D Kilbane
email: deirdre.kilbane@ucd.ie

Transition wavelengths and unresolved transition array statistics of ions with Z = 72 -89

Keywords: numbers: 31.10.+z, 32.30.Jc, 32.70.-n, 32.80.Aa

Potential extreme ultraviolet and soft X-ray radiation sources have been identified, using the flexible atomic code (FAC), as emission peaks arising from 4d -4f and 4p -4d transitions in Pd-like to Rb-like ions of hafnium through actinium. The effects of configuration interaction are investigated and for increasing nuclear charge, these strong emitters are seen to separate and move to shorter wavelength. Each source is characterized using the unresolved transition array model. They are proposed to complement the currently used nitrogen and argon sources in the "water window", and as possible successors to tin in next generation lithography.

Introduction

The success of extreme ultraviolet lithography (EUVL) hinges on the ability to identify powerful radiation sources at 13.5 nm, a wavelength at which molybdenum / silicon mirrors are highly reflective [START_REF] Churilov | d 8 -(4d 7 5p+4d 7 4f+4p 5 4d 9 ) transitions in Xe XI[END_REF][START_REF] Bakshi | EUV sources for lithography[END_REF][START_REF] O'sullivan | Emission and absorption in laser produced plasmas: processes and applications[END_REF][START_REF] Sasaki | Modeling of radiative properties of Sn plasmas for extreme-ultraviolet source[END_REF]. To date, both tin and xenon laser produced plasmas (LPPs) and discharge plasmas are the sources of choice at this wavelength: Sn 8+ -Sn 13+ emission due to 4d -4f and 4p -4d transitions and Xe 10+ emission due to 4d -5p transitions [START_REF] Churilov | d 8 -(4d 7 5p+4d 7 4f+4p 5 4d 9 ) transitions in Xe XI[END_REF][START_REF] Sasaki | Modeling of radiative properties of Sn plasmas for extreme-ultraviolet source[END_REF][START_REF] Böwering | Extreme ultraviolet emission spectra of highly ionized xenon and their comparison with model calculations[END_REF][START_REF] Churilov | Analyses of the Sn IX -Sn XII spectra in the EUV region[END_REF][START_REF] Poirier | Modeling of EUV emission from xenon and tin plasma sources for nanolithography[END_REF][START_REF] De Gaufridy De Dortan | Influence of configuration interaction on satellite lines of xenon and tin in the EUV region[END_REF]. Many recent research efforts have concentrated on investigating next generation lithographic sources at shorter wavelengths e.g. gadolinium and terbium at 6.75 nm [START_REF] Churilov | EUV spectra of Gd and Tb ions excited in laser-produced and vacuum spark plasmas[END_REF][START_REF] Otsuka | Rare-earth plasma extreme ultraviolet sources at 6.5 -6.7 nm[END_REF][START_REF] Otsuka | Systematic investigation of self-absorption and conversion efficiency of 6.7 nm extreme ultraviolet sources[END_REF][START_REF] Kilbane | Extreme ultraviolet emission spectra of Gd and Tb ions[END_REF][START_REF] Sasaki | Theoretical investigation of the spectrum and conversion efficiency of short wavelength extreme-ultraviolet light sources based on terbium plasmas[END_REF]. In [START_REF] Kilbane | Ground-state configurations and unresolved transition arrays in extreme ultraviolet spectra of lanthanide ions[END_REF], extreme ultraviolet (EUV) and soft X-ray (SXR) sources were identified at numerous wavelengths with the prospect of being used in the event of a highly reflective mirror becoming available at any of these wavelengths.

Moving to shorter wavelengths, further into the SXR region, we enter what is known as the "water window" (2.3-4.4 nm), where live biological sampling is desirable [START_REF] Wachulak | Water window' compact, table-top laser plasma soft X-ray sources based on a gas puff target[END_REF][START_REF] Silva | X-ray laser microscopy of rat sperm nuclei[END_REF]. Current sources in this region are strong quasi-monochromatic emission at λ = 2.879 nm and λ = 2.478 nm wavelengths arising from 1s 2 -1s 2p in N 5+ and 1s -2p in N 6+ respectively, and broadband emission between λ = 2-4 nm from argon gas targets [START_REF] Wachulak | Water window' compact, table-top laser plasma soft X-ray sources based on a gas puff target[END_REF]. Quasi-monochromatic sources are suitable for biological imaging using diffractive optics while broadband emission sources can be used for contact microscopy. The development of compact, high repetition rate, table top SXR sources using these gas puff targets, which have the advantage of being debris free, provide much needed laboratory in situ alternatives to free electron lasers and synchrotrons and can be used in numerous experiments, e.g. microscopy, spectroscopy and metrology.

It is proposed that the sources identified in this work would be generated in laser produced plasmas (LPPs), where a solid target is irradiated with an intense laser pulse. This however leads to the production of debris which can cause major degradation of expensive EUV / SXR mirrors. Different techniques to minimize debris have been proposed [START_REF] Bakshi | EUV sources for lithography[END_REF] such as mixed composition targets [START_REF] Hayden | 5 nm extreme ultraviolet emission from tin based laser produced plasma sources[END_REF][START_REF] Rakowski | Characterization and optimization of the laser-produced plasma EUV source at 13.5 nm based on a double-stream Xe/He gas puff target[END_REF] and liquid tin mirrors as collectors [START_REF] Fahy | Robust liquid metal collector mirror for EUV and soft x-ray plasma sources[END_REF]. Indeed it may be possible to extend these approaches into the SXR region, e.g. the liquid tin mirror being developed for 13.5 nm could be a prototype for elements at other wavelengths.

The recording of the 4d -4f emission in cesium through lutetium showed that these relatively narrow regions of resonance-like emission became more complex and moved to shorter wavelength with increasing nuclear charge Z [START_REF] Carroll | 4d-4f emission resonances in laser-produced plasmas[END_REF][START_REF] Carroll | Ground-state configurations of ionic species I through XVI for Z=57-74 and the interpretation of 4d-4f emission resonances in laser-produced plasmas[END_REF]. In order to interpret such complicated spectra, Mandelbaum et al [START_REF] Mandelbaum | Interpretation of the quasicontinuum band emitted by highly ionized rare-earth elements in the 70 -100 -Å range[END_REF] employed the unresolved transition array (UTA) approach developed by Bauche-Arnoult, Bauche and Klapisch [START_REF] Bauche-Arnoult | Variance of the distributions of energy levels and of the transition arrays in atomic spectra[END_REF][START_REF] Bauche-Arnoult | Variance of the distributions of energy levels and of the transition arrays in atomic spectra. II. Configurations with more than two open subshells[END_REF][START_REF] Bauche | Emissive zones of complex atomic configurations in highly ionized atoms[END_REF][START_REF] Bauche-Arnoult | Variance of the distributions of energy levels and of the transition arrays in atomic spectra. III. Case of spin-orbit-split arrays[END_REF] and concluded that interactions between the 4p 6 4d N -1 4f and 4p 5 4d N +1 configurations are responsible for narrowing the transition arrays and their superposition in adjacent ion stages. In this work we adopt the UTA approach to characterize the emission arising from 4d -4f and 4p -4d transitions in Pd-like to Rb-like ions of hafnium through actinium (Z = 72 -89). Of these elements tungsten and gold have received most attention to date owing to their use in the fusion community [START_REF] Mandelbaum | Analysis of quasi-continuum bands and line spectra of highly ionized W and Au[END_REF][START_REF] Biedermann | EUV spectroscopy of highly charged tungsten ions relevant to hot plasmas[END_REF][START_REF] Radtke | Line and band emission from tungsten ions with charge 21+ to 45+ in the 45 -70 -Å range[END_REF][START_REF] Zeng | Radiative opacity of gold plasmas studied by a detailed level-accounting method[END_REF][START_REF] Zeng | Spectrally resolved opacities and Rosseland and Planck mean opacities of lowly ionized gold plasmas: A detailed level-accounting investigation[END_REF]. Tungsten is currently being used in the walls of the divertor in many magnetic confinement fusion devices, such as tokamaks, to reduce tile erosion [START_REF] Radtke | Line and band emission from tungsten ions with charge 21+ to 45+ in the 45 -70 -Å range[END_REF][START_REF] Field A R, Fussmann | ASDEX-Upgrade Team 1995 Studies of divertor target plate erosion in the ASDEX-Upgrade tokamak[END_REF][START_REF] Naujoks | the ASDEX-Upgrade Team 1996 Tungsten as target material in fusion devices[END_REF]. However even small quantities of tungsten contaminating the core plasma (> 10 -5 [START_REF] Peacock | Diagnostics for Experimental Thermonuclear Fusion Reactors 291[END_REF]) can seriously limit the energy confinement in a fusion reactor. Knowledge of the W emission radiated from each ion stage in the plasma core is therefore essential to eradicate this degradation. Gold is used for indirectly driven inertial confinement fusion (ICF), where laser radiation heats the inside of a Au hohlraum producing a plasma which emits intense x-rays. Therefore knowledge of Au radiative opacity is crucial to the success of ICF [START_REF] Zeng | Radiative opacity of gold plasmas studied by a detailed level-accounting method[END_REF][START_REF] Zeng | Spectrally resolved opacities and Rosseland and Planck mean opacities of lowly ionized gold plasmas: A detailed level-accounting investigation[END_REF].

Recent experiments have employed the LPP technique using metal targets to generate EUV/SXR radiation. For example a bulk rhenium target was used to develop new SXR microscopy applications in the "carbon window"(λ ≈ 4.5 -5 nm) [START_REF] Artyukov | Soft X-ray imaging of thick carbon-based materials using the normal incidence multilayer optics[END_REF]. Also a solid gold-based LPP source was utilized in the design of an EUV source and optics setup to achieve high energy density and spatial resolution in a compact setup [START_REF] Bayer | Applications of compact laserdriven EUV/XUV plasma sources X-Ray[END_REF]. In [START_REF] Mantouvalou | Note: Study of extreme ultraviolet and soft x-ray emission of metal targets produced by laser-plasma-interaction[END_REF] the emitted X-ray spectra of different target materials were recorded over the wavelength range of 0.8-18 nm. Low-Z elements (Cu, Ti, Fe, and Al) resulted in intense line emission, while continuum-like emission was detected from higher-Z materials (Mo and Ag). Therefore it is possible to optimize the brilliance of the LPP source for a specific x-ray emission range and a particular application of interest.

The outline of this paper is as follows: In section 2, 180 theoretical 4d -4f and 4p -4d unresolved transition array (UTA) spectra are presented, for ions with configurations 4p 6 4d N -4p 6 4d N -1 4f and 4p 6 4d N -4p 5 4d N +1 , N = 1..10. The effects of configuration interaction (CI), and overall trends in the position and intensity of UTA peaks from hafnium to actinium are discussed. In section 3 UTA statistics, namely, mean wavelength λgA and spectral width ∆λ gA for the above ions are calculated which allows us to characterize these possible EUV / SXR sources. The gradual separation of the 4d -4f UTA from the 4p -4d UTA is highlighted. Finally in section 4 we conclude with a summary of this work.

Unresolved Transition Arrays of Ions with Z = 72 -89

As noted in [START_REF] Kilbane | Ground-state configurations and unresolved transition arrays in extreme ultraviolet spectra of lanthanide ions[END_REF] the strongest lines occurring in the EUV result from 4d -4f and 4p -4d transitions in ion stages with open 4d subshells. Calculations were performed with the flexible atomic code (FAC) [START_REF] Gu | Indirect X-ray line-formation processes in iron L-shell ions[END_REF][START_REF] Gu | The flexible atomic code Can[END_REF], a complete software package developed for the computation of various atomic collisional and radiative processes to model spectral emission from astrophysical plasmas. Since its introduction however it has been successfully applied in numerous additional fields e.g. magnetic fusion and laser produced plasmas. FAC combines the strengths of existing atomic codes e.g. ATOM [START_REF] Amusia | Computation of atomic processes: A handbook for ATOM programs[END_REF], HULLAC [START_REF] Bar-Shalom | HULLAC, an integrated computer package for atomic processes in plasmas[END_REF] and SZ [START_REF] Sampson | A Dirac-Fock-Slater approach to atomic structure for highly charged ions[END_REF][START_REF] Zhang | Fully relativistic and quasirelativistic distortedwave methods for calculating collision strengths for highly charged ions[END_REF]; it uses a fully relativistic approach based on the Dirac equation, and implements efficient methods for distorted wave approximation, thus allowing its application to ions with large values of nuclear charge. The following basis set was used: 4p 6 4d N , 4p 6 4d N -1 nl and 4p 5 4d N +1 where n≤ 8, l≤ 3 and 1 ≤ N ≤ 10. Figures 123show Pd-like through Rb-like spectra of hafnium through actinium including CI. 4d -4f transitions are shown in black, 4p -4d transitions are shown in orange (gray) and all other transitions are shown in yellow (light gray). It was noted in [START_REF] Kilbane | Ground-state configurations and unresolved transition arrays in extreme ultraviolet spectra of lanthanide ions[END_REF] that for a given element, the emission from 4d -4f and 4p -4d becomes stronger and more concentrated within a particular wavelength range when configuration interaction is included. It is well known that CI redistributes transitions providing a more accurate description of experimental spectra [START_REF] Cowan | The Theory of Atomic Structure and Spectra[END_REF]. However as Z increases there was little discernable difference between the spectra excluding CI and including CI. This point will be further developed under the UTA framework in section 3. Figure 4 displays the maximum peak emission (in terms of gA values) for (a) 4d -4f transitions and (b) 4p -4d transitions for each element as a function of wavelength. As was observed for the lanthanides [START_REF] Kilbane | Ground-state configurations and unresolved transition arrays in extreme ultraviolet spectra of lanthanide ions[END_REF][START_REF] Carroll | 4d-4f emission resonances in laser-produced plasmas[END_REF][START_REF] Carroll | Ground-state configurations of ionic species I through XVI for Z=57-74 and the interpretation of 4d-4f emission resonances in laser-produced plasmas[END_REF][START_REF] Mandelbaum | Interpretation of the quasicontinuum band emitted by highly ionized rare-earth elements in the 70 -100 -Å range[END_REF], both 4d -4f and 4p -4d emission moves to shorter wavelength on increasing Z. The dependence of peak transition energies on atomic number Z is presented in Figure 4 (c). This shows that on increasing Z, the 4d -4f emission peak and the 4p -4d emission peak clearly separate. However, the fact that the maximum gA value remains relatively comparable for each transition type over the range of Z = 72 -89, implies that these elements could be potential radiation sources over the wavelength range λ = 2.5 -6 nm. This point is further quantified with the aid of UTA statistical analysis in the following section.

Unresolved Transition Array Statistics of Ions with Z = 72 -89

In order to quantify the emission from the 4d -4f and 4p -4d transitions in these ions, we adopted the unresolved transition array (UTA) approach developed by Bauche-Arnoult, Bauche and Klapisch [START_REF] Bauche-Arnoult | Variance of the distributions of energy levels and of the transition arrays in atomic spectra[END_REF][START_REF] Bauche-Arnoult | Variance of the distributions of energy levels and of the transition arrays in atomic spectra. II. Configurations with more than two open subshells[END_REF][START_REF] Bauche | Emissive zones of complex atomic configurations in highly ionized atoms[END_REF][START_REF] Bauche-Arnoult | Variance of the distributions of energy levels and of the transition arrays in atomic spectra. III. Case of spin-orbit-split arrays[END_REF]. UTAs were introduced originally to aid interpretation of low resolution soft X-ray spectra emitted by hot plasmas. They are currently widely used to approximate complex atomic spectra in plasma opacity and emissivity calculations which reduces computation times in large-scale radiation hydrodynamic simulations of plasma dynamics. In the UTA model, the discrete line spectra are replaced by a continuous function (usually Gaussian) such that each configuration-configuration transition array is characterized by the average quantities such as total intensity, average transition energy and variance. The average and variance of the transition energies ( Ē and σ 2 respectively) can be expressed as the gA-weighted sums Ē = j,i<j

g j A ji E ij j,i<j g j A ji ( 1
)
and

σ 2 = j,i<j g j A ji ( Ē -E ij ) 2 j,i<j g j A ji , (2)
where A ji is the Einstein coefficient for spontaneous emission from level j to level i, and g j is the statistical weight of the upper level. The mean wavelength λgA and the spectral width ∆λ gA of the transition array can be defined as follows λgA = 10 8 / Ē, (3)

∆λ gA = √ 8ln2 × 10 8 σ/ Ē2 , (4)
where Ē and σ are expressed in cm -1 and λgA and ∆λ gA in Å. Before applying the above UTA analysis, it is worthwhile to demonstrate the accuracy of the FAC code and the validity of its use in the current work. As an example, the 4p 6 4d -4p 5 4d 2 and 4p 6 4d -4p 6 4f transition wavelengths and UTA statistics of the Rb-like tungsten ion, W 37+ are presented in Table 1. This is a suitable choice owing to recent interest in it by members of the fusion community. As such wavelengths from a number of sources are available and it can be seen that the results of the current theoretical work compare favorably with experimentally recorded values. The 4p 6 4d -4p 5 4d 2 and 4p 6 4d -4p 6 4f transitions were first identified by Radtke et al [START_REF] Radtke | Line and band emission from tungsten ions with charge 21+ to 45+ in the 45 -70 -Å range[END_REF] using an electron-beam ion trap at a wavelength uncertainty of ±0.05 Å. Later Utter et al [START_REF] Utter | Electron-beam ion-trap spectra of tungsten in the EUV Can[END_REF] improved on this by recording at a higher spectral resolution with wavelength uncertainties of between ±0.004 and ±0.02 Å. In the work of Radtke et al [START_REF] Radtke | Line and band emission from tungsten ions with charge 21+ to 45+ in the 45 -70 -Å range[END_REF], ab initio theoretical calculations were carried out using the multiconfigurational relativistic HULLAC code [46,47] and the difference between measured and calculated wavelengths was found to be as much as 1 Å. The current calculations employ the FAC code which has an accuracy of 10 -30 m Å at 10 Å [48]. The wavelengths calculated with the FAC code are in closer agreement with experimental values, where the largest difference is found to be 0.6 Å. Also presented in Table 1 is the UTA analysis for W 37+ where (i) denotes the results of Radtke et al [START_REF] Radtke | Line and band emission from tungsten ions with charge 21+ to 45+ in the 45 -70 -Å range[END_REF] obtained using the HULLAC code, and (ii) refers to results obtained by application of 1-4 to the FAC data produced in the current work. Agreement is good with a maximum difference of 0.16 Å in the spectral width of the mixed CI transition array (4p 6 4d -[4p 5 4d 2 + 4p 6 4f]). This can be attributed to the use of a larger basis set in the current work, as described in Section 2. As was the approach in [START_REF] Kilbane | Extreme ultraviolet emission spectra of Gd and Tb ions[END_REF], this basis set was chosen for consistency when surveying such a large number of ions. Inclusion of various configurations will continue as and when more detailed experimental energy levels, transition wavelengths and line strengths of these ions become available for comparison.

The above discussion indicates that the use of the FAC code is appropriate for producing the 180 theoretical ionic spectra considered here. It also shows that these spectra can be adequately described using the UTA approach. The 4d -4f and 4p -4d UTAs were extracted from the CI spectra and the above statistical analysis was applied to each separately. The results are displayed in Figure 5, excluding CI (left) and including CI (right), and Tables 2 and3. Figure 5 shows the dependence of mean wavelength ( λgA ) on ion stage for each element with Z = 72 -89 (4d -4f in black and 4p -4d in red). A number of observations can be made from these results: (i) CI effects are more dramatic in lower Z elements, (ii) CI concentrates the 4p -4d UTA at a particular wavelength for this series of ions, for example, in hafnium which is plotted in Figure 5 (c) non-CI and (d) CI. For the 4p -4d UTA, λgA varies from 5.29 -5.86 nm in the non-CI case compared to 5.17 -4.82 nm in the CI case. For the 4d -4f UTA, λgA varies from 6.43 -5.09 nm in the non-CI case compared to 6.26 -5.06 nm in the CI case. (iii) CI effects the 4p -4d UTA positioning more than the 4d -4f in hafnium. This can be observed for many of the lower Z elements in this study. Gradually the effects of CI are diminished as can be seen by comparing Figure 5 (a) and (b). Finally by actinium we see very little variation between CI and non-CI results. For the 4p -4d UTA, λgA varies from 2.79 -2.98 nm in the non-CI case compared to 2.77 -2.54 nm in the CI case. For the 4d -4f UTA, λgA varies from 4.27 -3.40 nm in the non-CI case compared to 3.88 -3.39 nm in the CI case. These results are summarized for the CI case only in Tables 2 and3. The mean wavelength λgA and spectral width ∆λ gA for 4d -4f and 4p -4d UTAs calculated from the CI spectra are presented. From these values and the results presented in Figure 4, it is possible to identify strong emitters for almost all wavelengths between approximately λ = 2.5 -6 nm. Indeed in some cases it is possible that a target of mixed composition may give broad band emission across this wavelength range encompassing the "water window" and soft X-ray region.

Conclusion

Possible extreme ultraviolet and soft X-ray radiation sources were identified, using the FAC relativistic code, as emission peaks arising from 4d -4f and 4p -4d transitions in Pd-like to Rb-like ions of hafnium through actinium. For increasing nuclear charge, these strong emitters are seen to separate and move to shorter wavelength. The emission was comparable from each source and was characterized using the unresolved transition array model. The laser produced plasma technique may be employed in future work to generate these radiation sources which find application in numerous exciting fields, e.g. microscopy, spectroscopy and lithography. Table 1. Transition wavelengths and UTA statistics of Rb-like tungsten, W 37+ . (a) Transition wavelengths: In column (i) λ exp and λ th respectively, denote the experimental and theoretical transition wavelengths obtained by Radtke et al [START_REF] Radtke | Line and band emission from tungsten ions with charge 21+ to 45+ in the 45 -70 -Å range[END_REF]. The theoretical values were calculated using the HULLAC code. In column (ii) λ th denotes theoretical wavelengths calculated in the present work using the FAC code. In column (iii) λ exp refers to transition wavelengths recorded experimentally by Utter et al [START_REF] Utter | Electron-beam ion-trap spectra of tungsten in the EUV Can[END_REF]

Figure 2 .

 2 Figure 2. (Color online) Pd-like through Rb-like spectra of platinum through bismuth computed with the FAC code including CI. Black denotes 4d -4f transitions, orange (gray) denotes 4p -4d transitions and yellow (light gray) denotes all transitions.

Figure 3 .

 3 Figure 3. (Color online) Pd-like through Rb-like spectra of polonium through actinium computed with the FAC code including CI. Black denotes 4d -4f transitions, orange (gray) denotes 4p -4d transitions and yellow (light gray) denotes all transitions.

Figure 4 .Figure 5 .

 45 Figure 4. (Color online) Maximum peak emission from (a) 4d -4f and (b) 4p -4d UTAs (including CI) in elements with Z = 72 -89. (c) Dependence of UTA transition energies on atomic number Z, 4d -4f (black open circles), 4p -4d (red crosses).

)

  Figure 1. (Color online) Pd-like through Rb-like spectra of hafnium through iridium computed with the FAC code including CI. Black denotes 4d -4f transitions, orange (gray) denotes 4p -4d transitions and yellow (light gray) denotes all transitions.

	[46] Bar-Shalom A, Klapisch M and Goldstein W H 1988 The HULLAC code for atomic physics (unpublished) [47] Bar-Shalom A, Klapisch M and Oreg J 1988 Electron collision excitations in complex spectra of ionized heavy atoms Phys. Rev. A 38 1773 [48] Gu M F 2003 FAC 1.1.0 manual [49] Kramida A E and Shirai T 2009 Energy levels and spectral lines of tungsten, W III through W LXXIV At. Data and Nucl. Data Tables 95 305 2 3 4 5 6 7 0 7 7 7 7 7 7 7 7 7 7 Hf 26+ Hf 27+ Hf 28+ Hf 29+ Hf 30+ Hf 31+ Hf 32+ Hf 33+ Hf 34+ Hf 35+ gA (10 13 s -1 ) 2 3 4 5 6 7 0 8 8 8 8 8 8 8 8 8 8 Re 29+ Re 30+ Re 31+ Re 32+ Re 33+ Re 34+ Re 35+ Re 36+ Re 37+ Re 38+ gA (10 13 s -1 ) 2 3 4 5 6 7 0 8 8 8 8 8 8 8 8 8 8 Pt 32+ Pt 33+ Pt 34+ Pt 35+ Pt 36+ Pt 37+ Pt 38+ Pt 39+ Pt 40+ Pt 41+ gA (10 13 s -1 ) 2 3 4 5 6 7 0 9 9 9 9 9 9 9 9 9 9 Tl 35+ Tl 36+ Tl 37+ Tl 39+ Tl 40+ Tl 41+ Tl 44+ Tl 43+ Tl 42+ gA (10 13 Tl 38+ s -1 )
			Wavelength (nm) Wavelength (nm)			Wavelength (nm) Wavelength (nm)
	gA (10 13 s -1 ) gA (10 13 s -1 )	2 0 7 7 7 7 7 7 7 7 7 7 2 0 8 8 8 8 8 8 8 8 8 8	3 3	4 4	5 5	6 Ta 27+ Ta 28+ Ta 29+ Ta 30+ Ta 31+ Ta 32+ Ta 33+ Ta 34+ Ta 35+ Ta 36+ 6 Au 33+ 7 7 Au 34+ Au 35+ Au 36+ Au 37+ Au 38+ Au 39+ Au 40+ Au 41+ Au 42+	gA (10 13 s -1 ) gA (10 13 s -1 )	2 0 8 8 8 8 8 8 8 8 8 8 2 0 9 9 9 9 9 9 9 9 9 9	3 3	4 4	5 5	6 Os 30+ 7 Os 31+ Os 32+ Os 33+ Os 34+ Os 35+ Os 36+ Os 37+ Os 38+ Os 39+ 6 7 Pb 36+ Pb 37+ Pb 38+ Pb 40+ Pb 41+ Pb 42+ Pb 45+ Pb 44+ Pb 43+ Pb 39+
			Wavelength (nm) Wavelength (nm)			Wavelength (nm) Wavelength (nm)
	gA (10 13 s -1 ) gA (10 13 s -1 )	2 0 7 7 7 7 7 7 7 7 7 7 2 0 8 8 8 8 8 8 8 8 8 8	3 3	4 4	5 5	6 W 28+ W 29+ W 30+ W 31+ W 32+ W 33+ W 34+ W 35+ W 36+ W 37+ 6 Hg 34+ 7 7 Hg 35+ Hg 36+ Hg 37+ Hg 38+ Hg 39+ Hg 40+ Hg 41+ Hg 42+ Hg 43+	gA (10 13 s -1 gA (10	2 0 8 8 8 8 8 8 8 8 8 8 2 0 9 9 9 9 9 9 9 9 9 9	3 3	4 4	5 5	6 Ir 31+ Ir 32+ Ir 33+ Ir 34+ Ir 35+ Ir 36+ Ir 37+ Ir 38+ Ir 39+ Ir 40+ 6 Bi 37+ Bi 38+ Bi 39+ Bi 40+ Bi 41+ Bi 42+ Bi 43+ Bi 46+ Bi 45+ Bi 44+	7 7
			Wavelength (nm) Wavelength (nm)			Wavelength (nm) Wavelength (nm)

Table 2 .

 2 and presented by Kramida and Shirai in[49]. (b) UTA statistics: Calculated mean wavelength λgA and spectral width ∆λ gA for the unresolved transition arrays. Column (i) denotes values from Radtke et al[START_REF] Radtke | Line and band emission from tungsten ions with charge 21+ to 45+ in the 45 -70 -Å range[END_REF] and column (ii) denotes values obtained by applying 1-4 to the FAC data produced in the current work. Wavelengths throughout the table are given in Å. Calculated mean wavelength λgA and spectral width ∆λ gA for the configuration interaction unresolved transition arrays of hafnium through mercury ions: Rb-like to Rh-like ions for the 4p -4d arrays and Rb-like to Pd-like ions for the 4d -4f arrays. Wavelengths throughout the table are given in Å. .28 45.66 5.23 44.20 5.43 42.41 4.97 40.65 4.15 38.94 2.78 37.49 2.70 36.09 2.53 Ru-like 48.47 5.17 46.81 5.24 45.06 4.85 43.19 3.02 41.70 3.52 40.19 3.34 38.70 2.66 37.29 2.51 35.92 2.35 Rh-like 48.24 7.28 46.65 7.36 44.56 5.26 43.01 5.11 41.51 4.95 40.04 4.77 38.62 4.57 37.23 4.35 35.88 4.09

	(a)

Table 3 .

 3 Calculated mean wavelength λgA and spectral width ∆λ gA for the configuration interaction unresolved transition arrays of thallium through actinium ions: Rb-like to Rh-like ions for the 4p -4d arrays and Rb-like to Pd-like ions for the 4d -4f arrays. Wavelengths throughout the table are given in Å. Rb-like 36.90 10.93 35.54 10.85 34.20 10.71 32.92 10.58 32.30 12.09 31.08 11.91 29.91 11.72 28.76 11.50 27.66 11.28 Sr-like 36.36 9.21 35.09 9.40 33.68 9.06 32.61 9.58 31.42 9.52 30.19 9.29 29.16 9.49 28.26 9.93 27.18 9.74 Y-like 35.59 6.08 34.35 6.51 33.19 7.16 31.90 6.93 30.67 6.85 29.54 6.98 28.63 7.69 27.80 8.42 26.65 8.09 Zr-like 35.35 5.12 34.14 5.61 32.77 5.21 31.58 5.49 30.44 5.70 29.43 6.28 28.36 6.42 27.58 7.34 26.35 6.77 Nb-like 35.08 3.62 33.78 3.73 32.44 3.22 31.29 3.83 30.24 4.69 29.06 4.55 28.06 5.04 26.90 4.58 36.06 5.50 Mo-like 34.90 2.83 33.57 2.71 32.36 3.15 31.11 2.78 29.98 3.15 29.12 4.65 27.72 3.15 26.81 3.91 25.83 4.30 Tc-like 34.74 2.34 33.42 2.01 32.15 1.71 30.97 1.92 29.77 1.59 28.65 1.54 27.54 1.31 26.67 3.17 25.74 3.61 Ru-like 34.59 2.17 33.28 1.33 32.04 1.25 30.85 1.18 29.69 1.11 28.58 1.04 27.50 0.98 26.45 0.92 25.44 0.86 Rh-like 34.57 3.78 33.29 3.42 31.91 0.00 30.74 0.00 29.75 3.11 28.65 3.11 27.58 3.02 26.53 2.84 25.41 0.00

		gA		gA
	Ac	∆λ	Ac	∆λ
		λgA		λgA
		gA		gA
	Ra	∆λ	Ra	∆λ
		λgA		λgA
		gA		gA
	Fr	∆λ	Fr	∆λ
		λgA		λgA
		gA		gA
	Rn	∆λ	Rn	∆λ
		λgA		λgA
		gA		gA
	At	∆λ	At	∆λ
		λgA		λgA
		gA		gA
	Po	∆λ	Po	∆λ
		λgA		λgA
		gA		gA
	Bi	∆λ	Bi	∆λ
		λgA		λgA
		gA		gA
	Pb	∆λ	Pb	∆λ
		λgA		λgA
		gA		gA
	Tl	∆λ	Tl	∆λ
		λgA		λgA
	4p -4d	Ion	4d -4f	Ion

Acknowledgments

This work was supported by Science Foundation Ireland under Principal Investigator research grant 07/IN.1/B1771. The author thanks G. O'Sullivan for helpful discussions.


Pierrick Coupé

José V Manjon

Elias Gedamu

Douglas Arnold

Montserrat Robles

D Louis Collins

Object-based Rician Noise Estimation for MR Images

Introduction

The MR images noise variance is an important measure used for many applications such as denoising, registration or image quality assessment. The real and the imaginary parts of the MR complex raw data are considered as corrupted by white additive Gaussian noises with the same variance [START_REF] Gudbjartsson | The Rician distribution of noisy MRI data[END_REF]. By taking the magnitude of the complex data, the noise is transformed into Rician noise. This noise is usually described by Rayleigh distribution in the background [START_REF] Gudbjartsson | The Rician distribution of noisy MRI data[END_REF] and approximated by Gaussian noise in the foreground when Signal Noise Ratio (SNR) is high enough. These descriptions of noise distribution has been used in the majority of noise estimation methods. Nevertheless, the Rayleigh model of the background can fail when ghosting artefact are presents (i.e. signal different to zeros) [START_REF] Sijbers | Automatic estimation of the noise variance from the histogram of a magnetic resonance image[END_REF], and the Gaussian approximation of foreground is no longer valid for low SNR images [START_REF] Sijbers | Automatic estimation of the noise variance from the histogram of a magnetic resonance image[END_REF]. To overcome these limitations an object-based estimation taking into account the Rician nature of the noise.

Method

The MAD estimator in wavelet domain [START_REF] Donoho | Ideal spatial adaptation by wavelet shrinkage[END_REF] is widely used to estimate the variance of Gaussian noise.

To obtain an unbiased estimation of s for all the SNR values, we proposed to use the correction procedure introduced in [START_REF] Koay | Analytically exact correction scheme for signal extraction from noisy magnitude MR signals[END_REF]. This analytical correction is based on an iterative estimation of the SNR in presence of Rician noise (see Figure 1). First, the object is extracted in the wavelet domain by k-means segmentation in the lowest subband. After, the MAD estimator is computed on wavelet coefficients corresponding to the object in the highest subband. Then, this first estimation s^ is iteratively corrected with equation given in Figure 1. Finally, the estimation sn^ of the Rician noise variance is obtained.

Result

Synthetic Data: To evaluate our method, synthetic T1-weighted MR data of Brainweb database [START_REF] Collins | Design and construction of a realistic digital brain phantom[END_REF] was corrupted with different levels of Rician noise (2 to 15%). Ghosting artefacts and inhomogeneity have been also added (see Fig 2). To estimate the accuracy of the method, the ratio between the estimated standard deviation and the applied standard deviation is computed for all the levels of noise. Our method was compared with the Maximum likelihood (ML) background method described in [START_REF] Sijbers | Automatic estimation of the noise variance from the histogram of a magnetic resonance image[END_REF] and the original MAD estimator. The results are given in Figure 3. T1-w data: Experiment on real data were performed on 23 T1-w MR images of 256x256x56 voxels acquired with the same sequence and device (1.5T Genesis Signa GE Medical system). The bronze standard is computed as the mean level of noise over all the data. The level of noise for a real data is obtain by extracting an artifact-free region in the background [START_REF] Gedamu | Automated quality control of brain MR images[END_REF]. The results are given in Figure 4.

Conclusion

The results on the BrainWeb phantom show that the proposed object-based method is more robust than background-based method. Moreover, the proposed Rician adaptation of the MAD produces better results than classical MAD estimator, especially for low SNR. Finally, experiments on real data show that the proposed method accurately estimate the variance of noise.

Figure 1 .

 1 Figure 1. Equation of the iterative estimation of Rician noise variance.

Figure 2 .

 2 Figure 2. Simulated ghosting artefacts on Brainweb with 20% of inhomogeneity.

Figure 3 .

 3 Figure 3. Results obtained by the compared methods for all the level of noise on synthetic image with ghosting artefacts and 20% of inhomogeneities.

Figure 4 .

 4 Figure 4. Results obtained by the compared methods for all the real data.


Stephan Merz

Hernán Vanzetto

Towards certification of TLA + proof obligations with SMT solvers

TLA + is a formal specification language that is based on Zermelo-Fränkel set theory and the Temporal Logic of Actions TLA. The TLA + proof system tlaps assists users in deductively verifying safety properties of TLA + specifications. tlaps is built around a proof manager, which interprets the TLA + proof language, generates corresponding proof obligations, and passes them to backend verifiers. In this paper we present a new backend for use with SMT solvers that supports elementary set theory, functions, arithmetic, tuples, and records. We introduce a typing discipline for TLA + proof obligations, which helps us to disambiguate the translation of expressions of (untyped) TLA + , while ensuring its soundness. Our work is a first step towards the certification of proofs generated by proof-producing SMT solvers in Isabelle/TLA + , which is intended to be the only trusted component of tlaps.

Introduction

TLA + [START_REF] Lamport | Specifying Systems, The TLA+ Language and Tools for Hardware and Software Engineers[END_REF] is a language for specifying and verifying concurrent and distributed algorithms and systems. It is based on a variant of Zermelo-Fränkel set theory for specifying the data structures, and on the Temporal Logic of Actions TLA for describing the dynamic system behavior. Recently, a first version of the TLA + proof system tlaps [START_REF] Chaudhuri | Verifying Safety Properties with the TLA + Proof System[END_REF] has been developed, in which users can deductively verify safety properties of TLA + specifications. TLA + contains a declarative language for writing hierarchical proofs, and tlaps is built around a proof manager, which interprets this proof language, expands the necessary module and operator definitions, generates corresponding proof obligations (POs), and passes them to backend verifiers, as illustrated in Figure 1. While tlaps is an interactive proof environment that relies on users guiding the proof effort, it integrates automatic backends to discharge proof obligations that users consider trivial.

The two main backends of the current version of tlaps are Zenon [START_REF] Bonichon | Zenon : An Extensible Automated Theorem Prover Producing Checkable Proofs[END_REF], a tableau prover for first-order logic and set theory, and Isabelle/TLA + , a faithful encoding of TLA + in the Isabelle [START_REF] Wenzel | The Isabelle Framework[END_REF] proof assistant, which provides automated proof methods based on first-order reasoning and rewriting. Zenon is not part of the trusted code base of tlaps, but outputs proof scripts in Isar syntax for the theorems that it proves. These proofs are passed to Isabelle for verification. In this way, Isabelle/TLA + is used both as a standalone backend prover and as the certification engine of proof scripts produced by other backends.

The currently available backends also include a generic translation to the input language of SMT solvers, focusing on quantifier-free formulas of linear arithmetic (not shown in Fig. 1). This SMT backend has occasionally been useful because the other verifiers perform quite poorly on obligations involving arithmetic reasoning. However, it covers a rather limited fragment of TLA + , which heavily relies on modeling data using sets and functions. Assertions mixing arithmetic, sets and functions arise frequently in TLA + proofs. In the work reported here we present a new SMT-based backend for (non-temporal) TLA + formulas that encompasses settheoretic expressions, functions, arithmetic, records, and tuples. By evaluating the performance of the backend over several existing TLA + proofs we show that it achieves good coverage for "trivial" proof obligations. The new modules comprising our backend appear shaded in Figure 1. We consider two target for our translation: SMT-LIB [START_REF] Barrett | The SMT-LIB Standard: Version 2[END_REF], the de facto standard input format for SMT solvers, and the native input language of the SMT solver Yices [START_REF] Dutertre | The Yices SMT solver[END_REF]. Using SMT-LIB as the target of our translation, tlaps can be independent of any particular solver. On the other hand, the Yices language provides useful concepts such as sub-typing or a direct representation of tuples and records. The considered TLA + formulas are translated to quantified first-order formulas over the theory of linear integer and real arithmetic, extended with free sort and function symbols. In particular, we make heavy use of uninterpreted functions, and we do not restrict ourselves to quantifier-free formulas. All SMT solvers mentioned in this paper produce proofs, although each one has its specific output format.

TLA + is an untyped language, which makes it very expressive and flexible, but also makes automated reasoning quite challenging [START_REF] Lamport | Should your specification language be typed?[END_REF]. Since TLA + variables can assume any value, it is customary (and recommended) to start any verification project by proving a so-called type invariant that constrains the values that variables of the specification may assume. Most higherlevel correctness proofs rely on the type invariant. It should be noted that TLA + type invariants frequently express more sophisticated properties than what could be ensured by a decidable type system. In contrast, our target languages are multi-sorted first-order languages, which are supported by dedicated decision procedures in SMT solvers. The first challenge is therefore to assign an SMT sort to each expression that appears in the proof obligation. We make use of this type assignment during the translation of expressions, which may depend on the types of the subexpressions involved. For example, equality between integer expressions will be handled differently from equality between sets or functions.

During a first phase, our translation attempts to infer types for all subexpressions of a proof obligation. This phase may fail because not every set-theoretic expression is typable according to our typing discipline, and in this case the backend aborts. Otherwise, the proof obligation is translated to an SMT formula. Observe that type inference is relevant for the soundness of the SMT backend: a proof obligation that is unprovable according to the semantics of untyped TLA + must not become provable due to incorrect type annotation. As a trivial example, consider the formula x + 0 = x, which should be provable only if x is known to be in an arithmetic domain.

Type inference essentially relies on assumptions that are present in the proof obligation and that constrain the values of symbols (variables or operators).

A type system for TLA + together with a concise description of the type inference algorithm is presented in the next section. Section 3 describes the translation. Finally, results for some case studies and conclusions will be given in Sections 4 and 5.

2 Type inference for TLA + We define a type system for TLA + expressions that underlies our SMT translation. We consider types τ according to the following grammar.

τ ::= i | o | Str | Nat | Int | Real | P τ | τ → τ | Rec {field i → τ i } | Tup [τ i ]
The atomic types are i (terms of unspecified type), o (propositions), strings, and natural, integer and real numbers. Complex types are sets (of base type τ ), functions, records (defined by a mapping from field names to types) and tuples (as a list of types).

For this type system, we define an inference algorithm that is based on a recursive, bottom-up operator [[e, ε]] I whose arguments are a TLA + expression e and an expected (minimum) type ε that e should at least have, according to a predefined partial order relation on types that includes relations such as Nat < Int or i < τ , for any type τ = i. The computation either returns the inferred type or aborts. The operator recurses over the structure of TLA + expressions, gathering information in a typing environment type : Id → τ , that maps each TLA + symbol to its type.

The operator [[•]] I is applied iteratively on the list of hypotheses H 1 , . . . , H n , until a fixpoint is reached: note that later hypotheses may provide additional information for symbols that occur in earlier ones.

Initially, we consider that every symbol has the unspecified type i, which can be thought of as a bottom type. Recursive calls to the operator [[•]] I may update the type of symbols as recorded in type by new types that are greater than the previous ones. A type assignment is definitive only when types for all expressions in the proof obligation have been successfully inferred. For example, consider the hypotheses S = {} and S ⊆ Int. After evaluating the first one, S will have type P i, but it will be updated to P Int when the second hypothesis is processed.

When type inference succeeds, the environment variable type will contain the resulting final type assignments. As we discuss below, there are two cases where the inference algorithm can fail: (1) when a symbol an expression depends on does not have an assigned type, and (2) when a constraint stating that two or more expressions need to be of the same type cannot be solved.

The operator [[•]] I assigns types to complex expressions based on the types of their constituents. Although expressions such as {a} ∪ 0 or 3 + true appear silly, they are allowed in TLA + , yet their meaning is unknown. Certain TLA + operators are defined in such a way that the result type is fixed, aiding type inference. For example, logical operators always return Boolean values, whatever are the types of their operands. Similarly, an expression S ∪ T is of type P i; if S and T are known to be sets of elements of the same type τ , then we obtain the more precise type P τ . Arithmetic operators guarantee the result type only if their arguments are in the arithmetic domain.

TLA + expressions such as e 1 = e 2 , e 1 ⊆ e 2 , if p then e 1 else e 2 are critical in the sense that their constituents e 1 and e 2 must have the same type in order to express them in the sorted format of the SMT solvers. Similarly, the expression e 1 ∈ e 2 requires that e 2 be of type P τ , and e 1 of type τ , for some τ . For similar reasons, we do not allow functions (or operators) to return values of different types. Type inference for this kind of expressions makes use of the function S([e 1 , . . . , e n ], ε), that given a list of expressions e 1 , . . . , e n and an expected type ε, returns their unique type and fails when some pair [[e 1 , ε]] I , . . . , [[e n , ε]] I cannot be equated.

As a concrete example, consider the proof obligation (¬¬P ) = P , where P is a defined operator whose definition is hidden (i.e., its definition cannot be used in the proof obligation). We know that ¬¬P is Boolean, and the typing rule for equality requires that the expressions on both sides must be of equal type. Without any information on the type of P , the obligation should be rejected by type checking. In particular, we are not allowed to infer that P is of type Boolean. Indeed, P might be defined as the integer 42, and we do not know if (¬¬42) = 42 is true in TLA + . We therefore derive the type assignment only from available facts, i.e., hypotheses of the proof obligation, of the forms id ⊗ e and ∀x ∈ S : id (x) ⊗ e, for ⊗ ∈ {=, ∈, ⊆}. In these expressions, id is a constant, variable or operator, and e is an expression whose type can already be inferred1 .

3 From TLA + to SMT Once we have determined a type assignment for a TLA + proof obligation, it can be translated to the input languages of SMT solvers. Our target languages (SMT-LIB and Yices) are similar, and the translation proceeds along the same lines. We define the operator [[•]] T that translates a TLA + expression, using the type information gathered previously, to its corresponding SMT formula extended with λ-terms. The typing discipline ensures that all resulting λ-abstractions can be β-reduced before obtaining the definitive translation of the original expression.

A symbol's translation depends on its inferred type. It is translated to an SMT variable or function name, which has to be declared in the output produced by the translation, together with its type. Purely arithmetic and first-order expressions are translated to the corresponding built-in operators of the target languages, except for quantified expressions, where quantified variables are introduced temporarily in the context with their types inferred accordingly. In this way, for the expression ∀x : e, the value [[e, o]] I is evaluated to obtain the type assignment for the variable x, so it can be properly declared in the translation of the quantified expression.

Sets and functions are translated to uninterpreted functions. The encoding of a set S represents its characteristic predicate ("is a member of set S"), allowing for the direct translation of the set membership relation. In this way, [[S]] T is a λ-abstraction which will be applied to a value to detect if it is an element of the set or not,

consequently [[e ∈ S]] T ≡ ([[S]] T [[e]] T ). Similarly, function application reduces to [[f [e]]] T ≡ ([[f ]] T [[e]] T ). A function [x ∈ S → exp],

whose domain is S, is translated to λy. [[exp(x/y)]] T , where x is replaced by y in the expression exp (the domain S is represented separately, as we will see below).

For example, a function f that returns a set of type P τ it will be translated as a λ-abstraction where the first parameter is the function's argument and the second is a value of type τ . Then the translation is defined as [[f ]] T ≡ λx y.(f x y), where f denotes the symbol's name in the SMT output. The translation of equality depends on the types of the two subexpressions, which must have the same type by type-inference. For example:

[[e 1 = e 2 ]] T ≡ [[∀x : x ∈ domain e 1 ⇔ x ∈ domain e 2 when e 1 is of type → ∧ (x ∈ domain e 1 ⇒ e 1 [x] = e 2 [x])]] T [[e 1 = e 2 ]] T ≡ [[∀f, g : (f ∈ e 1 ∧ g ∈ e 2 ) ⇒ f = g]] T
when e 1 is of type P ( → ) More generally, n-ary functions or operators returning a (simple) set are represented as predicates of arity n+1 with their last argument being the set members. We only consider simple sets, i.e. sets of individuals, in order to remain in the realm of first-order logic. Any hypotheses of a proof obligation that fall outside this class are discarded. For example, hypotheses of the form x ∈ S where S is of type PP τ are useful during type inference in order to determine the type of x but are then dropped during the SMT translation.

Because SMT-LIB has no notion of function domain, we associate with each function f a set domain f (as a characteristic predicate). For every function application that occurs in the proof obligation, we check that the argument values are in the domain: otherwise the value of the function application would be unspecified. To this end, we define an auxiliary operator [[•]] F that computes corresponding proof obligations. In particular, for function applications we let The main difference between the SMT-LIB and Yices translations is the encoding of tuples and records. In Yices, they are natively supported, whereas SMT-LIB currently does not have a pre-defined theory. To encode (fixed-size) tuples and records in the first-order logic of SMT-LIB, we treat each of their components separately. A symbol t of type Tup[τ i ] is translated as λi. t i, introducing the new symbols t i with types τ i to the context, corresponding to the i-th component of the tuple t. For example, [[t = e 1 , e 2 ]] T ≡ [[t 1 = e 1 ∧ t 2 = e 2 ]] T . The translation of records is analogous, with field names taking the place of tuple indexes. Currently, all constituents of tuples and records must be of basic types.

[[f [e]]] F ≡ [[f ]] F ∧ [[e]] F ∧ e ∈ domain f and [[Q x ∈ S : e]] F ≡ ∀x ∈ [[S]] T : [[e]] F for Q ∈ {∀,

Experimental results

We have used our new backend with good success on several examples taken from the tlaps distribution. For example, the non-temporal part of the invariant proof for the well-known Nprocess Bakery algorithm [START_REF] Lamport | A new solution of Dijkstra's concurrent programming problem[END_REF], which mainly uses set theory, functions and arithmetic over the natural numbers, could be reduced from around 320 lines of interactive proof to a completely automatic proof. The resulting obligation generates an SMT-LIB file containing 105 quantifiers (many of them nested), which has been proved by the CVC3 [START_REF] Barrett | CVC3[END_REF] SMT solver in around 10 seconds and by Z3 [START_REF] De Moura | Z3: An efficient SMT solver[END_REF] in less than a second on a standard laptop (whereas the original, interactive proof takes around 24 seconds to process). On the other hand, Yices cannot handle the entire proof obligation at once, and it was necessary to split the theorem into separate cases per subaction; it then takes about 8 seconds to prove the resulting obligations.

More interestingly, the Yices backend (with better support for records) could handle significant parts of the type and safety invariant proofs of the Memoir system [START_REF] Parno | Memoir: Practical State Continuity for Protected Modules[END_REF], a generic framework for executing modules of code in a protected environment. The proofs were almost fully automated, except for three sub-proof that required manual Skolemization of second-order quantifiers. In terms of lines of proof, they were reduced to around 10% of the original size. In particular, the original 2400 lines of proof for the complete type invariant theorems were reduced to 208 lines.

Conclusions

We defined a translation of certain TLA + proof obligations into SMT-LIB 2 and into the native language of Yices. The translation relies on a typing discipline for TLA + , which is untyped, and a corresponding type inference algorithm. This discipline restricts the class of TLA + expressions that can be translated. Nevertheless, a significant fragment of the source language can be handled. In particular, we support first-order logic, elementary set theory, functions, integer and real arithmetic, tuples and records. Sets and functions are represented as lambda-abstractions, which works quite efficiently but excludes handling second-order expressions involving sets of sets or quantification over complex types. Universal set quantifiers that occur at the outermost level can easily be removed by the user of tlaps, by introducing Skolem constants. An automatic pre-processing of such terms would further improve the backend. The current SMT-LIB backend provides only limited support for tuples and records.

In future work, we intend to study the question of interpreting proofs provided by SMT solvers for reconstructing them (as well as the type assignment) in the trusted object logic of Isabelle/TLA + . We also envisage extending our translation to the native input languages of other SMT solvers such as Z3 [START_REF] De Moura | Z3: An efficient SMT solver[END_REF]. These are similar to the ones that we considered here, and therefore their translation will be straightforward once the types are assigned.

  ∃}. The remaining clauses in the definition of [[•]] F collect all function applications that occur in an expression. In particular, [[x]] F ≡ true for an atomic expression x. An expression e depending on subexpressions e 1 , . . . , e n , is evaluated as [[e(e 1 , . . . , e n )]] F ≡ [[e 1 ]] F ∧ . . . ∧ [[e n ]] F .

For variables, these kind of expressions are usually given by the type invariant. Our backend requires similar type-correctness lemmas for hidden operators.


Pierrick Coupé

José V Manjon

Montserrat Robles

D Louis Collins

Adaptive Multiresolution Denoising Filter for 3D MR Images

Introduction

In MR imaging, denoising is an important issue in some clinical uses. Recently a new filter has been introduced by Buades [START_REF] Buades | A review of image denoising algorithms, with a new one[END_REF]: the Non-Local (NL-) means filter. Based on the natural redundancy of the images, this filter uses comparison of image patches in order to achieve the denoising. Due to its simplicity and high denoising performance, this filter is one of the best state-of-the-art methods [START_REF] Buades | A review of image denoising algorithms, with a new one[END_REF][START_REF] Coupe | An Optimized Blockwise NonLocal Means Denoising Filter for 3-D magnétique Resonance Images[END_REF]. Nevertheless, according to the nature of the underlying structures the filtering parameters should be adapted. To implicitly optimize the amount of denoising according to the underlying structure, an adaptive multiresolution framework is proposed.

Method

By combining in the wavelet domain the results obtained with different sets of filtering parameters, the denoising can be improved [START_REF] Lukin | A multiresolution approach for improving quality of image denoising algorithms[END_REF]. We propose to automatically adapt the mixing procedure according to the noise level and the spatial information contained in the detailed subbands.

The original image I is denoised with two sets of filtering parameters: one adapted to the image features preservation and an another adapted to the noise components removal (see [START_REF] Coupe | An Optimized Blockwise NonLocal Means Denoising Filter for 3-D magnétique Resonance Images[END_REF] and [START_REF] Wiest-Daessle | Rician noise removal by non-local means filtering for low signal-to-noise ratio mri: Applications to DT-MRI[END_REF] for details). This yields two images: Iu with efficient feature preservation and Io with efficient noise removal (see Fig. 1). After wavelet transformation, the signed distances between absolute values of wavelet coefficients of I and the threshold T are computed. This signed distance is used to perform a soft mixing between coefficients obtained from the two denoised images (see Fig. 1). When signed distance is superior to zero, the corresponding coefficient is mainly considered as data, thus wavelet coefficient from Iu should be privileged. Inversely, when signed distance is inferior to zero, the corresponding coefficient is mainly considered as noise, thus wavelet coefficient from Io should be privileged. The threshold T is the BayesSkrink described in [START_REF] Chang | Adaptive wavelet thresholding for image denoising and compression[END_REF] and the mixing function a sigmoid.

Result

Synthetic Data: The proposed multiresolution NL-means was tested on the BrainWeb database [START_REF] Collins | Design and construction of a realistic digital brain phantom[END_REF] corrupted with Rician noise at different levels. A comparison between the NL-means [START_REF] Coupe | An Optimized Blockwise NonLocal Means Denoising Filter for 3-D magnétique Resonance Images[END_REF][START_REF] Wiest-Daessle | Rician noise removal by non-local means filtering for low signal-to-noise ratio mri: Applications to DT-MRI[END_REF] and the proposed filter was performed by computing Peak Noise Ratio on T1-w phantom. For all the noise levels the proposed multiresolution filter obtained better results (see Fig. 2).

T1-w data: The proposed method has been applied on a T1-w MR image of 170x256x256 voxels acquired on a 1.5T Philips Gyroscan. Figure 3 shows the denoised image and the residual image. DTI data with high b-value: The DW-MRI with a b=3000 s/mm2 was composed of 8 B0 images and 100 gradient directions of 63x128x128 voxels acquired on a 3T Siemens TimeTrio. As proposed in [START_REF] Wiest-Daessle | Rician noise removal by non-local means filtering for low signal-to-noise ratio mri: Applications to DT-MRI[END_REF] each image and gradient direction has been denoised separately. Figure 4 shows the denoised images.

Conclusion

The results on the BrainWeb MR phantom shows that the proposed Multiresolution NL-means filter outperforms the implementations presented in [START_REF] Coupe | An Optimized Blockwise NonLocal Means Denoising Filter for 3-D magnétique Resonance Images[END_REF]. Experiments on real data emphasized the potential of the proposed filter on images with high levels of noise such as DW-MRI with high b-values. In all the cases, the proposed adaptive multiresolution filter obtained the best result.

Reference

Figure 1 .

 1 Figure 1. Adaptive Soft Wavelet Coefficient Mixing Procedure.

Figure 2 .

 2 Figure 2. PSNR values of the compared filters on the T1-w phantom for all the levels of Rician noise.In all the cases, the proposed adaptive multiresolution filter obtained the best result.

Figure 3 .

 3 Figure 3. Left: Original T1-w image. Middle: Denoised image obtained with the proposed filter. Right: Residual image (i.e. removed noise).

Figure 4 .

 4 Figure 4. Fractional Anisotropy (FA) map computed on the DW-MRI dataset with high b-value. Left: FA map computed with noisy DW-MRI. Middle: FA map computed on denoised DW-MRI with NLM filter. Right: FA map computed on denoised DW-MRI with the proposed filter.


Pierrick Coupé

Vladimir Fonov

José V Manjon

D Louis Collins

Template Construction using a Patch-based Robust Estimator

Introduction

In MR image analysis, template construction is often required to perform voxel-based morphometry analysis [START_REF] Ashburner | Voxel-Based Morphometry-The Methods[END_REF][START_REF] Senjem | Comparison of different methodological implementations of voxel-based morphometry in neurodegenerative disease[END_REF] in order to create a reference used to register a population of different subjects. Usually, after a non-linear registration step used to align the MR images together, simple voxel-wise averaging is used to construct the template. The averaging approach has the advantage of being fast and simple. However this technique is not robust to registration errors, and tends to degrade small structures and contrast between tissues producing an over-smoothed result. To overcome these limitations, we propose to use a robust patch-based estimation which is less sensitive to outliers during template construction and preserves image structure continuity.

Methods

In estimation problems, robust approaches such as median or M-estimators are usually preferred to sample mean, due to their ability to tolerate incorrect observations without biasing their estimations. These robust point-wise estimators are well-suited for unstructured data, but in image processing, it is possible to improve processing tasks by adding some priors on the local data organization. Recently, efficient priors have been proposed for MR denoising within a patch-based estimation scheme [START_REF] Coupe | An Optimized Blockwise NonLocal Means Denoising Filter for 3-D magnetic Resonance Images[END_REF]. The main idea of patch-based approaches is to use the similarity between patches instead of voxel intensities to perform a robust comparison of the samples. This way, estimation is driven by local geometry of the image, enabling better preservation of small structures and contrast between tissues. We propose to combine the advantages of robust statistics and patch-based approaches within an iterative template reconstruction scheme.

First, all the N subjects of the database are non-linearly co-registered and their intensities are normalized. The median of each voxel is then computed over the N subjects of the database to create the initial template. Afterward, the procedure consists in computing the intensity based distance (L2norm) between each 3D patch within the template centered on location x and the 3D patches of each registered subject S i at the same location. This distance is used to assign a weight for each sample within a robust function. The parameter of the robust function h(x) is locally estimated as the median of the 3D patch distances at the location x. Finally, this procedure is repeated until convergence is reached. The proposed method is mathematically described in Fig. 1.

The proposed estimator minimizes the effect of outliers (e.g. registration error or normalization inaccuracy) as well as preserves structure continuity by taking into account the local geometry of the image.

Results

During our experiments, a database of 20 T1-w images of 176x256x256 voxels of 1 mm 3 acquired from 20 healthy volunteers were used. These images were normalized with N3 [Zijdenbos et al. 1998] and intensity mapped into the range of the MNI152 template. Linear registration was then performed to the MNI152 space. Afterwards, the Minimum Deformation Template algorithm [START_REF] Fonov | Unbiased Average Age-Appropriate Atlas for Pediatric Studies[END_REF] was applied to create the template, using ANIMAL [START_REF] Collins | Animal: Validation and Applications of Nonlinear Registration-Based Segmentation[END_REF]] within a multi-resolution framework (non-linear deformation step size of 32, 16, 8, 4 and 2mm). For each resolution step, the template was been created using both the traditional mean and the proposed method with a 3x3x3 patch size.

An efficient template construction method should provide a template with a histogram close to the histogram of the images in database. To evaluate the method, histogram similarity is estimated with the Kullback-Leibler divergence metric [START_REF] Kullback | On Information and Sufficiency[END_REF]; smaller values of the divergence indicate more similar distributions. For all the resolution steps, the proposed approach produces a template with a histogram closer to the histograms of images in database (cf Fig. 2). The distance between template distribution and subject distributions decreases as the resolution of the deformation field increases.

The contrast between white and gray matter has been also computed. Ten anatomical locations for both tissue types were manually picked by an expert on the 2 mm template obtained by classical averaging. The location of these points was used to compute the normalized Michelson contrast [Michelson,1927] for each constructed template. For all the registration steps, the template obtained with the proposed approach has better contrast than the traditional mean-valued template method (cf Fig. 3).

Finally, visual inspection shows that the proposed method produces a template with better contrast preservation (cf Fig. 4).

Conclusion

A patch-based robust estimator has been proposed for MR anatomical template construction. Compared to the classical approach using the mean, the proposed method enables the construction of a template with a histogram closer to the histogram of the images in the database. Moreover, the proposed estimator produces a template with higher contrast. Finally, as shown with our validation on multi-resolution registration, the proposed method is more robust to registration error than the mean estimator. Further work on the impact the proposed method on analysis or processing tasks such as registration and segmentation will be the subject of investigation.

Figure 1 .

 1 Figure 1. Description of the proposed method

Fig 2 :

 2 Fig 2: Comparison between the histogram of constructed templates and the histograms of each image of in database at different non-linear registration step size (32mm, 16mm, 8mm, 4mm and 2mm).

Fig 3 :

 3 Fig 3: Contrast between White matter / Gray matter for templates obtained by using the Mean estimator and the proposed Patch-based robust estimator for each resolution.


Cedric Clouchoux

Pierrick Coupé

Jose Manjon

Nicolas Guizard

Marine Bouyssi-Kobar

Mylene Lefebvre

Adre Du Plessis

Alan Evans

Catherine Limperopoulos

A novel approach for high-resolution image reconstruction for in-vivo fetal brain MRI

Introduction:

In the present study, we propose an automatic framework for obtaining an accurate representation of the in-vivo fetal brain for the quantification of cerebral volume and cortical surface. One of the biggest challenges in performing non-sedated fetal MRI studies is fetal motion, resulting in motion artifact and poor image quality [Limperopoulos09,Prayer06]. To date, most conventional fetal MRI acquisitions will have a 1x1x2mm3 resolution. The challenge is then to recover a high image quality from such volumes, in order to reliably characterize cortical development during intra-uterine life. Typically, three fetal scans are acquired, one in each direction, providing high in-plane resolution slices in each direction. The problem is then to create a single, isotropic high-resolution volume. This reconstruction is detailed herein, as well as results based on synthetic and real fetal data. An example of fetal cortical surface rendering is also presented.

Method:

The input data is composed of three low-resolution (LR) T2-weighted volumes (1.2x1.2x2 mm3 voxel resolution), acquired in each direction (coronal, sagittal and axial). After the normalization and the coregistration of the three input volumes, a high-resolution (HR) reconstruction is achieved. This step consists of reconstructing one isotropic HR volume from 3 LR anisotropic volumes (fig. 1). In order to optimize the information derived from the input volumes, we propose an iterative super-resolution method based on bilateral filtering [Tomasi98], without any HR reference image. This process takes into account important and specific issues including: LR volumes with different anisotropic resolution, volumes not perfectly registered and, inconsistent image contrast. The algorithm proceeds as follows: first, a nearestneighbors interpolation is performed on the three input volumes, in order to have the same voxel size as the desired HR output. An initial volume is then computed, which is the average of these input volumes. Each interpolated volume is then used to compute an estimation of the reconstructed value by bilateral filtering along the HR axis. For each voxel, the local mean of its 2D neighborhoods in the HR plane is matched with their corresponding local means within the reconstructed volume in order to be robust to normalization error between the LR volumes. Finally, the estimations obtained from each direction are aggregated within a robust function to be less sensitive to misalignments. The resulting HR volume is then considered as the reference volume for the next iteration, and the process is repeated until convergence has been reached.

Results:

The HR reconstruction method was evaluated on simulated T2 data from the Brainweb database [Cocosco97], before being applied to clinical data. We first down-sampled a HR volume (1mm3) according to the three directions, to recover similar inputs that have the same resolution as the clinical data. Different reconstruction methods were then applied (fig. 2). The proposed method achieved better reconstruction than the other techniques, without the blurring induced by a cubic interpolation or block artifacts introduced by a nearest neighborhood interpolation. The cubic interpolation method showed greater differences with the ground truth than the proposed method, which was particularly evident with the contours (fig. 2). Peak signal-to-noise ratio (PSNR) was measured for the different methods. The PSNR for each direction of the LR images was as follows: axial=22.93 dB; sagittal=21.37 dB; coronal= 21.89 dB. For the reconstructed HR volumes, the results of the three methods are presented below: -Nearest-neighbors interpolated volume average: 25.14 dB -Cubic interpolated volume average: 26.02 dB -The proposed method volume average: 28.47 dB. In other words, in an ideal case (no registration or contrast errors) where interpolation-based methods [Rousseau06, Jiang07] should be optimal, the proposed method produced better results. The method was then applied to data from a 30-week gestation age (GA) fetus. Three single shot fast spin echo T2weighted imaging sequences were acquired, one for each direction (sagittal, coronal and axial) using the following parameters: TEeff=120ms;TR=12500ms, number of signal averages 0.625; field of view 330mm; slice thickness 2mm with no interslice gap, acquisition matrix 256x204 (1.5T scanner, 5-channel phased array cardiac coil). Figure 3 shows the resulting HR volume, and figure 4 illustrates the resulting cortical surface mesh extraction in a 28-week GA fetus.

Conclusion:

This novel and robust automatic methodological framework enables the quantification of fetal brain volumes using an innovative HR fetal MR image reconstruction. This methodology was developed specifically to address the unique obstacles of non-sedated fetal MR image acquisitions, validated on synthetic data and tested on clinical data. The proposed reconstruction method successfully overcame the inherent challenge of LR images resulting from ultra fast imaging sequences and fetal motion, and demonstrated a superior PSNR compared to existing interpolation-based techniques. The pipeline is entirely automated, allowing for the processing of large fetal MRI datasets. The quantification of fetal cortical brain development in healthy and high-risk fetuses over the second and third trimester is currently underway.

Figures: