domain: math shs

text:
Johanna Carvajal
email: j.benitez.14@ucl.ac.uk

Exploring affective reactions of children to Scratch programming and mathematics: The case of Kim

Keywords: Affect, Scratch programming, mathematics

Background

Around four decades ago the use of programming as a medium for mathematical learning became the focus of research for some mathematics educators [START_REF] Papert | Mindstorms. New York: Basic Books[END_REF][START_REF] Noss | Looking back and looking forward[END_REF]. The new computing curriculum in the UK with a strong focus on programming inspired the ScratchMaths project which rescues this idea and aims to exploit the potential of programming in Scratch to support mathematical thinking [START_REF] Benton | Building mathematical knowledge with programming: insights from the ScratchMaths project[END_REF]. Using this project's designed materials the purpose of my research has been to investigate the kind of affective reactions that emerge from the experience of Year 5 children working with the activities during their computing lessons. Given the connection of the materials with mathematics, and the relationship between mathematical and computational skills, those children that underperform in both subjects have been of particular interest for my study. Three Year 5 classes (sixty 9-10 years old in total) from a primary school in London and taught by the same teacher were observed for sixteen weeks during computing lessons. 11 children were purposively chosen to participate in the research.

Building mainly on the literature on affect and mathematics education [START_REF] Mcleod | Research on affect in mathematics education: A reconceptualization[END_REF]Hannula, 2012) and based on results from the pilot study, three categories concerning efficacy beliefs and emotions were explored: perceived efficacy, general emotion and emotion enjoyment. The three main research questions were:

-What kind of experiences do participants have when working with the activities?

-What changes can be observed (positive/negative evolution of the learner's perspectives)?

-On what might a positive or negative evolution of the learner's perspective depend on?

Methodology

The research used a case study approach and combined different methods for data collection: questionnaires (a short questionnaire administered lesson by lesson during 12 weeks and the attitudes towards Scratch questionnaire), field notes, students' work and critical incident (CI) interviews. The short questionnaire allowed the researcher to capture systematic information along the three categories explored regarding different types of tasks. Indicators above 5 were considered positive reactions, below 5 were considered negative reactions, and equal to 5 were considered neutral. Key moments were identified and were then triangulated with data from other sources.

Results

Kim's responses to the short questionnaire (Figure 1, left) suggest that for her, positive affective reactions tended to happen when the activities were of an exploratory nature and negative reactions Thematic Working Group 08 Proceedings of CERME10 when the activities had a more explicit mathematical focus (Figure 1, right). Nonetheless, she had positive or neutral experiences around 80% of the times that the questionnaire was applied. Data from the affect towards Scratch questionnaire administered in January and June of 2016 indicates that she had had a negative evolution of perspectives as a learner in this context. Data triangulation from all sources suggests that Kim had noticed a connection between the activities and mathematics and that this connection might have played a crucial role in her learning experience.

Children who struggle with mathematics may be in disadvantage when tackling some of the ScratchMaths activities on their own in classroom conditions. In the case of Kim, the involvement of mathematical skills in the activities seemed to trigger negative affective reactions and the belief that she just could not do it. However, when extra material and individual support was given during the lesson she was able to solve the task, help other students in the class, and moreover, she regarded that particular lesson as her best moment working with the activities.

Figure 1 :

 1 Figure 1: On the left, graph of Kim's response to the short questionnaire. On the right, examples of a task with mathematical focus (lesson 5) and a more playful and exploratory task (lesson 6)


text:
Hanna Viitala
email: hanna.l.viitala@uia.no

A case study on Finnish pupils' mathematical thinking: Problem solving and view of mathematics

Keywords: Problem solving, view of mathematics, affect, metacognition, meta-affect

In this article, the mathematical thinking of four Finnish pupils is reported using two temporally different data sets: problem-solving processes and view of mathematics. While the pupils seem similar on the surface level (high achievers, successful problem solvers, enjoy mathematics, motivated to learn mathematics), a closer look at their problem-solving processes and view of mathematics reveal very different strengths and weaknesses in their mathematical thinking. Most of the similarities in this study were found in individual pupils' problem-solving processes and view of mathematics.

Introduction

Developing mathematical thinking is one of the key tasks for mathematics instruction in the Finnish curriculum [START_REF] Fnbe | Perusopetuksen opetussuunnitelman perusteet 2014 [National core curriculum for basic education 2014[END_REF](FNBE, , 2004)). And indeed, Finnish pupils have succeeded well in international studies that assess pupils' mathematical thinking (PISA and TIMSS; see e.g. OECD, 2014; [START_REF] Mullis | PISA 2012 results: What students know and can do -Student performance in mathematics, reading and science[END_REF]. However, the most recent national and international studies show that the mathematics performance of Finnish pupils is descending (e.g. [START_REF] Välijärvi | Osaaminen kestävällä perustalla -Suomen PISA-tulosten kehitys vuosina 2000-2009. Tilannekatsaus helmikuu 2014[END_REF][START_REF] Rautopuro | Hyödyllinen pakkolasku. Matematiikan oppimistulokset peruskoulun päättövaiheessa 2012 [Useful forced landing[END_REF]. Additionally to the alarming trend in mathematics performance, we know very little about Finnish comprehensive school pupils' mathematical thinking that go beyond paper tests. Thus, a quantitative research study was conducted with the aim of describing what characterises Finnish 15year-old pupils' mathematical thinking.

On the way to describe what characterises Finnish pupils' mathematical thinking, the study reported in this article examines four high-achieving Finnish pupils' mathematical thinking through the intertwined relationships of problem-solving processes and view of mathematics. While some of the results of individual pupils' mathematical thinking have been discussed in previous publications [START_REF] Viitala | Alex's world of mathematics[END_REF][START_REF] Viitala | Two Finnish girls and mathematics: Similar achievement level, same core curriculum, different competences[END_REF]2016a), the purpose of this paper is to bring the results together, and answer what similarities and differences related to mathematical thinking can be found between these pupils. With this question, we can reveal some of the possible trends in skills and competences that the Finnish high-achieving pupils might have in their mathematical thinking.

Theoretical framework

Developing pupils' mathematical thinking is in the heart of mathematics education, also according to the Finnish curriculum [START_REF] Fnbe | Perusopetuksen opetussuunnitelman perusteet 2014 [National core curriculum for basic education 2014[END_REF]. While research in mathematics education does not seem to have a common understanding of the meaning of mathematical thinking, [START_REF] Schoenfeld | Learning to think mathematically: Problem solving, metacognition and sense making in mathematics[END_REF] recognised five aspects that are important in a study on mathematical thinking: the knowledge base, problem-solving strategies, monitoring and control, beliefs and affects, and practices. Similar findings have also been found in connection to literature on problem-solving performance [START_REF] Lester | Musings about mathematical problem solving research: 1970-1994[END_REF], and are also listed as part of final-assessment criteria in the Finnish curriculum (see FNBE 2014, pp. 433-434).

Similarly as the most recent theories on affect, mathematical thinking can be viewed through two temporally different aspects: state and trait (cf. [START_REF] Hannula | The structure and dynamics of affect in mathematical thinking and learning[END_REF]2012). On one hand, mathematical thinking is always situational (state). Following [START_REF] Schoenfeld | Learning to think mathematically: Problem solving, metacognition and sense making in mathematics[END_REF] categorisation, it is influenced by the pupils' knowledge base and heuristics, and guided by their metacognitive skills, affects and classroom practices. In this study, mathematical thinking is studied through problem-solving processes. In other words, 'pupils' activities, actions and explanations during problem solving are interpreted as visible signs or expressions of their mathematical thinking' (Viitala, 2015, p. 138).

Pupils' problem-solving behaviour is influenced by pupils' metacognition, affect and meta-affect that occur in a problem-solving situation. The successful application of problem-solving activities at the correct moment is a result of metacognitive skilfulness (e.g. van der Stel, Veenman, Deelen, & Haenen, 2010), affect influence the problem-solving situation for instance through the feeling of confidence, and meta-affect transforms individuals' emotional feelings [START_REF] Debellis | Affect and meta-affect in mathematical problem solving: A representational perspective[END_REF] and directs problem solving behaviour [START_REF] Carlson | The cyclic nature of problem solving: An emergent multidimensional problem-solving framework[END_REF].

On the other hand, problem-solving situations can show patterns of thought that can be interpreted as signs of more stable ways of thinking. Some of these patterns can also be revealed through pupils' view of mathematics (see e.g. Viitala, 2016a). View of mathematics draws from psychological theories. It is a mixture of cognitive, motivational and emotional processes that include for instance beliefs, attitudes, values, feelings and motivation [START_REF] Hannula | The structure and dynamics of affect in mathematical thinking and learning[END_REF]2012). In this study, view of mathematics is studied through four components: mathematics (as science and as a school subject), oneself as a learner and user of mathematics, learning mathematics, and teaching mathematics [START_REF] Pehkonen | Pupils' view of mathematics: Initial report for an international comparison project[END_REF], cf. Op't Eynde, de Corte, & Verschaffel, 2002).

Methods

Data collection

At the time of data collection, the four pupils (Alex, Daniel, Emma and Nora) were 15 years old and in their 9 th and final year of compulsory school in Finland. Additionally, they were all high achievers (mathematics grades between 9 and 10 on a whole number scale of 4 to 10).

The data was collected in three cycles over the course of three months. In each cycle, one mathematical task was solved in an ordinary classroom situation as a 'main task'. The pupils solved the tasks individually but they were allowed to talk about the tasks with a friend or ask for help from the teacher. In each cycle, the pupils were video recorded while they solved the task(s) in class and their solutions on paper were collected. Below, there is an example of a main task (School Excursion, OECD, 2006, p. 87).

A school class wants to rent a coach for an excursion, and three companies are contacted for information about prices.

Company A charges an initial rate of 375 zed plus 0.5 zed per kilometre driven. Company B charges an initial rate of 250 zed plus 0.75 zed per kilometre driven. Company C charges a flat rate of 350 zed up to 200 kilometres, plus 1.02 zed per kilometre beyond 200 km.

Which company should the class choose, if the excursion involves a total travel distance of somewhere between 400 and 600 km?

Thematic Working Group 08

In each cycle, the pupils were interviewed individually. The interviews took place either on the same day, or on the next day after solving the task in the classroom. The interviews contained two parts. The first part concentrated on affective traits and treated the following themes: pupil's background, mathematical thinking, and pupil's view of mathematics (following the categorization of [START_REF] Pehkonen | Pupils' view of mathematics: Initial report for an international comparison project[END_REF] see example questions in Table 1, Viitala, 2016aViitala, , p. 1295)). This part of the interview was semi-structured and focused [START_REF] Kvale | InterViews[END_REF]. The second part of the interview was about problem solving. The classroom data was used as stimuli when the pupil's problem-solving process was discussed. The pupils were asked to explain their thinking and actions during the problem-solving situation and additional questions were asked (e.g. what are you thinking now? Why are you doing so? What did you feel when you read the task? Did you think about your own thinking when solving the task?).

Finally, in each interview, the pupils were asked to assess their confidence before, during and after solving the problem, as well as their confidence in school mathematics using a 10 cm line segment (scale from 'I couldn't do it at all' to 'I could do it perfectly'). All interviews were video recorded.

Analysis

Following the state and trait aspects of the study, the analysis was divided into two sections: problem solving and view of mathematics. The problem-solving processes were analysed first by going through the problem-solving phases introduced by [START_REF] Carlson | The cyclic nature of problem solving: An emergent multidimensional problem-solving framework[END_REF]: orienting, planning, executing and checking (cf. [START_REF] Polya | How to solve it[END_REF]. Then the results on problem-solving behaviour were complemented with metacognitive activities (orientation, planning, evaluating and elaboration van der [START_REF] Van Der Stel | The increasing role of metacognitive skills in math: A cross-sectional study from a developmental perspective[END_REF], affect (state and trait, as well as cognition, emotion, motivation; [START_REF] Hannula | The structure and dynamics of affect in mathematical thinking and learning[END_REF]2012) and meta-affect [START_REF] Debellis | Affect and meta-affect in mathematical problem solving: A representational perspective[END_REF] emerging in problem-solving processes. Finally, the pupils' confidence to solve the problems was analysed.

The first analysis of the pupils' view of mathematics followed the themes of data collection [START_REF] Pehkonen | Pupils' view of mathematics: Initial report for an international comparison project[END_REF]. After condensing the results, a pupil profile was created to be used as background information about the pupil. Pupil profile is a short description of the pupil that is based on the pupil's mathematics grade, motivation to learn mathematics, and the core of his view of himself as a learner of mathematics (ability, success, difficulty of mathematics, and enjoyment of mathematics, following [START_REF] Rösken | Dimensions of students' views of themselves as learners of mathematics[END_REF].

In the end, the results of problem solving and view of mathematics were compared to see if there are similarities in pupil's problem-solving skills (state) and competences found through pupil's view of mathematics (trait). More details of the methods used in the study are reported for instance in Viitala (in press).

Results

On a surface level, Alex, Daniel, Emma and Nora seem quite similar: they are all high achievers in mathematics, they enjoy mathematics, and they are motivated to learn mathematics (see excerpts in Table 2). They are also successful problem solvers, that is, they could solve all the problems given to them in the study and justify their answers and solutions. However, a deeper look at their problem solving and view of mathematics introduce four pupils with a very different skills and competences. In the following, the key results of each pupil will be introduced individually.

Alex is very fluent and thorough mathematics learner and problem solver. He can move naturally between different phases of problem solving. He is aware of his own thinking and fluent in explaining and justifying his cognitive and metacognitive actions in problem solving. Similarly, when explaining his learning of mathematics, he says he is actively seeking for connections between new knowledge and prior knowledge, and he is able to spontaneously give examples of this behaviour. He says he trusts his own thinking more than his calculations, and shows to be able to direct his behaviour according to his affects in problem solving. He is confident in school mathematics but in the interviews, he constantly compares his abilities to mathematics as a science and recognises that there is much more than school mathematics (more results in [START_REF] Viitala | Alex's world of mathematics[END_REF][START_REF] Viitala | A framework for studying students' mathematical thinking at different ages: A longitudinal case study of Alex[END_REF].

Whereas Alex seems to be very fluent in every aspect of mathematical thinking studied in this research project, from a similar starting point, Daniel shows somewhat different strengths in mathematics. Unlike any of the three other pupils, he is extremely confident in mathematics. He says that mathematics is easy for him, and he shows to be very aware of his success in mathematics.

His confidence seems to guide also his problem-solving processes. He is able to move fluently back and forth between problem-solving phases and is skilful in performing metacognitive acts. However, even though (or because of) learning mathematics and solving problems are easy for him, he cannot explain the processes he goes through in or for learning, and he has problems in explaining his problem-solving actions after the problem-solving situation. An illustrative example of this situation is Daniel's explanation about how he learns mathematics: pieces just click together or things become familiar (more results in Viitala, in press).

Similarly as in Daniel's case, also Emma's learning of mathematics and problem solving are strongly influenced by her confidence in mathematics, or more precisely, her lack of confidence.

Because of the uncertainty in mathematics, for Emma, learning takes time and effort. She says she learns every topic as a separate entity, and she is able explain the steps that are needed for her to learn a new thing. Similarly, she uses a considerable amount of time for orienting and planning in problem solving. After understanding the problem and the given data, she is able to follow her plan through and check her solution. It seems that Emma's uncertainty in mathematics makes her work harder, and through hard work, she succeeds in mathematics. Moreover, she says that succeeding in mathematics and understanding it, makes it worthwhile studying. On the other hand, affect can also be an obstacle in her problem solving, since she does not seem to have efficient tools to overcome the feeling of getting stuck (more results in [START_REF] Viitala | Two Finnish girls and mathematics: Similar achievement level, same core curriculum, different competences[END_REF]2016a).

Also for Nora, learning mathematics takes time and effort but after learning something, applying is easy. She says that she is quite confident in mathematics and likes learning mathematics very much.

She is capable in explaining her thinking and problem solving, and connecting mathematics to her own life. She also has a diverse view of mathematics as a science. In problem solving, she is flexible in directing her actions based on the affective states occurring in problem-solving situations. She is also fluent in moving between orienting, planning and executing in problem solving. However, given the choices she had made while planning, she is happy with the first answer she gets, and does not check her results (more results in [START_REF] Viitala | Two Finnish girls and mathematics: Similar achievement level, same core curriculum, different competences[END_REF].

Ability and success

Some reflections of the results

In addition to forming descriptions of pupils' mathematical thinking, and showing pupils' strengths, the study also revealed issues that pupils could work with in order to develop their mathematical thinking. For instance, even though Alex was fluent in problem solving and school mathematics, he did not relate the problems to real life and his view of mathematics outside school was quite limited (see [START_REF] Viitala | Alex's world of mathematics[END_REF][START_REF] Viitala | A framework for studying students' mathematical thinking at different ages: A longitudinal case study of Alex[END_REF]. Recognising mathematics more in his own life could enrich Alex's view of mathematics, and through that, also his understanding of school mathematics might develop. Daniel, on the other hand, had problems explaining his thinking after the problem-solving situation and had similar problems with explaining his mathematics learning (see Viitala, in press). Problem solving and learning mathematics might be easy for Daniel in compulsory school, but what happens if (when) the situation changes? Becoming aware of his own learning and problem-solving processes could help him cope in new situations and develop his metacognitive skills not only in mathematics but also in other school subjects.

Emma's weak point was her uncertainty which she had turned into success in problem solving and learning of mathematics. She had overcome some of the uncertainty with the support of her family (see Viitala, 2016a). However, because she was not confident in mathematics, she learnt every topic in mathematics as its own entity, and did not connect it to prior knowledge. This might also hinder her learning. Hence, supporting Emma emotionally could open doors to more thorough learning and understanding of mathematics. Finally, Nora's results were not always correct, and both her activities and explanations showed that she does not evaluate her problem-solving process or check her results (see [START_REF] Viitala | Two Finnish girls and mathematics: Similar achievement level, same core curriculum, different competences[END_REF]. Supporting her to look back, and perhaps exposing her more to, for instance, open problems, might help her to become more reflective user and learner of mathematics.

Summary and discussion

The purpose of the paper was to answer the question what similarities and differences related to mathematical thinking can be found between the four Finnish high-achieving pupils. Mathematical thinking was studied through two temporally different data sets: problem-solving processes (state) and view of mathematics (trait). The results showed that the similarities between the pupils were found to be mainly on a surface level: all the pupils liked mathematics, were motivated to learn it, enjoyed doing mathematics and were successful problem solvers. However, after a deeper look into their problem-solving processes and view of mathematics, the study revealed a great deal of differences between the pupils, and showed different competences: Alex is a very conscious thinker and learner of mathematics, and excellent in justifying his thinking and actions in mathematics.

Daniel is extremely confident and metacognitive skills are prominent in his problem solving. Emma is an unsure but very thorough problem solver and learner of mathematics. Nora is fluent in expressing her thoughts and connecting mathematics to real life.

In addition to the strengths found in these four pupils, the framework also revealed some of their weaknesses. The strengths, together with the weaknesses can be used to support individual pupils' development in mathematics. For instance, Alex seemed to see mathematics only as a tool to solve something and his view of mathematics outside school was quite limited (see [START_REF] Viitala | Alex's world of mathematics[END_REF][START_REF] Viitala | A framework for studying students' mathematical thinking at different ages: A longitudinal case study of Alex[END_REF]. This knowledge can be used to develop pupil's mathematical thinking. Four years after the data collection of this research project, I met Alex again. At this point, Alex was as a university student. He explained that only after realising the tool value that mathematics had for him, and learning that mathematics is not just calculations but also ways of thinking, he began to see mathematics everywhere in his real life, and he began to use his mathematical thinking more creatively (see [START_REF] Viitala | A framework for studying students' mathematical thinking at different ages: A longitudinal case study of Alex[END_REF].

All in all, the results showed that even though the pupils seem similar on the surface level, on a closer look, they have very different skills and competences in mathematics. This is an indication that the framework allows different pupils to show different strengths, and also different weaknesses in problem solving and learning of mathematics. Hence, the framework could assist also teachers to pay attention to the aspects that pupils might need help with in developing their mathematical thinking, which in turn can help the pupils to recognise the knowledge, skills and affects that might need further developing (cf. FNBE, 2014, p. 377;Viitala, in press; see also [START_REF] Viitala | Two Finnish girls and mathematics: Similar achievement level, same core curriculum, different competences[END_REF]. An example of how teachers can use this framework to support their teaching is presented in Viitala (in press).

Table 1 : Interview themes and example questions.

 1

	Theme	Example questions
	Background	Tell me about your family.
	Mathematical thinking	What does mathematical thinking mean? / How do you recognise it?
	Mathematics	What is mathematics as a science? / Does it exist outside of school? (How? Where?)
	Oneself and mathematics	Is mathematics important to you? / Does it help you think logically? (How?)
	Learning mathematics	How do you learn mathematics? / Is it most important to get a correct answer?
	Teaching mathematics	Does teaching matter to your learning? (How?) / What is good teaching?

Table 2 : Examples of pupils' own statements about their view of mathematics (cf. pupil profile).

 2

Proceedings of CERME10


Frédéric Sur

A Non-Local Dual-Domain Approach to Cartoon and Texture Decomposition

Keywords: Cartoon and texture decomposition, dual-domain algorithm, non-local estimation, Fourier domain

This paper addresses the problem of cartoon and texture decomposition. Microtextures being characterized by their power spectrum, we propose to extract cartoon and texture components from the information provided by the power spectrum of image patches. The contribution of texture to the spectrum of a patch is detected as statistically significant spectral components with respect to a null hypothesis modeling the power spectrum of a non-textured patch. The null-hypothesis model is built upon a coarse cartoon representation obtained by a basic yet fast filtering algorithm of the literature. Hence the term "dual domain": the coarse decomposition is obtained in the spatial domain and is an input of the proposed spectral approach. The statistical model is also built upon the power spectrum of patches with similar textures across the image. The proposed approach therefore falls within the family of non-local methods. Experimental results are shown in various application areas, including canvas pattern removal in fine arts painting, or periodic noise removal in remote sensing imaging.

Introduction

Decomposing an image as the sum of geometric and textural components is a problem of image analysis that has lead to a great deal of work. In this problem, known as cartoon and texture decomposition, the cartoon component is piecewise smooth, made of the geometric shapes of the images, and the texture component is made of stationary or quasi-stationary oscillatory patterns filling the shapes. Of course, the problem, based on perceptual clues, is intrinsically illposed and any definition of cartoon and texture is subject to discussion. After an overview of the literature, we introduce a new algorithm for cartoon and texture decomposition based on a dualdomain approach [START_REF] Knaus | Dual-domain filtering[END_REF]: a spatial domain filtering gives a coarse decomposition, which permits estimating statistics of the power spectrum of cartoon patches. Estimating these statistics is based on the similarity of textured patches across the image, in the same spirit as the non-local approaches [START_REF] Buades | A review of image denoising algorithms, with a new one[END_REF]. Therefore, the resulting algorithm performs a non-local dual-domain cartoon and texture decomposition.

Related works

Most authors define the cartoon component as a bounded-variation function whose total variation is controlled, which discards oscillatory patterns and enforces its piecewise constant nature, giving cartoon components with sharp edges. In contrast to the cartoon component, the texture component is the subject of discussions and a huge literature has emerged in the last decades. Variational approaches aim at decomposing the input image as the sum of the cartoon and texture component by minimizing some energy, often defined as the sum of the total variation (or some variant of it) of the cartoon part and of some norm of the texture part. This latter norm may be defined as the L 2 -norm [START_REF] Rudin | Nonlinear total variation based noise removal algorithms[END_REF], L 1 -norm [START_REF] Chan | Aspects of total variation regularized L1 function approximation[END_REF], Meyer's G-norm [START_REF] Gilles | Properties of BV -G structures + textures decomposition models. application to road detection in satellite images[END_REF][START_REF] Meyer | Oscillating Patterns in Image Processing and Nonlinear Evolution Equations: The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures[END_REF], Gabor-norm [START_REF] Aujol | Structure-texture image decompositionmodeling, algorithms, and parameter selection[END_REF], or norms enforcing a sparse representation in an adapted basis [START_REF] Starck | Image decomposition via the combination of sparse representations and a variational approach[END_REF], just to cite a few. The authors of [START_REF] Fehrenbach | Variational algorithms to remove stationary noise: Applications to microscopy imaging[END_REF] address the problem of structured noise, which can be seen as a texture covering the whole image domain. Structured noise is indeed modeled as a stationary process and a variational approach to denoising is proposed in [START_REF] Fehrenbach | Variational algorithms to remove stationary noise: Applications to microscopy imaging[END_REF]. Recently, it has been proposed [START_REF] Schaeffer | A low patch-rank interpretation of texture[END_REF] to model the texture part as the sum of a small number of base textures in a low patch-rank approach. According to the authors of [START_REF] Ono | Cartoon-texture image decomposition using blockwise low-rank texture characterization[END_REF], low patch-rank outperforms previous approaches which smooth the cartoon component by transferring high-frequency information from edges to the texture instead of the cartoon component. It is, however, a global approach which is not suited to natural images where textures show different patterns of different nature. A local approach based on the block nuclear norm is proposed in [START_REF] Ono | Cartoon-texture image decomposition using blockwise low-rank texture characterization[END_REF]. The interested reader is asked to refer to the complete discussion in [START_REF] Aujol | Structure-texture image decompositionmodeling, algorithms, and parameter selection[END_REF][START_REF] Schaeffer | A low patch-rank interpretation of texture[END_REF][START_REF] Ono | Cartoon-texture image decomposition using blockwise low-rank texture characterization[END_REF][START_REF] Buades | Fast cartoon + texture image filters[END_REF].

All these methods are slow, because it is difficult to find a numerical solution of the abovementioned variational problems. A fast non-linear filter for cartoon and texture decomposition is proposed by the authors of [START_REF] Buades | Fast cartoon + texture image filters[END_REF] (software available in [START_REF] Buades | Cartoon+texture image decomposition[END_REF]). It is based on the relative reduction rate of the local total-variation across scale, which is a good feature to measure whether an image point belongs to a textured region or not. This spatial filter giving quite a blurry cartoon part, a follow-up paper [START_REF] Buades | Directional filters for color cartoon+texture image and video decomposition[END_REF] (software in [START_REF] Buades | Directional filters for cartoon + texture image decomposition[END_REF]) makes use of banks of filters adapted to image edges. A scale factor governs the trade-off between textures and shapes, large-size texture details being considered as shapes rather than texture. Recent papers investigate the use of a guidance signal for image filtering [START_REF] Ham | Robust guided image filtering using nonconvex potentials[END_REF][START_REF] Sun | Image structure retrieval via L0 minimization[END_REF][START_REF] Zhang | Rolling guidance filter[END_REF] (after [START_REF] He | Guided image filtering[END_REF]), which makes it possible to remove small-scale details without affecting other structures. Applications of this approach include texture removal, the difference with the aforementioned methods is probably that the retrieved cartoon component is quite sketchy.

Many of these works are based on the fact that textures are well represented in the Fourier domain. It is indeed well known that microtextures are characterized by their power spectrum, as discussed in, e.g., [START_REF] Wang | Texture classification using texture spectrum[END_REF] concerning texture feature extraction, and in [START_REF] Galerne | Random phase textures: Theory and synthesis[END_REF], after [START_REF] Van Wijk | Spot noise texture synthesis for data visualization[END_REF], for texture rendering. The power spectrum of a texture even generally reduces to a few spikes of various amplitudes, which justifies the sparse representation of [START_REF] Starck | Image decomposition via the combination of sparse representations and a variational approach[END_REF][START_REF] Schaeffer | A low patch-rank interpretation of texture[END_REF]. Some authors make use of such a property to design notch filter to remove structured noise covering the whole image domain [START_REF] Aizenberg | A windowed Gaussian notch filter for quasi-periodic noise removal[END_REF][START_REF] Cannon | Background pattern removal by power spectral filtering[END_REF][START_REF] Hudhud | Digital removal of power frequency artifacts using a Fourier space median filter[END_REF][START_REF] Sur | Automated removal of quasiperiodic noise using frequency domain statistics[END_REF][START_REF] Sur | An a-contrario approach to quasi-periodic noise removal[END_REF]. In these papers, the difficulty is to automatize the detection of texture spikes. It is proposed in [START_REF] Sur | Automated removal of quasiperiodic noise using frequency domain statistics[END_REF] to detect texture spikes as deviation of the expected power spectrum of natural images, which decreases proportionally to some power of the inverse of the frequency [START_REF] Van Der Schaaf | Modelling the power spectra of natural images: Statistics and information[END_REF]. The expected power spectrum is simply obtained by averaging patches covering the whole image as in [START_REF] Cannon | Background pattern removal by power spectral filtering[END_REF], which is not possible when considering localized textures.

The method was adapted to a-contrario detection in [START_REF] Sur | An a-contrario approach to quasi-periodic noise removal[END_REF]. Removing canvas patterns in fine art paintings can also be considered as a form of textured noise removal. For instance, the authors of [START_REF] Cornelis | Removal of canvas patterns in digital acquisitions of paintings[END_REF] remove canvas patterns in paintings by fitting a parametric model of the canvas to the texture part given by the algorithm of [START_REF] Buades | Fast cartoon + texture image filters[END_REF].

Contribution

The proposed contribution consists in locally detecting the texture components as statistically significant Fourier coefficients under a null hypothesis modeling non-textured patches. Our statistical model is based on two ingredients: first, the Gaussian nature of image patches, which is the ground of several recent papers on denoising [START_REF] Arias | Video denoising via empirical Bayesian estimation of space-time patches[END_REF][START_REF] Houdard | High-dimensional mixture models for unsupervised image denoising (HDMI)[END_REF][START_REF] Lebrun | A nonlocal Bayesian image denoising algorithm[END_REF][START_REF] Yu | Solving inverse problems with piecewise linear estimators: From Gaussian mixture models to structured sparsity[END_REF]; second, the use of a coarse cartoon and texture decomposition which permits estimating statistics of non-textured patches in a non-local way. Section 2 gives the expectation and variance of the power spectrum components of a random Gaussian non-textured patch. Since the probability distribution of patches is not constant across the whole image domain, these statistics are empirically estimated in a non-local way. We select a sample of patches with similar power spectra and extract the cartoon parts with the coarse yet fast algorithm of [START_REF] Buades | Fast cartoon + texture image filters[END_REF][START_REF] Buades | Cartoon+texture image decomposition[END_REF] in order to estimate the above-mentioned statistics. This approach is in line with the famous non-local means methodology [START_REF] Buades | A review of image denoising algorithms, with a new one[END_REF] in which similar patches are assumed to be independent realizations of the same random patch.

Section 3 explains how to build a statistical test to detect texture spectrum spikes in each image patch, based on the null model of Section 2. A non-linear filter separating cartoon and texture components is thus designed, and implemented using the discrete windowed Fourier transform.

Since texture is extracted in the Fourier domain from a coarse information coming from a spatial filter, this is a dual-domain approach, a term coined by the authors of [START_REF] Knaus | Dual-domain filtering[END_REF] when proposing to refine denoising in the frequency domain after a first estimation in the spatial domain. Note that the present contribution is not, strictly speaking, a dual-domain filter in the sense of [START_REF] Knaus | Dual-domain filtering[END_REF].

Experiments are discussed in Section 4. In order to reproduce the experiments presented here, we make a Matlab code available on the following webpage, together with a companion document showing additional experiments: https://members.loria.fr/FSur/software/NoLoDuDoCT/

Notation and reminder

The Fourier transform of any image patch p defined over the pixel domain D p = {1, . . . , L} × {1, . . . , L} is denoted by p and is such that for any ξ = (ξ, η) ∈ {1, . . . , L} × {1, . . . , L},

p(ξ) = x∈Dp p(x)e -2iπ<x,ξ> (1)
where x = (x, y) ∈ D p and < x, ξ >= (xξ + yη)/L. The modulus and conjugate of any complex number z are denoted by |z| and z. The power spectrum of a patch p is | p| 2 .

The expectation and variance of a random variable V are denoted by E(V ) and var(V ), respectively. The covariance of two random variables V and V is denoted by cov(V, V ). The empirical estimation of a statistic S is denoted by S.

We recall that if n is a 0-mean noise patch of variance σ 2 (x) at pixel x, then the Fourier transform of n satisfies, for any (ξ, η), E( n(ξ, η)) = 0, and E(| n(ξ, η)| 2 ) = σ 2 where σ 2 = x∈Dp σ 2 (x) is the total variance calculated over the patch domain.

The power spectrum of Gaussian patches

The Fourier transform of a Gaussian patch follows a complex normal distribution, and the power spectrum coefficients follow a non-central generalized chi-square distribution. While accurate approximations of this distribution are available [START_REF] Imhof | Computing the distribution of quadratic forms in normal variables[END_REF][START_REF] Liu | A new chi-square approximation to the distribution of non-negative definite quadratic forms in non-central normal variables[END_REF], their computational cost is not suitable for our use. It is a standard trick to use the Gaussian approximation which is more tractable and justified by the central limit theorem.

Expectation and variance of the power spectrum

We assume that any random patch p 0 = c+n is the sum of a random non-textured component c and a Gaussian noise n of variance σ 2 (x) at pixel x, statistically independent from c(x).

Since we aim at building a statistical model for the absence of texture (null hypothesis), we assume that p 0 does not show any texture, although this does not impact the calculation of the expectation and variance in this section.

The Fourier transform being linear, p 0 = c + n, and the power spectrum of the patch p thus writes, for any frequency ξ = (ξ, η):

| p 0 (ξ)| 2 = | c(ξ)| 2 + | n(ξ)| 2 + c(ξ) n(ξ) + c(ξ) n(ξ) (2)
We now calculate the expectation and variance of | p 0 (ξ)| 2 , for any ξ. For better readability, we drop ξ in the following equations.

Expectation

Since c and n are independent, E( n) = 0, and E(| n| 2 ) = σ 2 , the following equation holds:

E(| p 0 | 2 ) =E(| c| 2 ) + E(| n| 2 ) + E( c)E( n) + E( c)E( n) =E(| c| 2 ) + σ 2 (3)

Variance

Variance algebra gives:

var(| p 0 | 2 ) = var(| c| 2 ) + var(| n| 2 ) + var( c n + c n) + 2 cov(| c| 2 , | n| 2 ) + 2 cov(| c| 2 , c n + c n) + 2 cov(| n| 2 , c n + c n) (4)
We successively calculate:

• cov(| c| 2 , | n| 2 ) = 0 since c and n are independent. • cov(| c| 2 , c n + c n) = E((| c| 2 -E(| c| 2 ))( c n + c n)) = E((| c| 2 -E(| c| 2 )) c)E( n) + E((| c| 2 - E(| c| 2 )) c)E( n) = 0 since c and n are independent and E( n) = 0. • cov(| n| 2 , c n + c n) = E((| n| 2 -E(| n| 2 ))( c n + c n)) = E(| n| 2 n)E( c) + E(| n| 2 n)E( c) - E(| n| 2 )E( c)E( n) -E(| n| 2 )E( c)E( n) = 0 since E(| n| 2 n) = E( n) = 0. • var( c n+ c n) = E(( c n+ c n) 2 ) = 2Re(E( c 2 )E( n 2 ))+2E(| c| 2 )E(| n| 2 ) ≤ 4E(| c| 2 )E(| n| 2 ) since
for any complex number z, Re(z) ≤ |z|, and for any random variable

V , |E(V )| ≤ E(|V |). • concerning var(| n| 2 ) = E(| n| 4 ) -E(| n| 2 ) 2 . By definition of n, E(| n| 4 (ξ)) = E(n x,y n x ,y n x ,y n x ,y )e -2iπ x-x +x -x L ξ+ y-y +y -y L η (5)
where the summation is over (x, y), (x , y ), (x , y ), and (x , y ) spanning the image domain. Now, E(n x,y n x ,y n x ,y n x ,y ) = σ 4 if (x, y, x , y ) = (x , y , x , y ), or if (x, y, x , y ) = (x , y , x , y ), or if (x, y, x , y ) = (x , y , x , y ), and it simplifies into E(n x,y n x ,y n x ,y n x ,y ) = 0 otherwise. Thus,

E(| n(ξ)| 4 ) = x,x 2 + e -4iπ<x-x ,ξ> σ 2 (x)σ 2 (x ) (6) = 2σ 4 + x σ 2 (x)e -4iπ<x,ξ> 2 (7) Since var(| n| 2 ) = E(| n| 4 ) -σ 4 , var(| n| 2 (ξ)) = σ 4 + σ 2 (ξ) 2 ≤ 2σ 4 (8)
the latter inequation holding thanks to Bessel inequality.

To sum up the discussion, we can conclude that:

var(| p 0 | 2 ) = var(| c| 2 ) + σ 4 + V (9)
where

V ≤ 4σ 2 E(| c| 2 ) + σ 4 .

Estimating the statistics of a non-textured random patch

In Section 3, texture components are detected in the power spectrum of a patch as deviations to a Gaussian model of the power spectrum of non-textured random patches. We first need to empirically estimate the expectation and variance of the power spectrum of a non-textured patch p 0 . We can see from Equation 3 and 9 that this requires to estimate the expectation and variance of the power spectrum of the cartoon component c, and the noise total variance σ 2 =

x σ 2 (x).

Estimating cartoon statistics

Although the approach of [START_REF] Buades | Fast cartoon + texture image filters[END_REF] gives a coarse cartoon output which is often quite smooth, it is fast to compute. We therefore take it as an initial guess of the cartoon component of any patch in the image. Smoothness is even an advantage here. Indeed, it makes it possible to assume that noise has been removed from the cartoon component. In order to estimate both the expectation E(| c| 2 ) and the variance var(| c| 2 ) for any random patch p = c + n, we make use of a non-local estimation approach. More precisely, for a patch p x centered at any pixel x, we extract from the input image the set S of patches such that their power spectrum are the N nearest neighbors of | p x | 2 in the sense of the Euclidean norm. These patches are thus likely to share the same texture and to be realizations of the same random process. Weighted sample estimates of the expectation E(| c| 2 ) and variance var(| c| 2 ) are calculated based on the cartoon components of the patches in S. The weights take into account dissimilarities and are proportional to e -||| c| 2 -| s| 2 ||/λ(x) for any patch s ∈ S, λ(x) being a normalizing parameter defined later.

In practice, the proposed procedure is made of two steps. First, the power spectra of image patches and of the corresponding cartoon components are extracted from the whole image. In order to deal with shape edges and avoid mixing texture information from two adjacent regions, the contribution of each pixel is weighted as a non-decreasing function of the similarity to the central pixel in the cartoon domain. Indeed, it has been shown in several papers [START_REF] Deledalle | Non-local methods with shape-adaptive patches (NLM-SAP)[END_REF][START_REF] Foi | Pointwise shape-adaptive DCT for high-quality denoising and deblocking of grayscale and color images[END_REF] that adapting patches to the shape is a good way to keep non-local methods from mixing information from overlapping shapes. Moreover, in order to deal with border discontinuities and avoid spectral leakage, we also multiply patches by a Gaussian window. Therefore, denoting by I the input image, by D = {1 . . . X} × {1 . . . Y } its domain, and by C c its coarse cartoon representation obtained by [START_REF] Buades | Fast cartoon + texture image filters[END_REF], we actually compute modified windowed Fourier transforms defined at each pixel x as:

I x (ξ) = y∈D g α (x -y) g β (C c (x) -C c (y)) I(y) e -2iπ<y,ξ> (10)
C x (ξ) = y∈D g α (x -y) g β (C c (x) -C c (y)) C c (y) e -2iπ<y,ξ> (11)
where g α and g β are Gaussian functions of standard deviation α and β, g α being truncated to define a L×L patch. The corresponding power spectra are denoted by P I x and P C x . By definition,

P I x (ξ) = |I x (ξ)| 2 and P C x (ξ) = |C x (ξ)| 2
for any frequency ξ = (ξ, η) ∈ {1, . . . , L} × {1, . . . , L}. We extract these power spectra in such a way that their overlap is limited in order to avoid bias when estimating sample mean and sample variance. In practice, we impose that the patch center x lies on a subsampled grid D s made of regularly distributed pixel locations at a distance of s pixels.

The second step consists in estimating at any pixel x ∈ D the expectation and variance of the power spectrum of cartoon components based on the P I

x i and P C x i calculated above for x i ∈ D s . At any x ∈ D, these statistics are estimated over the set of the N cartoon power spectra

P C x i , x i ∈ D s , 1 ≤ i ≤ N , (12)
with the x i defined such that

P I x i , x i ∈ D s , 1 ≤ i ≤ N (13)
is the set of the N -nearest neighbors of the query power spectrum P I x . We use weighted estimates to take the dissimilarity of the nearest power spectra into account. For any ξ,

E(P C x (ξ)) = 1 N i=1 w i N i=1 w i P C x i (ξ) (14) var(P C x (ξ)) = 1 N i=1 w i N i=1 w i P C x i (ξ) -E(P C x (ξ)) 2 (15)
where the weights w i satisfy

w i = e -||P I x -P I x i || 2 /λ(x) 2 (16)
|| • || being the Euclidean norm of the components of frequency larger than 2/L cycles per pixel (to remove the effect of very low frequency changes), and λ(x) being a normalization parameter equal to the median of the distances of the N nearest neighbors of P I

x . The goal of this normalization, similarly to the so-called contextual dissimilarity measure [START_REF] Jégou | Accurate image search using the contextual dissimilarity measure[END_REF], is to adapt the weights to the non-homogeneous distribution of the power spectra.

The nearest neighbors of the current power spectrum are found over the whole image by using the efficient FLANN library [START_REF] Muja | Scalable nearest neighbor algorithms for high dimensional data[END_REF].

It should be noted that, patch being here the result of the pixelwise multiplication of the underlying image and an analysis window, it is necessary to assume the noise amplitude to vary from pixel to pixel even if a Gaussian white noise affects the image to be decomposed.

Estimating noise variance

We see from Equation 3 that the noise total variance σ 2 can be estimated as the empirical mean of | p 0 (ξ)| 2 -| c(ξ)| 2 for any ξ. The non-textured patch p 0 is unknown, and only a coarse estimation of c is available. As a consequence, for any ξ, we assume that σ 2 E(P I x (ξ)) - E(P C x (ξ)). This equation mainly differs from the preceding one in that the power spectrum P I

x is affected by texture components. Since textures have a sparse Fourier representation, these components only affect a few Fourier components, thus averaging the power spectrum difference over a range of frequencies smoothes them out. Moreover, we average over high frequencies since low and medium frequencies are expected to have residual image content not purely made of noise, cf. [START_REF] Ponomarenko | An automatic approach to lossy compression of AVIRIS images[END_REF][START_REF] Colom | Analysis and extension of the Ponomarenko et al. method, estimating a noise curve from a single image[END_REF]. Consequently, we estimate σ 2 as:

σ 2 = 1 M ||ξ||>H E(P I x (ξ)) -E(P C x (ξ)) ( 17
)
where H is a frequency threshold and M is the number of discrete frequencies above H. In practice, we take H = 0.5 cycle per pixel.

Estimating statistics of the power spectrum of a noisy non-textured patch

Based on Equation 3, we eventually estimate the expectation of the power spectrum P x of a nontextured noisy patch. Concerning the variance, we use the upper bound given by Equation 9.

Consequently,

E(P x (ξ)) = E(P C x (ξ)) + σ 2 (18)
var(P x (ξ)) = var(P C x (ξ)) + 2 σ 4 + 4 σ 2 E(P x (ξ)) (19)
3 A non-local dual-domain cartoon and texture decomposition

The preceding section gives the expectation and variance of the power spectrum components of non-textured noisy patches, empirically estimated with a tentative coarse cartoon and texture decomposition (Equations 18 and 19). We are therefore able to detect texture components as deviations to this model, since these components have a larger amplitude than expected under a null hypothesis assuming a non-textured patch.

Statistical hypothesis testing

Given an image patch p, a significant spectrum component is detected at frequency ξ provided that the p-value

Pr H 0 (| p(ξ)| 2 ≥ | p 0 (ξ)| 2
) is below some level of confidence α. The nullhypothesis H 0 is that of a non-textured random patch p 0 . Under a Gaussian distribution assumption, a texture component is therefore detected at ξ as soon as:

G P I x (ξ) -E(P x (ξ)) var(P x (ξ)) ≤ α (20)
where P I x is the power spectrum of the image patch centered at x, E(P x (ξ)) and var(P x (ξ)) are given by Equations 18 and 19, respectively, and G is the tail distribution function (complementary cumulative distribution function) of the standard normal distribution. Here, the estimation of the variance being an upper bound of the actual variance, some significant detections may be missed.

Since a statistical test is performed for any frequency, we use the Bonferroni correction [START_REF] Bonferroni | Teoria statistica delle classi e calcolo delle probabilità[END_REF] to control the false alarm over each patch of size L × L under ε, thus:

α = ε L 2 ( 21
)
where ε is the familywise error-rate, set here equal to 5%. We do not take into account the correlation matrix of the power spectrum components, since estimating the correlations would require a large number of samples, which is out of reach here. It should be noted that a recent paper addresses the estimation of the correlation matrix of patches [START_REF] Arias | Video denoising via empirical Bayesian estimation of space-time patches[END_REF].

Cartoon and texture components

Section 3.1 gives a framework to detect texture components in the spectrum of any patch. For any patch centered at pixel x, we thus define a very simple bandpass filter M x : M x (ξ) = 1 if a significant texture component is detected at frequency ξ, and M x (ξ) = 0 otherwise. This gives a non-linear bandpass filter through a windowed Fourier transform of the input image I. If I(x, ξ) denotes the windowed Fourier transform of I, such that:

I(x, ξ) = y∈D I(x)g α (x -y)e -2iπ<x,ξ> (22)
then the texture component T is retrieved as the inverse windowed Fourier transform of the product M x (ξ) I(x, ξ). The cartoon component C is simply such that I = C + T .

Data: An input image I, domain D = {1 . . . X} × {1 . . . Y }. Parameters: σ CT , α, β, N .

1 Compute a tentative coarse cartoon representation (denoted by C c ) from I with [START_REF] Buades | Fast cartoon + texture image filters[END_REF], scale-factor parameter σ CT ; 2 for any pixel x = (x, y) ∈ D s do 3 Calculate the modified windowed Fourier transforms for any frequency ξ: Detect texture components at ξ through statistical hypothesis testing:

4 I x (ξ) = y∈D g α (x -y)g β (C c (x) -C c (y))I(y)e -2iπ<y,ξ> 5 C x (ξ) = y∈D g α (x -y)g β (C c (x) -C c (y))C c (y)e -
14 M x (ξ) =    1 if G P I x (ξ)-E(Px(ξ))

Algorithm and parameters

Algorithm 1 summarizes the proposed non-local dual-domain cartoon and texture composition. Parameter σ CT in [START_REF] Buades | Fast cartoon + texture image filters[END_REF] to obtain the tentative coarse cartoon is set so that textures are discarded from the cartoon but is small enough to prevent discarding high frequency details. In Section 4, σ CT = 2 unless otherwise specified. Patch width L and parameter α satisfy α = L/5; their value define the trade-off between shape and texture. Any pattern showing a low frequency with respect to 1/α will not be detected as a texture, basically because it does not give a spike in patch power spectra. A trade-off has to be made between the ability to detect low-period patterns (which requires large L) and the need to have a large enough sample of non-overlapping patch for statistical estimation (which bounds L). We extract patches on a subsampling grid D s of step s = L/4. Concerning the experiments of Section 4, a typical value for L will be L = 32 pixels, which permits detecting texture patterns of frequency larger than 1/16 pixel -1 . Note that such a patch size is much larger than typical values in patch methods dedicated to noise removal. Parameter β prevents mixing texture information with shapes with a gray value larger or smaller than the current shape. Because a Gaussian function is used, a difference of the cartoon components larger than 2β indeed multiply the corresponding gray value by e -2 . The value of β obviously depends on the contrast of the image, low-contrasted images indeed require smaller values. We set β = 20 in Section 4 unless otherwise stated. The number of nearest neighbors N is set to 20.

Experimental results

This section presents some experiments. We deal with color images by designing the bandpass filter of Section 3.2 in the luminance channel, the same filter being eventually used in the RGB channels. The texture components shown in this section actually correspond to the 8-bit image 128 + 3T so that they appear much salient in the figures. The reader is asked to zoom in on the images in the electronic version of the paper.

Since the cartoon and texture decomposition problem is based on perceptual clues, no ground truth is available. Overall, we would like the cartoon edges to be preserved and the stationary patterns to be in the texture part. As mentioned in [START_REF] Buades | Fast cartoon + texture image filters[END_REF], the distinction between texture and cartoon is not very clear because at a certain scale a texture element becomes a geometric shape. We visually compare the proposed algorithm with two state-of-the-art algorithms, namely the output of the block nuclear norm based algorithm [START_REF] Ono | Cartoon-texture image decomposition using blockwise low-rank texture characterization[END_REF], which is claimed to outperform variational methods such as [START_REF] Schaeffer | A low patch-rank interpretation of texture[END_REF], and the directional non-linear filter proposed in [START_REF] Buades | Directional filters for color cartoon+texture image and video decomposition[END_REF], which certainly achieves the best trade-off between the quality of the cartoon / texture separation and computation time. We use the software implementation kindly provided by the authors of [START_REF] Ono | Cartoon-texture image decomposition using blockwise low-rank texture characterization[END_REF] and [START_REF] Buades | Directional filters for cartoon + texture image decomposition[END_REF]. Note that the implementation of [START_REF] Ono | Cartoon-texture image decomposition using blockwise low-rank texture characterization[END_REF] only gives gray-level outputs. The default parameters of [START_REF] Ono | Cartoon-texture image decomposition using blockwise low-rank texture characterization[END_REF] are used, and the scale-factor parameter σ CT in [START_REF] Buades | Directional filters for cartoon + texture image decomposition[END_REF] is adjusted to obtain the best visual results. Our algorithm is implemented as a publicly available Matlab code.

To begin with, we illustrate the proposed algorithm with a famous image containing several different textures. Figure 1 shows the original image, the cartoon component given by the fast non-linear filter of [START_REF] Buades | Fast cartoon + texture image filters[END_REF] (this is the coarse decomposition used in our algorithm) and the output of the proposed approach. As we can see, while the texture component has mostly been taken out, the cartoon component of [START_REF] Buades | Fast cartoon + texture image filters[END_REF] is quite blurry. Comparing with the output of [START_REF] Ono | Cartoon-texture image decomposition using blockwise low-rank texture characterization[END_REF] and [START_REF] Buades | Directional filters for cartoon + texture image decomposition[END_REF], we can see that the output of the proposed algorithm tends to better separate texture and cartoon, with less blurry shape edges and a larger part of the texture component well separated from the cartoon as, for instance, in the tablecloth, in the chair on the background, and even in narrow shapes as in the folds of the scarf. Most edges cannot be seen in the texture part, except for periodic patterns on the face for instance. It is a positive feature of the proposed algorithm, which means that high frequency components corresponding to edges are correctly detected as non-significant spectrum components. Here, the scale-factor parameter σ CT of [START_REF] Buades | Directional filters for cartoon + texture image decomposition[END_REF] is difficult to set: σ CT = 2 keeps textures in the cartoon part, and σ CT = 3 (not shown here, see the companion document) gives a quite smooth cartoon with edge information in the texture.

Figure 2 illustrates the behavior of the algorithm for several patches. We can see that similar patches (in the sense of power spectrum similarity) are weighted according to the texture similarity. The numerical values indicated here are the w i / w i of Equation 16; they would be equal to 1/N = 0.05 if the patches were evenly distributed. Moreover, we can see that the expected cartoon power spectrum actually varies from patch to patch, which illustrates that they come from different Gaussian processes. The hypothesis testing framework allows obtaining the localization of the texture spikes in the Fourier domain, and detects no significant texture components if no textures are present, as in the third patch. power spectrum of the query patch | p| 2 , map M x of the significant coefficients detected in the power spectrum of the query patch (in yellow). To ease comparison, the same logarithmic scale is used for the spectra of each patch.

Figure 3 is another experiment with textured building facades 1 . Compared to the other methods, our cartoon component appears more realistic, probably because the noise component is kept in it, while the other methods minimize the total variation of the cartoon part, either explicitly or implicitly, thus smoothing out the noise. Figure 5 shows a detail of Norman Rockwell's painting, Our Heritage3 . It is an application of cartoon and texture decomposition to canvas pattern removal. The canvas of the paintings are sometimes visible and need to be removed for further processing, as explained in [START_REF] Cornelis | Removal of canvas patterns in digital acquisitions of paintings[END_REF]. Although this problem has been the subject of works from the eighties [START_REF] Heitz | Application of autoregressive models to fine arts painting analysis[END_REF], it is still an open question [START_REF] Cornelis | Removal of canvas patterns in digital acquisitions of paintings[END_REF][START_REF] Cornelis | Digital canvas removal in paintings[END_REF]. It can be seen as a cartoon and texture problem as the canvas is locally not visible if the painting layer is too thick. Here, σ CT has been set to 3 for obtaining the tentative coarse cartoon with [START_REF] Buades | Fast cartoon + texture image filters[END_REF], and β = 30 since the image is quite contrasted. As we can see, the proposed algorithm is interesting in this application domain since it removes the quasi-periodic component of the canvas pattern (contrary to [START_REF] Ono | Cartoon-texture image decomposition using blockwise low-rank texture characterization[END_REF] and in a better way than [START_REF] Buades | Fast cartoon + texture image filters[END_REF]), and it still keeps sharp edges. Figure 6 shows an application to remote sensing image processing4 . Remote sensing image processing dates back to the dawn of robotic probe imaging [START_REF] Rindfleisch | Digital processing of the Mariner 6 and 7 pictures[END_REF]. Such images are often impaired by electrical interferences during image acquisition, miscalibrated sensors, or missing data, causing line dropout, striping, banding, or more complex background noise. In this image, the size of the patches L has been set to 64, since we want to capture texture information such as the checkerboard noise pattern which has a periodicity of approximately 20 pixels. The contrast being low, β is set to 10. As we can see, the proposed algorithm removes most of the background noise (contrary to [START_REF] Buades | Directional filters for cartoon + texture image decomposition[END_REF]) while keeping a quite sharp output (in contrast to [START_REF] Ono | Cartoon-texture image decomposition using blockwise low-rank texture characterization[END_REF]).

As explained in Section 1, assessing the separation between cartoon and texture is difficult since it depends on perceptual clues. Although such an experiment may look somewhat contrived, it is possible to test the algorithms discussed here on synthetic data. Figure 7 shows cartoon and texture decompositions of a synthetic image, built by adding to a piecewise constant 8-bit image5 a texture image whose quadrants are made of three sine-wave textures, the fourth quadrant being kept untextured. As we can see, the outputs of [START_REF] Ono | Cartoon-texture image decomposition using blockwise low-rank texture characterization[END_REF] and [START_REF] Buades | Directional filters for cartoon + texture image decomposition[END_REF] show somewhat mixed cartoon and texture, cartoon edges being noticeable in the retrieved texture. This Figure 6: Mariner6. The contrast is augmented for a better visibility. First row: original image, output of [START_REF] Buades | Fast cartoon + texture image filters[END_REF], details of the original image with the "checkerboard" noise clearly visible. Second row: output of the proposed algorithm. Third row: output of [START_REF] Ono | Cartoon-texture image decomposition using blockwise low-rank texture characterization[END_REF]. Fourth row: output of [START_REF] Buades | Directional filters for cartoon + texture image decomposition[END_REF]. Second, third, and fourth row, from left to right: cartoon, texture, and close-up view of the cartoon component. [START_REF] Ono | Cartoon-texture image decomposition using blockwise low-rank texture characterization[END_REF], texture retrieved by [START_REF] Ono | Cartoon-texture image decomposition using blockwise low-rank texture characterization[END_REF], cartoon retrieved by [START_REF] Buades | Directional filters for color cartoon+texture image and video decomposition[END_REF], texture retrieved by [START_REF] Buades | Directional filters for color cartoon+texture image and video decomposition[END_REF].

phenomenon is less visible with the proposed algorithm. Here, the proposed algorithm and the one of [START_REF] Ono | Cartoon-texture image decomposition using blockwise low-rank texture characterization[END_REF] are used with their default parameters, and σ CT = 2 in [START_REF] Buades | Directional filters for cartoon + texture image decomposition[END_REF]. Table 1 gives a quantitative assessment with the root-mean-square error (RMSE) between the retrieved cartoon (resp. texture) component and ground-truth cartoon (resp. texture) component. The RMSE criterion confirms the visual impression (the lower the values, the better the decomposition). Although the cartoon component of [START_REF] Ono | Cartoon-texture image decomposition using blockwise low-rank texture characterization[END_REF] looks sharper, some parts have a grayer aspect than in the original image, which explains the larger RMSE. With the proposed algorithm, RMSE are equal for cartoon and texture components because the sum of both components exactly matches the input image. Texture normalization used in the software program [START_REF] Buades | Directional filters for cartoon + texture image decomposition[END_REF] may explain why RMSE of cartoon and texture differ for this method in this particular experiment. To conclude this discussion, Table 2 gives computation times on a desktop equipped with a 12-core Intel Xeon E5-2650 v4 @ 2.20GHz CPU and 64 Gb memory. We can see that the proposed method, implemented in a non-optimized Matlab software, is much faster than the one of [START_REF] Ono | Cartoon-texture image decomposition using blockwise low-rank texture characterization[END_REF]. In all cases, the computation time of the preprocessing step [START_REF] Buades | Fast cartoon + texture image filters[END_REF] in the proposed algorithm is less than 0.3 seconds.

Table 2: Computation times in seconds

Image

Size proposed [START_REF] Ono | Cartoon-texture image decomposition using blockwise low-rank texture characterization[END_REF] [15] (in pixels) method

Barbara (Fig. 1) 567 × 787 35.5 332.1 6.5 Manhattan (Fig. 3) 575 × 1024 43.7 518.9 9.2 Kodim08 (Fig. 4) 512 × 768 29.7 375.5 6.3 Rockwell (Fig. 5) 1600 × 1060 124.3 967.9 47.8 Mariner6 (Fig. 6) 461 × 471 15.1 168.2 1.2 Synthetic (Fig. 7) 512 × 512 20.8 172.3 3.4

Conclusion

This paper presents a novel patch-based approach to cartoon and texture decomposition. Fourier components of textures are detected in patches through statistical hypothesis testing, the null model being estimated in a non-local way over a tentative coarse cartoon representation. The proposed algorithm achieves a good compromise between computation time and accuracy.

16 1 L 2 Algorithm 1 :

 16121 Apply the bandpass filter M x (ξ) to the windowed Fourier transform of I to retrieve texture T and cartoon C = I -T , i.e., T (x) = y∈D ξ∈D T (y, ξ)e 2iπ<x,ξ> where T (x, ξ) = M x (ξ) y∈{1...X}×{1...Y } g α (x -y)I(y)e -2iπ<y,ξ> . Result: A cartoon component C and a texture component T . Proposed algorithm for non-local dual-domain cartoon and texture decomposition.

Figure 1 :Figure 2 :

 12 Figure 1: Barbara. First row: original image and output of[START_REF] Buades | Fast cartoon + texture image filters[END_REF]. Second row: output of the proposed algorithm. Third row: output of[START_REF] Ono | Cartoon-texture image decomposition using blockwise low-rank texture characterization[END_REF]. Fourth row: output of[START_REF] Buades | Directional filters for cartoon + texture image decomposition[END_REF] (σ CT = 2). From the left to the right: cartoon, texture (except for first row), and close-up.

Figure 3 :

 3 Figure 3: Manhattan. From top to bottom: original image, output of the proposed algorithm, output of[START_REF] Ono | Cartoon-texture image decomposition using blockwise low-rank texture characterization[END_REF], output of[START_REF] Buades | Directional filters for cartoon + texture image decomposition[END_REF].

Figure 4 2

 4 Figure42 also illustrates that our method retrieves texture details that other methods miss, such as tiles on roofs.

Figure 4 :

 4 Figure 4: Kodim08. Top left: original image. Top right: output of the proposed algorithm. Bottom left: output of [11]. Bottom right: output of [15].

Figure 5 :

 5 Figure 5: Rockwell. Top left: original image. Top right: output of the proposed algorithm. Bottom left: output of [11]. Bottom right: output of [15].

Figure 7 :

 7 Figure 7: Synthetic. First row: ground truth cartoon component, input image, cartoon retrieved by the proposed algorithm, texture retrieved by the proposed algorithm. Second row: cartoon retrieved by[START_REF] Ono | Cartoon-texture image decomposition using blockwise low-rank texture characterization[END_REF], texture retrieved by[START_REF] Ono | Cartoon-texture image decomposition using blockwise low-rank texture characterization[END_REF], cartoon retrieved by[START_REF] Buades | Directional filters for color cartoon+texture image and video decomposition[END_REF], texture retrieved by[START_REF] Buades | Directional filters for color cartoon+texture image and video decomposition[END_REF].

  Calculate estimation of total variance σ 2 (Eq. 17);

		2iπ<y,ξ>
	6	and the corresponding power spectra P I x (ξ) = |I x (ξ)| 2 and P C x (ξ) = |C x (ξ)| 2 .
	7 end
	8 for any pixel x ∈ D do
	9	Find the N nearest image-patch power spectra P I x 1 , . . . , P I x N to P I x with the FLANN
		library [39];
	10	Calculate sample mean E(P C x (ξ)) (Eq. 14) and sample variance var(P C x (ξ)) (Eq. 15)
		based on the x i ;
	12	Calculate estimations of expectation E(P x (ξ)) (Eq. 18) and variance var(P x (ξ))
		(Eq. 19) of a non-textured patch;
	13

11

Table 1 :

 1 RMSE with synthetic data

		proposed [11] [15]
		method
	cartoon	2.72	4.51 6.15
	texture	2.72	6.04 11.42

Image source: Kodak Image Suite. http://r0k.us/graphics/kodak/

Image source: http://muddycolors.blogspot.fr/2013/10/norman-rockwell-american-originals. html

Image source: http://www.imageprocessingplace.com/[START_REF] Gonzalez | Digital Image Processing[END_REF], see also[START_REF] Rindfleisch | Digital processing of the Mariner 6 and 7 pictures[END_REF]

Image source: http://sipi.usc.edu/database/

Acknowledgment

The authors of [11], [12] and [14] are gratefully acknowledged for making their software publicly available.


text:
Attila Szabo
email: attila.szabo@stockholm.se

Mathematical memory revisited: Mathematical problem solving by high-achieving students

Keywords: High-achievers, mathematical memory, mathematical abilities, problem solving

The present study deals with the role of the mathematical memory in problem solving. To examine that, two problem-solving activities of high achieving students from secondary school were observed one year apart -the proposed tasks were non-routine for the students, but could be solved with similar methods. The study shows that even if not recalling the previously solved task, the participants' individual ways of approaching both tasks were identical. Moreover, the study displays that the participants used their mathematical memory mainly at the initial phase and during a small fragment of the problem-solving process, and indicates that students who apply algebraic methods are more successful than those who use numerical approaches.

Introduction and background

Despite a growing emphasis on the identification and teaching of mathematically able pupils, much remains unknown about the abilities they display when solving mathematical problems. For reasons of social justice and equality, research has typically focused on low achieving pupils [START_REF] Swanson | Math disabilities: A selective meta-analysis of the literature[END_REF] while relatively few studies have observed the abilities of the gifted and highachievers (e.g. [START_REF] Vilkomir | Using components of mathematical ability for initial development and identification of mathematically promising students[END_REF] or addressed those pupils' memory functions during mathematical activities [START_REF] Leikin | Memory abilities in generally gifted and excellingin-mathematics adolescents[END_REF][START_REF] Raghubar | Working memory and mathematics: A review of developmental, individual difference, and cognitive approaches[END_REF]. In particular, just a few studies (e.g. [START_REF] Krutetskii | The psychology of mathematical abilities in schoolchildren[END_REF][START_REF] Szabo | Examining the interaction of mathematical abilities and mathematical memory: A study of problem-solving activity of high-achieving Swedish upper secondary students[END_REF] examined the role of the mathematical memory in gifted and talented students' problem-solving activities.

Mathematical abilities

Our innate ability to estimate quantities, known as the approximate number system, is extremely limited [START_REF] Dehaene | The number sense[END_REF], but an active contact with the subject may, under favourable conditions, generate mathematical abilities that are both complex and structured [START_REF] Krutetskii | The psychology of mathematical abilities in schoolchildren[END_REF]. The nature of mathematical abilities has engaged researchers for more than 120 years; already at the end of the 19th century, [START_REF] Calkins | A study of the mathematical consciousness[END_REF] presented, based on observations of Harvard students, significant information about the way mathematicians approached the subject. However, the research on mathematical abilitiesmainly because of the dominance of psychometric approaches, and thereby considering abilities as innate and statichas not delivered widely accepted results during the first half of the 20th century [START_REF] Vilkomir | Using components of mathematical ability for initial development and identification of mathematically promising students[END_REF]. Therefore, of importance for the present paper is the research of [START_REF] Krutetskii | The psychology of mathematical abilities in schoolchildren[END_REF], whose longitudinal observational study analysed the problem-solving activities of around 200 pupils. He concluded that able pupils' mathematical ability, while complex and dynamic, typically comprises four broad abilities. These are  the ability to obtain and formalise mathematical information (e.g. formalised perception of mathematic material),

 the ability to process mathematical information (e.g. logical thought, flexible mental processes, clear and simple solutions, generalized mathematical relations),  the ability to retain mathematical information, that is, mathematical memory (a generalized memory for mathematical relationships) and  a general synthetic component (mathematical cast of mind) (Krutetskii, 1976, pp. 350-351). However, while these abilities have frequently been associated with mathematical giftedness, Krutetskii (1976, pp. 67-70) argues they can also be displayed properly by high-achievers.

Mathematical memory

It is largely agreed that memory plays an essential role both in the learning of mathematics and in mathematical problem solving (e.g. [START_REF] Leikin | Memory abilities in generally gifted and excellingin-mathematics adolescents[END_REF][START_REF] Raghubar | Working memory and mathematics: A review of developmental, individual difference, and cognitive approaches[END_REF]. Thus, what seems to be crucial "is not whether memory plays a role in understanding mathematics but what it is that is remembered and how it is remembered by those who understand it" (Byers & Erlwanger, 1985, p. 261). Calkins' (1894) early study showed that the memories of mathematicians are more concrete than verbal, that mathematics students do not memorise facts more easily than other students and that, when performing mathematics, there is no difference between men and women. Some decades later, [START_REF] Katona | Organizing and memorizing: Studies in the psychology of learning and teaching[END_REF] found that rational methods are easier to memorise that random digits, while [START_REF] Bruner | The process of education[END_REF] showed that simple interrelated representations are effective when recalling detailed knowledge. However, [START_REF] Krutetskii | The psychology of mathematical abilities in schoolchildren[END_REF] distinguished mathematical memory from the mechanical recalling of numbers or algorithms, by stressing that it is a memory consisting of generalized methods for problem solving. Hence, the mathematical memory does not retain "all of the mathematical information that enters it, but primarily that which is 'refined' of concrete data and which represents generalized and curtailed structures" (Krutetskii, 1976, p. 300). Moreover, he found that able students usually retain the contextual facts of a problem only during problemsolving and forgot it instantly afterwards, but remember several months later the methods they applied. Conversely, low-achievers often remember contextual facts but rarely the problem-solving methods [START_REF] Krutetskii | The psychology of mathematical abilities in schoolchildren[END_REF]. Cognitive psychology studies (e.g. [START_REF] Sternberg | Cognitive Psychology[END_REF] indicate important distinctions between different memory systems; that is, long term memory can be divided into implicit and explicit memory, based on the type of the stored information. In a mathematical context, the implicit memory contains automatized procedures and algorithms, while explicit memory retains information about experiences and facts which can be consciously recalled and explained, such as schemas for problem-solving. Thus, according to the cognitive model, we may assume that mathematical memory, as defined by Krutetskii, is explicit. Besides, it is a memory formed at later stages (e.g. [START_REF] Davis | A memory-based model for aspects of mathematics teaching[END_REF] based on the ability to generalize mathematical material, because at young able pupils "the relevant and the irrelevant, the necessary and the unnecessary are retained side by side in their memories" (Krutetskii, 1976, p. 339).

The study

The present study had two aims, both based on [START_REF] Krutetskii | The psychology of mathematical abilities in schoolchildren[END_REF] definitions of mathematical ability. The first was to identify the structure of mathematical abilities when high-achieving students solve non-routine but structurally similar problems. The second was to examine the role of mathematical memory during problem-solving activity.

Participants

Because young children and low-achievers rarely exhibit mathematical memory [START_REF] Krutetskii | The psychology of mathematical abilities in schoolchildren[END_REF], participants were 16-17 years old volunteers from an advanced mathematics programme in Swedish upper secondary school who had achieved the highest grade in the Swedish national test. Prior to data collection study, to familiarize students with the study, I spent 30 hours, over a period of four months, as a participant observer in their mathematics classroom. During this period, they came to trust me as an observer of their problem-solving activities. At the end of this process, after consulting their teacher, 6 students, 3 boys and 3 girls, were invited to participate.

Tasks

The theoretical background indicates that an appropriate way to identify the distinct structure of the mathematical ability is to analyse the problem-solving activities of the individuals (e.g. [START_REF] Krutetskii | The psychology of mathematical abilities in schoolchildren[END_REF]. Moreover, the structure of a mathematical problem reveals the mathematical thinking which is required to solve it, because problem solving "is an activity requiring the individual to engage in a variety of cognitive actions, each of which requires some knowledge and skill, and some of which are not routine" (Cai & Lester, 2005, p. 221). However, able students typically forget the context of a problem shortly after solving it, but, as an impact of their mathematical memory, they are several months later able to recall the methods applied to solve it. Thus, in order to complete the aims of the study, the participants solved two problems approximately one year apart. At the first observation, in order to avoid as far as possible the influence of previous experiences, the main criterion was to select a challenging non-routine task, Task 1 (T1). When selecting Task 2 (T2)in order to emphasize the role of the mathematical memorythe main criterion was to propose a task which was non-routine, but could be solved by methods similar to those used previously. Both tasks underwent substantial a-priori testing with corresponding groups of high-achievers, confirming that they were well-suited for the study and for the mathematical knowledge of the participants. This test confirmed that the students solved the proposed tasks with similar methods, that is, by applying the formulae for perimeters of circles and squares.

Observations and interviews

To avoid confounding factors during classroom interaction, which may affect pupils' thought process [START_REF] Norris | The implication of visual research for discourse analysis: transcription[END_REF], every participant was observed individually and, to avoid stress, given unlimited time to solve each task. T1 was solved and approximately one year later T2 was solved. In order to avoid participants' memories being activated mostly because of recalling the circumstances for the first observation as an unusual element in their daily activitiesthat is, not because of recalling the previously solved task -I continued to interact with them during their mathematics classes between the two observations. The students were invited to solve the tasks in a think-aloud manner and encouraged to describe every step in the process. To minimise participants' influence on each other, the tasks were solved during single days. The observations took place in a private room at their school and, when needed, supplementary questions were posed in order to facilitate the process. If a student neither wrote nor spoke for a while, the following questions were posed: What is bothering you? Why do you do that? What do you want to do and why? What are you thinking about? Pupils generally are not used to verbalise their problem-solving process [START_REF] Ginsburg | The clinical interview in psychological research on mathematical thinking: aims, rationales, techniques[END_REF], thus, in order to avoid the risk that essential parts of their cognitive activities would not be communicated, every observation was followed by a reflective interview. The purpose of the interviews was to display the hidden cognitive processes at problem-solving and to evaluate the levels of competence in those processes [START_REF] Ginsburg | The clinical interview in psychological research on mathematical thinking: aims, rationales, techniques[END_REF]. Each observation was recorded using a technology that digitises both speech and handwritten notes; the audio recordings were transcribed verbatim. Although they were given unlimited time, no participant needed more than 14 minutes to complete a single task.

Data analysis

The piloting of the tasks on corresponding groups of high-achievers indicated that the general synthetic componenta typical ability of gifted students (Krutetskii, 1976, p. 351)was unlikely to be observed during problem-solving; consequently, this ability was excluded from the analysis. The analytical framework for this study contained the following abilities from Krutetskii's framework: obtaining and formalizing mathematical information (O), processing mathematical information (P), generalizing mathematical relations and operations (G), and mathematical memory (M).

The digital recordings resulted in an exact linear reproduction of the students' actions, which was especially beneficial when performing qualitative content analysis of the material, inspired by van [START_REF] Van Leeuwen | Introducing social semiotics[END_REF]. The participants´ actions were analysed by identifying, coding and categorising the basic patterns in the empirical content. This method highlighted the abilities that were directly expressed in the empirical material; each episode lasting at least one second in written solutions and verbal utterances was scrutinised for the presence of the focused abilities. Next, the data from observations were combined with data from the interviews. I exemplify this with data from Linda, who, when solving T2, didn't say or wrote anything during the initial 62 seconds, before stating: Linda:

I would like to write down, start with writing a… some nice little estimations… After this episode she drew three squares with sides a, b and c, and wrote "a + b = c". Thus, based on the observation, the presence of O was certain, but it was not possible to decide if other abilities were also present in the actual episode. Yet, the following sentences from the reflective interview proved that she recalled another task which could be solved with similar methods:

Linda: I got blocked until I remember similar tasks, because it's a lot more difficult to solve this kind of tasks if one doesn't have a determined way to approach it… I believe I will bring up the same task as last time, with triangles and squares.

The utterances "a determined way to approach it" and "the same task as last time" indicate that Linda recalled a different task and its methods, thereby validating the presence of both O and M in the actual episode. In this way, the analysis revealed both the structure and the sequential order of the focused abilities, that is, every ability which occurred during the 12 problem-solving activities was displayed in a matrix. However, as exemplified above, some abilities (e.g. O and M) occurred closely interrelated during certain episodes and were extremely hard to differentiate.

Results

When asked, each participant confirmed that both tasks were non-routine, this being a prerequisite for the study. The analysis concluded in a matrix, with every episode of the process related to the focused abilities. As mentioned, certain abilities were closely interrelated during some episodes. As displayed (Table 1), M is presentsolitary or interrelatedat 16% during the first and at 10.5% during the second observation. The most manifested ability is P, which increased from 53% to 67% a year later, while O, the second most exposed ability, decreased from 47% to 31.5%. According to the a-priori testing of the tasks, G could be revealed when numerical resultsthat is, solutions for particular caseswere developed into general, algebraic solutions. Thus, every student who offered purely numerical resultsnamely Erin, Sebastian and Larrywas encouraged to consider general solutions. Yet, when solving T1 and asked if their numerical results apply also for arbitrary semicircles, none of them could generalize (G) their findings:

Erin: I don't know how I should prove this … if I have to do some general method.

Sebastian: I don't know if I shall demonstrate that it should be the same thing there, for every measure. But now in my head it sounds like that it should be so.

Larry: Yes, I suppose, but I don't know how to confirm it, it only feels that way.

Thus, the increase of G from 0% to 2% (Table 1) occurred because during the reflective interview connected to T2, when offered additional opportunities to reflect over the patterns in her numerical results, Erin performed a successful generalization of the obtained solutions, and stated:

Erin: I've never made a general solution like this ... But it was fun ... Especially when it concluded in something.

When concerning the efficiency of the applied methods, the analysis shows that Earl, Linda and Heather solved both tasks properly by applying general, algebraic methods. Conversely, purely numerical approaches didn't lead to fully acceptable results. The most efficient solutions were offered by Linda, who applied the same algebraic model (and its identical steps) at both tasks.

The role of mathematical memory

The recalling of the applied methods several months after solving a problem is a typical display of mathematical memory [START_REF] Krutetskii | The psychology of mathematical abilities in schoolchildren[END_REF]. Thus, another main criterion for the study was that both tasks could be solved with similar methods. However, only Earl and Larry associated T2 to T1:

Earl:

We got a very similar task last year, when we had the circle and that semicircle.

Larry:

We did a pretty similar task last time, when it was something like this, something with the radius or diameter on them.

Earl and Larry applied identical methods at the individual level when approaching both tasks. That is, Earl solved both tasks by using the same algebraic method, while Larry approached both tasks with the same numerical method. However, Earl's algebraic method gave accurate solutions while Larry couldn't solve any task properly. The other four students said that they didn't associate T2 to T1. But even though not recalling T1, they approached both tasks in identical ways at the individual level. For example, when Linda solved T2, despite stating that she didn't think at all of T1, she applied the same general method as a year before:

Linda: I will bring up the same task as last time, with triangles and squares. It is a bit the same thing ... I connect very often geometrical tasks to that. I have written that solution many times and I can see every step in the process in front of me.

As seen above, Linda refers to a generalized method which she associates to a geometrical taskabout finding the side of a square drawn in a right trianglewhich differs considerably from the proposed tasks. Yet, influenced by her mathematical memory (Krutetskii, 1976, p. 300) she states that "It is a bit the same thing" and applies the same method when solving both T1 and T2.

Heather as well used identical algebraic approaches for both tasks a year apart:

Heather (T1): I needed a common variable. Otherwise it will be difficult to calculate.

Heather (T2): I needed some relation among these sides in that and the large square's sides. Otherwise it will be difficult.

Also the individual approaches of Erin and Sebastian were respectively identical; Erin approached both tasks by reasoning, testing numerical values and applying particular solutions, while Sebastian reasoned carefully before requesting the use of numerical values at both occasions. Thus, every participant approached both tasks identically at the individual level. The analysis also shows that M is displayed mainly at the beginning of the process, for recalling mathematical relations and problem-solving methods; moreover, none of participants modified the initially selected methods.

Discussion

One of the aims of this study was to display the role of the mathematical memory (M) when highachieving students solve non-routine tasks, which can be solved with similar methods. Despite its small proportion, M seems to play a pivotal role in problem-solving because the participants selected their methods at the early stages of the process and the methods were not changed later. Thus, by confirming earlier results (e.g. [START_REF] Szabo | Examining the interaction of mathematical abilities and mathematical memory: A study of problem-solving activity of high-achieving Swedish upper secondary students[END_REF], it seems that the choice of methods is directly influenced by M and it is critical for the success of the problem-solving.

However, unexpectedly, only two of the six participants recalled the solution process to the earlier task, contradicting [START_REF] Krutetskii | The psychology of mathematical abilities in schoolchildren[END_REF] finding that able students recall the process but not the context of earlier problems. But even when not recalling T1, every participant approached both tasks in the same individual way. For example, Linda's method, connected to a square in a triangle and apparently very different from what is predicted, is a general approach that she uses for nonroutine geometrical tasks. And even though the individual approaches of Erin, Larry and Sebastian were not successful when solving T1, they were repeated a year later. Thus, it seems that the participants rely on methods which appear to be inflexible and applied regardless of their success.

The general structure of the participants' mathematical abilities indicates that O and M decrease while P increases at the second observation. Hence, it is not unreasonable to assume that the displayed stability of the individual approaches made O and M more efficient at T2, and therefore students had a larger focus on P. And even if none of the students could generalize numerical results at the first observation, Erin generalized her results during the interview after T2. Thus, when additional opportunities were offered, by evolving the patterns in her numerical results, which may be interpreted as a form of convergent thinking [START_REF] Tan | Convergence in creativity development for mathematical capacity[END_REF], Erin could improve the quality of her problem solving. These findings may suggest that some participants are unlikely to have experienced teaching focused on methods of generalization, because the individual structure of the mathematical ability depends on received instructions [START_REF] Krutetskii | The psychology of mathematical abilities in schoolchildren[END_REF]. However, by confirming earlier studies (e.g. [START_REF] Krutetskii | The psychology of mathematical abilities in schoolchildren[END_REF], the methods of this study were not able to differentiate M from O during episodes when students did not say or write anything; thus, a better investigation of the mathematical memory requires further studies. In addition, the study shows that mathematical memory has a key role during the early stages of problem-solving (e.g. [START_REF] Szabo | Examining the interaction of mathematical abilities and mathematical memory: A study of problem-solving activity of high-achieving Swedish upper secondary students[END_REF] and that individual problem-solving methods seem to be very stable and apparently independent of their efficiency when high-achievers solve non-routine tasks. Finally, the study indicates that, if given additionally opportunities to reflect over their numerical solutions, some students might be able to display their ability to generalize mathematical relations and operations.

Task 1 :

 1 In a semicircle we draw two additional semicircles, according to the figure. Is the length of the large semicircle longer, shorter or equal to the sum of the lengths of the two smaller semicircles? Justify your answer. Task 2: In a square we draw two arbitrary contiguous squares, according to the figure. Is the perimeter of the large square longer, shorter or equal to the sum of the perimeters of the two smaller squares? Answer the question without measuring the figure. Justify your answer.

Table 1 : Average time for the focused mathematical abilities in the problem-solving process

 1

		O	O & P	O & M	P	G	M
	Task 1	31%	4%	12%	49%	0%	4%
	Task 2	20%	1.5%	10%	65.5%	2%	0.5%

Proceedings of CERME10