Data points: {Y i,: } M ′ i=1 ∈ R N , number of clusters: K, Bregman divergence: D Φ given by (5) Ensure: Clusters centers: {µ k } K k=1 , partition associated:{C k } K k=1 Initialize {µ k } K k=1 by randomly selecting K elements among {Y i,: } M ′ i=1 (The Assignment step: Cluster updates) 4: Set C k := ∅, 1 ≤ k ≤ K for each i = 1, 2, • • • , M ′ do 6: C k * := C k * ∪ {Y i,: } 7: k * = arg min k ′ =1,••• ,K D Φ (Y i,: ||µ ′ k )

Table 1

 1 PWPCA parameters and their values in the experiments shown in this articleParameters of the PWPCA method are listed in Table1, together with their values in the experiments shown here (Sections 4 and 5). The image piece size is defined as follows:

	Parameter	Meaning	Value
	M ′	Image piece size	See Eq. (11)
	N	Denoising patch size	400 (=20 × 20)
	l	Number of dictionary	4
		components (axes)
	K	Number of clusters	14
	N iter	Iteration limit	20
	ε stop	Stoping tolerance	10 -1
	3.4 Parameters
	√ M ′ =	    

257 P eak ≤ 0.3, 161 0.3 < P eak ≤ 0.8, 121 0.8 < P eak ≤ 3, 101 P eak > 3,

Table 2

 2 Comparison between PWPCA and NLPCA. ∆PSNR is The difference PSNR value between them. The "Average" means the average of PSNR values for all images denoised with the same method.

	Method	Peak Spots Galaxy Cells Barbara Peppers House Camera Bridge	Average
	NLPCA	0.5	28.02	24.59 22.81	18.50	18.77	20.89	19.36	18.49	21.43
	PWPCA	0.5	28.99	24.74 23.08	18.58	18.87	21.04	19.51	18.59	21.68
	∆PSNR	0.5	0.97	0.15	0.27	0.08	0.10	0.15	0.15	0.10	0.25
	NLPCA	1	30.95	25.37 24.13	19.19	19.64	22.19	20.34	18.98	22.60
	PWPCA	1	30.81	25.52 24.57	19.62	19.92	22.78	20.59	19.44	22.90
	∆PSNR	1	-0.14	0.15	0.44	0.43	0.28	0.59	0.25	0.46	0.30
	NLPCA	2	34.07	25.73 25.57	19.67	20.63	23.51	20.76	19.52	23.68
	PWPCA	2	34.82	26.16 25.92	20.63	21.05	24.23	21.47	20.03	24.29
	∆PSNR	2	0.75	0.43	0.35	0.96	0.42	0.72	0.71	0.51	0.61
	NLPCA	4	35.01	26.75 26.34	20.36	20.97	24.81	21.15	20.25	24.46
	PWPCA	4	36.24	27.60 27.01	21.78	22.41	26.21	22.09	20.95	25.54
	∆PSNR	4	1.23	0.85	0.67	1.42	1.44	1.40	0.94	0.70	1.08

Table 3

 3 Comparison of PWPCA against several other methods using synthetic Poisson-noise images with different Peak values and the PSNR measure (in decibels).The best PSNR results are marked in bold. The "Average" means the average of PSNR values for all images denoised with the same method.

	Method	Peak Spots Galaxy Cells Barbara Peppers House Camera Bridge	Average
	PURE-LET	0.5	25.93	24.13	22.19	18.11	18.18	20.11	18.67	18.30	20.70
	BM3D	0.5	23.82	24.75	22.66	18.04	18.48	20.36	18.66	18.36	20.64
	SPDA	0.5	28.34	25.30	23.48	18.53	18.66	20.59	18.80	18.54	21.53
	NLPCA	0.5	28.02	24.59	22.81	18.50	18.77	20.89	19.36	18.49	21.43
	PWPCA	0.5	28.99	24.74	23.08	18.58	18.87	21.04	19.51	18.59	21.68
	PURE-LET	1	27.34	24.98	23.45	18.87	19.33	21.37	19.83	19.06	21.78
	BM3D	1	24.84	25.13	24.36	18.92	20.01	22.34	20.49	19.18	21.91
	SPDA	1	31.14	25.62	24.76	19.39	19.99	22.63	20.17	19.21	22.86
	NLPCA	1	30.95	25.37	24.13	19.19	19.64	22.19	20.34	18.98	22.60
	PWPCA	1	30.81	25.52	24.57	19.62	19.92	22.78	20.59	19.44	22.90
	PURE-LET	2	29.59	26.20	24.88	19.71	20.59	22.37	20.86	19.87	23.00
	BM3D	2	27.59	26.24	25.68	20.49	22.10	23.95	22.23	20.26	23.56
	SPDA	2	32.60	26.47	25.98	19.98	21.23	25.05	21.49	20.14	24.11
	NLPCA	2	34.07	25.73	25.57	19.67	20.63	23.51	20.76	19.52	23.68
	PWPCA	2	34.82	26.16	25.92	20.63	21.05	24.23	21.47	20.03	24.29
	PURE-LET	4	31.60	27.39	26.00	20.46	21.86	23.90	22.12	20.96	24.29
	BM3D	4	30.82	27.59	27.06	22.24	24.19	26.13	24.06	21.62	25.46
	SPDA	4	34.25	27.22	26.71	20.52	22.09	26.02	21.87	20.55	24.90
	NLPCA	4	35.01	26.75	26.34	20.36	20.97	24.81	21.15	20.25	24.46
	PWPCA	4	36.24	27.60	27.01	21.78	22.41	26.21	22.09	20.95	25.54

IET Research Journals, pp. 1-12 c ⃝ The Institution of Engineering and Technology 2015

IET Research Journals, pp. 1-12

c ⃝ The Institution of Engineering and Technology 2015

Acknowledgements

This work was supported by a grant from the National Natural Science Foundation of China (No. 61661039,61661038,61661040),Natural Science Fund of Inner Mongolia Autonomous Region (No. 2016MS0107) and Scientific Research Projection of Higher Schools of Inner Mongolia (No. NJZY16017); and FOCUS for Establishing Supercomputing Center of Excellence (Japan). The authors would also like to thank Joseph Salmon, Zachary Harmany, Charles-Alban Deledalle, and Rebecca Willett for providing their NLPCA algorithm and test images, and Raja Giryes and Michael Elad for providing their SPDA algorithm and test images.

APPENDIX A. Newton Method for Minimizing L

The estimate of Fq is given by

where the elements of the matrix Fq are exp

q ∈ R M ′ ×l and V * q ∈ R l×N are found by minimizing, with respect to Uq and Vq, the loss function L given by

(15) We use Newton method [START_REF] Roy | Finding Approximate POMDP solutions Through Belief Compression[END_REF][START_REF] Gordon | Generalized 2 Linear 2 Models[END_REF] to minimize L. As L is not convex jointly in Uq and Vq, the minimization is done by solving two separate partial optimization problems (the problem is convex for one variable fixed and the other one varying).

The gradient of L(Uq, Vq) with respect to the matrix Uq is given by

and the gradient of L(Uq, Vq) with respect to the matrix Vq is given by

Using element-wise representation, this is equivalent to [START_REF] Foi | Practical poissoniangaussian noise modeling and fitting for single-image raw-data[END_REF] and

In element-wise form, the Hessian matrices lN diagonal matrix) are respectively written as follows:

and

To facilitate the calculation, let us introduce the function

) ⊤ that transforms a matrix into a single column (concatenates the columns) and the function Vect R Vect R :

that transforms a matrix into a single row (concatenates the rows). The Newton updating steps for Uq and Vq can then be written as follows:

and

q )).

Considering [START_REF] Foi | Practical poissoniangaussian noise modeling and fitting for single-image raw-data[END_REF], the ith row of U

, denoted by U (t+1) q,i,: , is updated as follows:

where D i is a diagonal matrix of size N × N i.e.

The term V (t+1) q,:,j , the jth column of V (t+1) q

, is updated in the same way as follows:

where E j is a diagonal matrix of size M ′ × M ′ i.e.

q ] M ′ ,j )).

APPENDIX B. Patch clustering algorithm

The K-means clustering algorithm can be generalized using Bregman divergences (Eq. 5), which was introduced in [START_REF] Banerjee | Clustering with bregman divergences[END_REF] and refereed to as Bregman clustering. We have adapted this algorithm to our piecewise processing approach as shown in Algorithm 2.


Assist. Professor Costas Tsolakidis

Phd Costas Kostalias

A Dynamic Web Site for the Distant Self-Assessment of Students

Keywords: evaluation, distance learning, self-test, environmental education

This work deals with the development of a freely accessible website system, useful for selfassessment of distance students of different levels. The system also offers the teachers the possibility to construct tests and to evaluate their students' progress. The system includes a database that is created in accordance to the theoretical framework that concerns evaluation techniques using new technologies. The database covers the cognitive material of different levels of primary and secondary education. The contact between the final users of the system and the database is achieved using ASP web pages. The database was designed using a Microsoft SQL Server. A pilot test of the system was applied for the topic of Environmental Education in primary and secondary education students.

Introduction

The development of Personal Computers (PCs) and the progressive reduction of their cost gave the possibility almost to everybody to obtain their own PC. Thus, more than ever before, children nowadays use electronic means at home for educational purposes. Educational software, e-encyclopaedias, the Internet are valuable sources that children are using to better prepare their homework. On the other hand, most of the teachers still prepare evaluation tests in handwriting manuscripts. This is so, because electronic evaluation tests adjusted to the school curriculum are not available. This report presents an e-system for the evaluation of students' performance. It is designed in three directions. The first refers to students' self-assessment. For this purpose the system contains a series of questions that students are asked to answer so as to assess their knowledge on selected subjects. Along the same direction, the system gives the students the possibility to check their progress in a number of subjects and guides them to the background theory that should be studied in order to improve their performance. The second direction is to help teachers develop or assemble the evaluation tests related to the subjects they teach. In this sense the teachers use the system as a tool for the development of students' evaluation tests and as a pool for drawing relevant material prepared for this purpose by other colleagues with access to the same system. Finally, the third direction refers to the use of the system as a diagnostic tool at the beginning of a learning process, or at any time that the teacher needs to obtain information about the cognitive level of the class. This system was designed so as to meet four basic characteristics, i.e., to be (a) dynamic, (b) friendly and unsophisticated to the user, (c) able to keep records of various kinds and (d) easily accessible to the users with no time and/or location restrictions.

System Design

The system is addressed to two categories of users -students and teachers-and to all levels of primary and secondary education. The thematic sections that can be selected are based on the school curriculum for different grades. The use of multimedia data attached to each question adds to the attractiveness of this self-assessment test tool. In this way the questions seem smart and intriguing and the quality of the test is high. safety reasons and for the smooth operation of the system since teachers should alone have direct access to their questions and answers.

The main core of the system consists of an evaluation model that was designed so as to allow for different levels of difficulty and a variable pass level, by introducing a random factor both in the choice and in the numbers of questions. The evaluation model is separated into (a) a self-assessment tool, where the result of the rate of success is presented and (b) a test tool where the grade of the examined student is calculated.

The system includes a database and a web site. These two parts are directly connected and interdependent. At the same time, being two separate parts, they can be easily altered and repaired.

The essential functions of the system include management of the accounts, identification of users, editing and pre-overview of questions, transport of multi-media files, evaluation of students and collection of results. The relevant operations concerning these functions can be performed and altered at a distance using the web pages.

The system provides for the possibility of interconnection with external databases or other sources such as the Web or a file in the local PC. This external source of data can be used in the process of student examined so that the necessary theory is referenced. Internally, the system's structure consists of the thematic sections, the levels and the classes, the difficulty levels of the questions and the types of the questions.

Figure1: The cross-correlation of the system's structures

The model developed for the calculation of the students' grade in the test process is based on the random choice of questions and on the difficulty level of each question.

The model adds up each Question's Difficulty Level (QDL) in order to produce the Total Difficulty Level of the test (TDL). The questions in each Test are picked by the system in a random manner hence the difficulty level of each Test has a variable value.

Using the Maximum Test Level (MTL) a grade B for each question is calculated depending on the difficulty of the question. Thus, in the beginning of each test process, the system knows the level of each question. The model is described as follows:

Subjects and thematic sections Unlimited in numbers

Level of difficulties of each question: easy, satisfactory, difficult

Question type. 7 types of questions

The grade of each question Β ν is calculated by the formula:

TDL QDL MTG ν ν * = Β
Where, ΜTG is the Maximum Test Grade.

It follows that

G ΜΤ = Β ∑ ν ν 1
Using the above formulae for the allocation of marks on each question, the following results are achieved:

1. A top score is attributed to students that answer correctly all the questions of the test 2. Difficult questions have a different weighting in the evaluation procedure.

3.

In accordance to (2), a student that answers 50% of the questions does not necessarily get 50% of the top score.

Technical Characteristics and Implementation

The system is based on SQL database where data of users, test questions and a comprehensive record of actions are stored. The database is connected to the interface, a number of dynamic WebPages developed using Visual Basic NET tool. In this way there is the potential to combine the dynamic content of VBScript language with an impressive and friendly Interface, such as HTTP (Diagram 1). The Web Server is connected to an SQL data base server. In addition, the SQL server handles the maintenance process of the data such as the condensing of the data for elimination of possible redundant data and the process of creating backup copies Diagram 2.

Subject for Implementation

Data was fed to the system to evaluate functionality and goal attainment. Environmental Education was selected as suitable subject for a pilot application. Among other reasons, it was preferred to other subjects because (a) it allows a wide use of multimedia, hence it gives grounds for their use; (b) it is a broad and popular subject that forming questions by the teachers is a relatively easy task; (c) it is not an unknown subject to any of the students who were participating in the test run, since all students have some background knowledge and information on Environmental Education from their family, the mass media and their social environment; and (d) it allows for the evaluation not only of knowledge but also of attitudes, stances, opinions, behaviour and values. Two characteristics of Environmental Education, both referring to interdisciplinarity, are worth mentioning here and played a significant role in selecting this subject for pilot purposes. First, Environmental Education functions as an axis along which many other subjects of education can be studied; hence it can be used in support of teaching other subjects. Second, it is a scientific subject that combines knowledge from many disciplines hence it is appropriate for the application of innovative teaching approaches -e.g. it is a subject often taught applying the project method.

INDICATIVE SCREENS OF INTERFACE ENVIRONMENT

Picture1: Indicative web page of the evaluation procedure Picture 2: Indicative web page of the format and the development of questions.

. Picture 3: Web page of the results of evaluation procedure

Test Run

The system was evaluated in two ways. The first was to put the system on a test run in schools

The visitors of the system completed a total of 396 evaluation sessions and answered a total of 5457 questions. The average score of the students in the evaluation sessions, regardless of the thematic modules was 49.6 %. The above shows that the system functions in a satisfactory way and serves a great number of users.

(b) The system was used by many students who did not belong to any of the schools that participated in the experiment. This implies that the experiment was discussed a lot among students and their friends, impressed them and drew their attention. The same result shows that the choice of Environmental Education was right.

(c) The fact that the system was used by students without any guidance from their teachers in relation to the large number of questions that were answered and the long average duration of the connection time indicate that the system is simple, friendly and easily accessible, in other words that it meets the desired characteristics that were set as targets.

(d) The majority of the students that used the system have visited it more than once and from different access points, most likely from their house. This is indicative of how important it is not to have technological restrictions with respect to the operational system, the connection to the Internet and specific software.

(e) From the teacher's point of view, the relatively small average duration of connection to the system (17,01 min) indicated that the design of the relevant tools and the instruction booklet helped them substantially to create their own questions quickly and without any particular problems.

(f) With respect to the types of answers, the greatest percentage of success (60.0%) appeared to questions in which the student had to choose one out of two different answers, probably because of the high probability to choose the correct answer by chance. However in some questions in which the task was the selection of the right picture out of two, the rate of success was relatively low (27.8%). The smallest percentage of success (11,54%) appeared in the question type "one number" or "one phrase" but this is justified by the fact that a. it is an open type question and b. it had a relative great degree of difficulty (30) as it was reported by the teachers that wrote the questions. The majority of the questions that no answer was given were the open type questions, "Give a number" (28,53%) and "Give a phrase" (21,63%), which was expectable. The large percent of success to question type "Give one phase" (37,88%) was impressive.

(g) From the use of the system some educational conclusions emerge that cannot be generalised but are worth mentioning and constitute issues that should be further searched. Thus, so far as attractiveness of sections is concerned the students showed a preference for the section of Air (75), Sea (48) and Forest (45), while the majority of students selected to be self -evaluated in the questions of all the thematic sections. The success order per section was: Forest (60,74%), Sea (56,85%) and Air (48,48%). Finally, no particular difficulty with respect to the percentage of unanswered questions, regarding the thematic section, since it is ranging from 3,73% to 5,76%.

The interview of the teachers was aiming (a) to discuss the conclusions and to investigate in depth their opinions of the process (b) to collect their observations and (c) to identify any possible weaknesses that they could figure out. These formed the basic sections of the interview.

With respect to these axes, the teachers agree that the system was helpful and friendly. They observed that there was a competition among their students as to who will manage to collect the highest score. This was why many students repeated the processes of self-assessment many times.

With respect to the specifications of the infrastructure of each school that participated, the answers differed among teachers. All school computer laboratories were connected through the national school network to the Internet using standard ISDN lines (speed of 64 kbps to 128 kbps). Although the speeds were low, the system responded satisfactorily. The proxy server technology implemented by all school laboratories helped in that. We were led to this conclusion, after a teacher observed that the system had a delay in the process to register new students and more specifically in the process to choose the level and the school unit. Discussing the matter, we realised that the laboratory of the specific school didn't use proxy server and after correction, the problem was cured.

The computer laboratories were relatively old with the age of computers ranging from four to eight years and the operational systems used were Windows 98, Window 2000 and Windows XP. All computers were using Microsoft Internet Explorer in various editions. The students worked in teams of two or three per PC and each one created his own user account in the system. This produced a minor problem since some students did not have enough time to work in the system, during one teaching hour.

A question regarding the weakest point of the system was asked to the teachers. The majority considered that the small number of questions in the thematic section 'Sea' has created problem since the questions were repeated. This was a known characteristic of the system and was confirmed. It is clear that the correct use of this tool requires a very large number of questions in each thematic section in order to have a reliable system of self-assessment after the random choice of questions.

Conclusions

In this work a dynamic network environment, connected with an appropriate database, for the distance self-assessment of the students was presented. The network's URL is http://www.rhodes.aegean.gr/etest. The system was based on a database accessed via an interface using ASP web pages. The system manager of the complete system, according to the rights given, was able to manage the users' accounts, to draw up questions and tests, to fill in sessions of self-assessment and tests and to tabulate repetitive results in time. The system changes the interface, depending on the level of each user and is calculating the grade of the session (test) based on a model that was designed specifically for this purpose. This model automatically calculates the scores allocated to each question of a test procedure, with respect to the level of difficulty and the maximum score from the test (taking into consideration the random selection of questions by the system).

During trial operation, teachers and students of primary and secondary education from the island of Rhodes participated and some encouraging conclusions were drawn as far as the capabilities and the usefulness of the system is concerned. Nevertheless, the system requires development, tests, extensions, modifications and feeding of data so that it can become a really effective tool for the educational community.

  Diagram1.: System and Interface Architecture Diagram 2.: Data Base Structure