Beuf

Capacitive approach to restore decoupling between channels for four-element MR coil array

Capacitive approach to restore decoupling between channels for four-element MR coil array.

A. L. Perrier, D. Grenier, N. Ravel, P. Litaudon and O. Beuf Multi-channel coil arrays are increasingly being used to improve Signalto-Noise Ratio (SNR) in Magnetic Resonance Imaging (MRI). The decoupling between coils is an important parameter in array design. Indeed coupling between elements affects resonance frequency of each single coil and decreases it sensitivity. Many solutions were developed to achieve decoupling between elements of multi-channel coil array. In this paper, we present a capacitive solution to restore channel decoupling of a specific four-channel receiver coil array using common conductors. The principle of an effective decoupling was first demonstrated by circuit simulations of |S|-parameters. A receive-only four-channel coil array was design for rat head MRI. Experimental |S|parameter measurements validated the proposed capacitive approach by restoring decoupling between elements and particularly between external loops.

Introduction:

The Nuclear Magnetic Resonance (NMR) multichannel coil arrays introduced in 1990 [START_REF] Roemer | The NMR phased array[END_REF] are used to increase sensitivity of MR coils while preserving a large volume of exploration. The basic idea is to use the juxtaposition or overlapping of several single loops. These loops are usually connected to low input impedance and low noise preamplifiers to reduce the coupling between the different elements of the array and to preserve SNR respectively. Recent works described other decoupling techniques such as strip transmission line array [START_REF] Adriany | A geometrically adjustable 16-channel transmit/receive transmission line array for improved RF efficiency and parallel imaging performance at 7 tesla[END_REF], overlap geometries [START_REF] Pinkerton | Noise properties of a NMR transceiver coil array[END_REF] and capacitive decoupling networks [START_REF] Pinkerton | Noise properties of a NMR transceiver coil array[END_REF][START_REF] Zhang | Design of a capacitively decoupled transmit/receive NMR phased array for high field microscopy at 14.1 T[END_REF]. A different solution for two-channel coil arrays was also described. In this setup, single elements of the array are not juxtaposed or overlapped but elements are electrically jointed with a common conductor [START_REF] Gareis | Transmit-receive coil-arrays at 17.6T, configurations for H-1, Na-23, and P-31 MRI[END_REF][START_REF] Dieringer | Design and Application of a Four-Channel Transmit/Receive Surface Coil for Functional Cardiac Imaging at 7T[END_REF][START_REF] Perrier | Design of a two-channel NMR coil using an impedance transformation approach[END_REF]. Based on this specific approach, a newly designed four-channel coil array was proposed [START_REF] Perrier | Design of a four-channel array coil for dual highresolution rat knee MR Imaging[END_REF].

Principle: The equivalent electrical circuit of a four-channel coil array based on common conductor architecture is presented in Fig. 1 where inductors are representing conductor sections. At the operating frequency, Cd and Cc capacitance values are chosen to compensate imaginary impedance parts of the Ld and Lc inductance values, respectively. At the operating frequency, without Lgnd parasitic inductances, these perfect compensations of impedance allow zeros of transmission between channels by deporting the ground plane to the black points labelled 3. This virtual ground plane between loops ensures the decoupling between channels [START_REF] Perrier | Design of a two-channel NMR coil using an impedance transformation approach[END_REF][START_REF] Perrier | Design of a four-channel array coil for dual highresolution rat knee MR Imaging[END_REF]. The three parasitic inductors Lgnd coming from the experimental design of the coil array deteriorate zeros of transmission and lead to coupling between elements. In this paper, we propose a solution to restore the decoupling between elements by adding a Cgnd capacitor in series with the central Lgnd inductance. This solution is illustrated in "Simulated results" section with circuit simulations of Fig. 1 topology and in "Experimental results" section with measurement of a four-channel coil array described in "Design" section. [START_REF] Pinkerton | Noise properties of a NMR transceiver coil array[END_REF] 3

2 2 1 1 1 Loop 2 Loop 1 Loop 3 Loop 4 R L c L c C c R C d C a C e L d Port 2 (50 Ω) Port 1 (50 Ω) C a C a C a Port 3 (50 Ω) Port 4 (50 Ω) L b L d C c R R L b C e C d L c C c C c L c C e C d L d L b C c C d L d L b C e L gnd L gnd C gnd L gnd

Fig. 1 Equivalent electrical circuit of a four channel coil array with common conductors.

Design: A specific four-channel receiver coil array was built on a flexible substrate and glued on a cylinder to fit the animal morphology. Loop inductors were realized with 35 µm thick and 4 mm width section of copper tape. Each element of the coil consists in a rectangular loop with 20x24 mm 2 internal and 28x32 mm 2 external dimensions. To decouple the receiver array from the transmitter coil, each loop integrates an active decoupling circuit made with two DH80055 PIN diodes. Four 50 Ω BNC cables were soldered at the four loops inputs to connect the coils with the acquisition data cabinet integrating transmit/receive switches and preamplifiers of MR system. The Ca and Ce were chosen to be adjustable elements for the four-channel coil array. Matching of each loop to 50 Ω at f0=300 MHz (proton Larmor frequency at 7 T static magnetic field) was realized using varicap diodes BB149. The capacitance values of these diodes ranged from 2 pF to 22 pF for a bias voltages between -30 V and 0 V. Fixed capacitors are non magnetic case A series 100 ATC capacitors. All fixe capacitors are experimentally adjusted to minimize all |Sij| dB parameters. A photograph of the coil array is shown in Fig. 2.

Fig. 2 Photograph of the four channel coil array built for rat head. to the three 10 nH parasitic inductances Lgnd: |Sij|dB parameters are increased at the operating frequency and curve minima are shifted. Fig. 3c and Fig. 3d present the restoration of decoupling between channels by additional Cgnd capacitors equal to 28.1 pF. It can be noticed on simulated results that a single Cgnd capacitor on central section (Fig. 3c) is sufficiency to restore an acceptable decoupling. This solution was chosen for practical realisation of the array.

Experimental results: |S|-parameter measurements of the four-channel coil array were carried out with an Agilent E5071C four-port VNA (Agilent Technologies Inc., Santa Clara, CA, USA). Results of coil array described in "Design" section are presented in Fig. 4a. |Sii|parameters perform better than -20 dB and characterizing each coil matching at the resonant frequency. |S14| parameter value is about -7 dB which is not sufficient to decouple external loops. This |S14|-value is accentuated by magnetic coupling compare to circuit simulation (Fig

Fig. 3

 3 Fig. 3 Simulations of the four-channel coil array. a Without parasitic inductance b With parasitic inductances Lgnd fixed to 10 nH c With capacitors Cgnd in series with Lgnd fixed to 28.1 pF d With one central capacitor Cgnd in series with the central Lgnd and fixed to 28.1 pF Simulated results: Circuit simulations realised with Designer software. The influence of Lgnd and Cgnd component values on the |S| parameters is shown in Fig.3. Simulations were achieved for an arbitrarily chosen value of coil inductors and for a 300 MHz operating frequency. The equivalent electrical circuit of Fig.1was simulated with all Lb, Lc, Ld and Lgnd values equal to 10 nH; R was fixed to 1.1 Ω. To compensate imaginary part of inductors at the operating frequency, capacitors Cc and Cd were fixed to 28.1 pF. Ca and Ce capacitors were fixed to 4.5 pF and 24.2 pF respectively. Fig.3ashows the ideal coil with perfect decoupling between channels at the operating frequency without Lgnd-Cgnd components: all |Sij| parameters are lower than -30 dB at the operating frequency. Fig.3bpresents the decoupling deterioration due

  3.b). All other coupling parameters are inferior to -16 dB at the centre frequency. In Fig. 4b a 24.8 pF capacitor is added in series with the central Lgnd of the coil array. At the operating frequency, the |Sij|parameters are below -20 dB. These values allow the use of the coil array for MR imaging without coupling between channels.

Fig. 4

 4 Fig. 4 |S| parameters measurements of the four-channel coil array. a Without Cgnd b With one central capacitor Cgnd = 24.8 pF Conclusion: A capacitive solution to achieve decoupling between elements of a particular four-channel coil array based on common conductors was first demonstrated by circuit simulations of an equivalent electrical circuit. The solution was validated by the experimental |S|-parameters measured on four-channel coil array dedicated to rat head: a coupling lower than -20 dB was obtained between elements at the operating frequency.

Acknowledgments: The authors thank the Institut Féfératif des Neurosciences de Lyon (IFNL) for financial support. This work was supported by an ANR grant (#ANR-07-NEURO-030). A. L. Perrier, D. Grenier and O. Beuf (Université de Lyon, CREATIS, CNRS UMR 5220, Inserm U1044, INSA-Lyon, Université Lyon1, Villeurbanne, France) E-mail: anne-laure.perrier@univ-lyon1.fr N. Ravel and P. Litaudon (Centre de Recherche en Neurosciences de Lyon, CNRS UMR 5292, INSERM U1028, Université Lyon 1, France)


Coherent interactions of light with collisionless atomic hydrogen.

Jacques Moret-Bailly June 11, 2018

Résumé L'objet de cet article est l'étude d'un système optique théorique constitué de corps noirs sphériques très chauds et distants (appelés étoiles), émettant des protons et des électrons dont le refroidissement par expansion adiabatique produit, sur des "sphères de Strömgren", des atomes d'hydrogène excités. De grands espaces entre ces systèmes élémentaires sont supposés contenir un gaz d'hydrogène atomique à très basse pression dont les interactions avec la lumière sont spatialement cohérentes. L'étude de ce système théorique utilise les règles établies par Einstein en 1917 et largement vérifiées dans l'étude des lasers à gaz. Des comparaisons discutables avec des observations astronomiques sont distinguées de l'étude théorique par des astérisques précédant les titres. De même que les lasers émettent des rayons dans les directions où l'amplification cohérente est maximale, la coquille d'hydrogène atomique excité qui entoure une sphère de Strömgren dessine le limbe de la sphère en un cercle éventuellement ponctué. Cette émission superradiante est associée à une super-absorption de la lumière excitatrce reçue de l'étoile qui par exemple dissimule SN1987A depuis l'allumage de SNR1987A. Un effet Raman impulsionnel stimulé (ISRS) réduit les fréquences des impulsions lumineuses formant la lumière naturelle au bénéfice du rayonnement froid pour accroitre l'entropie. Une excitation des atomes froids principalement dans le niveau 2P est nécessaire à l'ISRS, de sorte que le rougissement s'arrète lorsqu'une raie absorbée atteint la fréquence Lyman alpha, ce qui génère les rougissements remarquables de Karlsson 3K et 4K. La spectroscopie élémentaire remplace big bang, matière et énergie noirs, ne requiert pas de variation de la constante hyperfine, etc. La loi de Hubble évalue la densité de colonne d'hydrogène atomique excité, accroissant les distances dans les zones chaudes (galaxies) donc la taille des galaxies spirales stables sans matière noire, gonflant des bulles dans les cartes des galaxies.

Abstract The aim of this paper is a study of a theoretical optical system made up of distant, very hot spherical blackbodies (named stars) emitting light, protons and electrons cooled into Strömgren's shells of excited H atoms starting on Strömgren's spheres. Long distances between these elementary systems are assumed filled by very low pressure hydrogen atoms whose interactions with light are spatially coherent. This theoretical system is studied using rules of spectroscopy founded on Einstein's theory and laser experiments, widely verified in labs. Questionable astronomical comparisons are clearly distinguished from reliable spectroscopy by an asterisk * at title of sections. Lasers emit light into directions for which amplification is maximum. Likewise limbs of Strömgren's spheres show bright circles possibly split into dots. Superradiant emission is bound to a super-absorption of exciting light so that SN1987A disappearsed when its rings appeared. Impulsive Stimulated Raman Scattering (ISRS) shifts frequencies of light pulses to increase entropy. Theory shows that H atoms excited to 2P state shift very weakly frequencies of pulses making ordinary light. Lyman alpha absorption produces these atoms until absorbed lines are shifted to alpha frequencies, which produces remarkable Karlsson's redshifts 3K and 4K. Ordinary spectroscopy replaces big bang, dark energy and mater, variation of hyperfine constant, and so on. Hubble's law evaluates column densities of excited atomic hydrogen, increasing sizes of galaxies, so that spiral ones are stable without dark matter, and inflating bubbles in their maps. Keywords: 52.25.Os Emission, absorption, and scattering of electromagnetic radiation -52.38.Dx Laser light absorption in plasmas (collisional, parametric, etc.)-Super-emission/absorption -Impulsive Stimulated Raman Scattering (ISRS).

1 Introduction.

After publication of Einstein's 1917 paper, a long time was necessary to build lasers, and understand spatially coherent interactions of electromagnetic waves with matter, for instance, how energy provided by a super-absorption is transferred to the superradiant beam of a ruby or dye laser. Here, we study interaction of light with the simplest matter, collisionless atomic hydrogen.

Astrophysicists are suspicious because a lot of absurd experiments and theories were developped to understand astronomical observations. Menzel [START_REF] Menzel | The dilution of radiation in a nebula[END_REF] wrote "It is easily proved that so-called "stimulated" emissions are unimportant in the nebulae", but, in his demonstrations, he confounds radiance and irradiance. Here, we use only theories deduced from Einstein's discovery, and consequences of which are observed every day in labs of spectroscopy (or chemistry to follow fast evolutions of chemical reactions).

To avoid a confusion between spectroscopy and its questionable applications to astronomical observations, titles of questionable (sub)sections start with an asterisk.

All interactions of light with assumed low pressure gas are spatially coherent. Dipolar and quadrupolar interactions are very different, so that they will be studied separetely.

2 Dipolar, spatially-coherent gas-light interactions.

Einstein's and laser theories.

Thermodynamics provides two equations, one for conservation of energy, the other for stability or increase of entropy. Einstein could only show that variation dI of spectral radiance I of a single, monochromatic, plane, spatially coherent light (electromagnetic) beam along a path dx is dI=BIdx, where he introduced spectral coefficient B depending on medium state and frequency. Assuming a constant B, variation of I is exponential, so that a non-zero solution requires an initial non-zero radiance, possibly zero Kelvin radiance. This difficult, random start of coherent light beams provides flares generated, for instance, by an electric power in a very low pressure gas, or by electrons in polar aurora. Studying lasers pumped by light (ruby, neodyme YAG, dye lasers) spectroscopists observed that superradiance of a ray produces strong coherent absorbtions of most other rays in a "competition of modes" of light in matter.

An usual, semi-classical explanation of I factor in second member of Einstein's equation is a synchronization of molecular dipoles on sheets, wave surfaces of incident light. An obtained set of synchronous dipoles works as a single dipole. Emitted amplitude is proportional to the momentum of synchronous dipoles, radiated energy is proportional to the square.

Rays close to exciting one may have some coherence with dipoles and can be slightly amplified. But far modes have an incoherent but strong interaction with syncronized sheets of dipoles, they have a super-absorption. For a better study of super-absorptions of light by matter, and references, see Higgin's paper [START_REF] Higgins | Superabsorption of light via quantum engineering[END_REF].

Superabsorption plays an important role in light-pumped lasers (ruby, dye,..), active medium of which becomes dark if a superradiant (laser) ray is emitted.

Competition of modes and beams interactions with variously excited atoms is generally too complicated for a precise study, so that it remains only the two fundamental rules of thermodynamics: While energy flows, entropy increases. (Temperature of a polarized, monochromatic ray is deduced from its frequency and spectral radiance by Planck's law).

* Emission of a ring shows the limb of a Strömgren's

sphere.

Stars may emit a "stellar wind" made mainly of protons and electrons. These particles are cooled by expansion and combine on a "Strömgren's sphere". Outside, a Strömgren's shell contains mainly excited H atoms. Amplification of light in shell is superradiant for rays having a maximal path in this shell, that is tangentially to the sphere (figure 1). De-excited by super-radiance, then involved in super-absorption, atoms intensely absorb energy received from the star. This energy can not be stored notably in gas which works as medium of a "light pumped laser".

Picture 2 shows initial limbs of SNR 1987A in which absorption or scattering of light by equatorial planets or rings strangled Strömgren's sphere into a three limbs (to Earth) hourglass. Now, the star has eaten its planets. This picture proves superradiance, with a high probability: Two background stars are much brighter than a lot of other background stars, while they are seen through rings: Radiance of rays received from a star being large, their light is much more amplified than light initially generated in Strömgren's shell.

Coherent dipolar interactions explain that SN1987A disappeared when its rings appeared. At scale of propagation time of light, these effects are simultaneous because energy stored by atoms excitation is negligible. SNR1987A shell is an optical black hole: Inside rings, it remains only radiation of hydrogen Lyman lines probably emitted by rare atoms in external region of Strömgren's sphere. Corresponding energy cannot be absorbed through shell because a transfer of energy to rings would decrease entropy.

Spectroscopy of light-pumped lasers or optical black holes is complex. Thermodynamics is the main rule, almost all stars radiation energy being transferred to rings.

In sky, astronomers observe many limbs, (circles or arcs), that competition of modes may punctuate.

May we say that light-pumped lasers are black holes ? Do gravitational black holes exist? 3 Quadrupolar (Raman), spatially-coherent gaslight interactions.

3.1 Quadrupolar interactions of light with excited hydrogen atoms: redshifts. In a quasar spectrum, Patrick Petitjean [START_REF] Petitjean | Le contenu baryonique de l'univers révélé par les raies dábsorption dans le spectre des quasars[END_REF] found several absorption lines attributed to redshifted Ly α line of H atom, but no shifted Ly β or Ly γ lines. We searched for them by applying Rydberg and Doppler formulas to redshifted alpha lines and found that redshifted beta or gamma are mixed with differently redshifted alpha ones. This superposition explains Petitjean's observation and shows a quantization of redshifts according to Karlsson's formula because we notice that 3K (or 4K) (where K is empirical Karlsson's constant) is redshift that brings an absorbed beta (or gamma) line to alpha frequency. Why does an absorbed line at alpha frequency stop redshift, allowing strong, sharp absorption of gas lines ? The simplest explanation is that redshift requires alpha absorption, thus creation of atoms in an excited 2P state.

A weak, slow redshift resulting from a beta absorption may restart the process. This is possible as long as shifted emission spectrum of star has energy at beta frequency.

Picture 3 shows a construction of "Lyman alpha" lines of a quasar spectrum. It is mistakenly assumed that a redshift multiplies all frequencies of light by a constant, as in a Doppler shift. As redshift results from an interaction with H Figure 4: Amplitudes of blue and red sine waves are added in purple.

atoms, it has a dispersion, so that the scale of frequencies in picture 3 must be slightly modified, mostly at low frequencies.

Quadrupolar spatially coherent interactions: Impulsive Stimulated Raman Scattering (ISRS).

How do excited hydrogen atoms redshift light ?

Need of pulsed light.

Spectrum of monochromatic, incoherent light scattered by a gas shows generally new, Raman lines. Assume that, on figure 4, the red line represents time variation of amplitude of an exciting coherent light, the blue one amplitude of a Raman scattered line, and purple one their sum.

As long as beats are far (here around 3 periods), purple seems to be a monochromatic wave of intermediate period. Impulsive Stimulated Raman Scattering uses this result by use of short light pulses.

Quadrupolar (Raman) gas-light interactions shift frequencies of light.

Except in bi-refringent crystals, two light frequencies propagating along a same ray have different speeds. Thus, coherent Raman scattered at different points have generally different phases at an observation point, they cancel for opposite phases, resulting Raman intensity remains low. G. L. Lamb [START_REF] Lamb | Analytical description of ultra-short optical pulse propagation in a resonant medium[END_REF] shows that powerful interactions are obtained using light pulses "shorter than all involved time constants". Thus, "Impulsive Stimulated Raman Scatterings" (ISRS) used to study fast evolutions of chemical reactions require femtosecond light pulses [START_REF] Yan | Impulsive Stimulated Scattering: General Importance in Femto-second Laser Pulse Interactions with Matter and Spectroscopic Applications[END_REF].

Use of temporally incoherent light.

ISRS is generally performed in laboratories using 10 femtoseconds laser pulses [START_REF] Yan | Impulsive Stimulated Scattering: General Importance in Femto-second Laser Pulse Interactions with Matter and Spectroscopic Applications[END_REF]. Usual light, of thermal origin, temporally-incoherent, is formed of roughly 1 nanosecond pulses, 10 5 times longer.

Compare roughly orders of magnitude of ISRS frequency shift using 1 ns rather than 10 fs light pulses. Taking into account only variations of time constants due to Lamb's conditions:

-Increase of collisional time by 10 5 divides the acceptable gas pressure, thus the shift by this factor.

-Reduction of Raman resonance period divides ISRS twice: directly by reduction of Raman quantum, and by Boltzman factor for difference of hyperfine levels populations at equilibrium.

Thus, an equal frequency shift requires a path whose order of magnitude is 10 15 times larger than in laboratory, an astronomical path. This explains the failure of experiments done to show a redshift of usual light by atomic hydrogen.

Hyperfine resonance at 1420 MHz of non-excited H atoms corresponds to a period less than 1 nanosecond, Lamb's conditions are not fulfilled. On the contrary, in excited states, hyperfine frequencies below 1 GHz produce an ISRS.

Hyperfine levels may be de-excited by another ISRS heating a thermal frequency without perturbation of isotropy. Such sets of ISRS are named "spatially Coherent Raman Effects on temporally Incoherent Lights" (CREIL). Presence or absence of Ly α absorption structures space into shells in which H atoms are excited or not. Previously described process provides a spectrum of so-called Ly α lines. But quasar spectrum shows many other absorbed, sharper, saturated lines which make "Lyman forests": Ly α absorption excites H atoms to 2P state, density of which increases in gas, that becomes able to amplify Ly α H line. Start of coherent "emission" which is an amplification requires an initial ray possibly provided by zero point field, or rays refracted from observed ray by fluctuations of gas density. Thus, a superradiant ray bursts as an observable flare which also produces a superabsorption of observed ray, so that a "Lyman forest" line is absorbed in observed spectrum. Gas is excited, and so on ... It seems difficult to compute frequencies of this relaxation process.

* Applications

* Correction of Hubble's law.

Hubble did not appreciate the common interpretation of his law. Thus we propose the following corrected interpretation: Redshifts of stars provide column densities (from stars to Earth) of atomic hydrogen in state 2P.

This sentence is not very precise because most excited states of H atoms, other atoms, may shift weakly frequencies of light, but it seems a good enough approximation for most uses.

* Redshifts of spiral galaxies.

Celestial mechanics provides a formula for stability of orbits of stars making spiral galaxies. With evaluation of distances by Hubble's law, that is the present evaluation of sizes of spiral galaxies, their masses must be increased by "dark matter". But it appears that search of "dark matter" inside laboratories buried under big mountains is negative. Thus it seems good to take into account an over-estimation of distances of these galaxies by an abundance of excited hydrogen emitted by hot stars, so that dark matter and energy are not needed.

* Dispersion of multiplets in spectra of far stars

As CREIL results from an interaction with matter as refraction, it depends on frequency. There is no need to modify fine structure constant. -A kernel of neutrons is surrounded by a decreasing temperature pseudo crystal (stabilized by pressure) made mainly of H atoms able to transfer energy between X rays by CREIL. Entropy is increased by exchange of energy between observed X rays emitted at increasing distances from center, thus have decreasing thermal emission frequencies. ( Such spectra are observed in X emissions of Sun.)

-An atmosphere of hydrogen has similar properties than Earth atmosphere, stratosphere, Heaviside noisy regions .... ** Anomalous accelerations of space probes: Radius of Strömgren's sphere of Sun is around 10 AU. Just outside, microwaves frequencies needed to compute distances and accelerations of probes may be increased by a CREIL transfer of energy from sunlight to colder microwaves, excited H atoms playing a role of catalyst. ** Maps of galaxies: Distances are exaggerated close to hot stars. Thus bubbles are inflated on maps, universe seems spongy. ** Isotropy of microwave background. Frequencies of these radiations are close to hyperfine frequencies: A strong CREIL inside background radiations increases entropy.

Conclusion

Interactions of light with interstellar gas are coherent.:

** Spatially coherent spectroscopy must be used in study of low pressure interstellar gas.

** Stars are surrounded by a shell of excited hydrogen atoms illuminating the limb of a Strömgren's sphere.

** Hubble's law does not evaluate distances, but column densities of excited atomic hydrogen. ** Dark matter and energy, big bang, are tales.

Figure 1 :

 1 Figure 1: Rays a and b cross the same number of sub-shells making Strömgren's shell of excited hydrogen. The path is, for each sub-shell, larger for more external b ray: The most amplified ray is tangent to Strömgren's sphere.

Figure 2 :

 2 Figure 2: SNR 1987A in 2012 (Hubble picture). The three rings are limbs of an hourglass of excited H atoms.

Figure 3 :

 3 Figure 3: Construction of "H atoms Lyman alpha" lines of a quasar spectrum.

3. 1 . 1 *

 11 Redshifts of "Lyman alpha lines" (Ly α lines) of quasars.

4. 1 *

 1 Lyman forests of quasars.

4. 5 *

 5 Very simplified descriptions of other applications. ** Model of quasar: Previous description of lines formation leads to a model of quasars:


Carl Tholin-Chittenden

Juan F P J Abascal

Manuchehr Soleimani

Juan Felipe

Perez-Juste Abascal

Automatic Parameter Selection of Image Reconstruction Algorithms for Planar Array Capacitive Imaging

Keywords:

  L'archive ouverte pluridisciplinaire

I. INTRODUCTION

Electrical Capacitance Tomography (ECT) is a non-invasive and non-destructive method of imaging through a material under test (MUT) [START_REF] Huang | Capacitance-based tomographic flow imaging system[END_REF]. By measuring the capacitance through the MUT, it is able to build up a permittivity distribution indicating areas of low or high permittivity comparatively. The process requires capacitance data acquisition, environment Engineering Tomography Lab (ETL), University of Bath, Department of Electronics & Electrical Engineering, Bath BA2 7AY, United Kingdom (C.Tholin-Chittenden@bath.ac.uk, M.Soleimani@bath.ac.uk) Univ Lyon, INSA Lyon, Universite Claude Bernard Lyon 1, UJM-Saint Etienne, CNRS, Inserm, CREATIS UMR 5220, U1206, Lyon, France (Juan.Abascal@creatis.insa-lyon.fr) modelling, simulation and image reconstruction. ECT systems are often 2-dimensional circular array devices [START_REF] Huang | Tomographic imaging of two-component flow using capacitance sensors[END_REF] as they are imaging a region which can be accessed from all sides. However when only 1 side is available, the sensor must be flattened into a co-planar array [START_REF] Mamishev | Interdigital sensors and transducers[END_REF]. The modelling must be done in 3 dimensions as capacitance is arching through a 3D space between the sensor electrodes. Due to their orientation the inter-electrode capacitance is reduced thus decreasing detectable distance from the sensor. Previous work on planar array ECT has been able to detect objects as far away from the sensor head as roughly the distance between the furthest apart electrodes [START_REF] Hu | Planar capacitive sensors â Ȃ Ş designs and applications[END_REF] which for a 200mm square grid-like sensor head was approximately 100mm [START_REF] Ye | Planar array 3D electrical capacitance tomography[END_REF]. However the work was vague on how distance is quantified from results and how image reconstruction quality is scored which makes performance comparisons difficult.

One possible application of planar array ECT is landmine detection in which the buried landmine is only detectable from above the ground and sensors cannot be placed all around the viewing region. Current landmine technologies, such as metal detectors, struggle to detect modern landmine technology as they contain almost no metal at all [6]. Theoretically, ECT would be able to detect any material which differed from the ground it was buried in as well as reconstructing a permittivity distribution of the different materials contained in the landmine. This makes it an attractive prospect for future landmine detection technology but in order to get it to a stage where it could realistically be used it needs to be improved in a number of ways. One of these is improving the depth detection range which is currently a big limiting factor of ECT [START_REF] Tholin-Chittenden | Planar Array Capacitive Imaging Sensor Design Optimization[END_REF]. In order to locate the landmine it is important to know the depth of the landmine and the shape of the landmine which could then also allow identification to be possible. These metrics need to be clearly defined and quantified.

A common method of image reconstruction in ECT is Tikhonov regularisation but Total Variation (TV) algorithms have also received a great deal of attention and many implementations are considered state of the art [START_REF] Xiao | A Fast Algorithm for Total Variation Image Reconstruction from Random Projections[END_REF] [9] for image reconstruction. These algorithms are used in many different applications such as medical [START_REF] Li | Medical images denoising based on total variation algorithm[END_REF] [START_REF] Lohvithee | Parameter selection in limited data cone-beam CT reconstruction using edge-preserving total variation algorithms[END_REF] and industrial [START_REF] Hosani | Imaging metallic samples using electrical capacitance tomography: forward modelling and reconstruction algorithms[END_REF] [START_REF] Chen | Image reconstruction algorithms for electrical capacitance tomography based on ROF model using new numerical techniques[END_REF] with various implementations. Among the available methods for TV minimization, splitting methods, such as proximal algorithms [START_REF] Combettes | Proximal splitting methods in signal processing[END_REF] and the split Bregman method [START_REF] Goldstein | The Split Bregman Method for L1-Regularized Problems[END_REF], [START_REF] Setzer | Split bregman algorithm, douglas-rachford splitting and frame shrinkage[END_REF], are efficient and widely used. However there is no clear winner as each application requires a different reconstruction approach with very different parameter selections.

Image reconstruction in an ill-posed problem requires optimal selection of image reconstruction parameters. For L2regularisation, the L-curve or the generalised cross validation are widely used for selecting the regularization parameter [START_REF] Hansen | The Use of the L-Curve in the Regularization of Discrete Ill-Posed Problems[END_REF], [START_REF] Abascal | Comparison of methods for optimal choice of the regularization parameter for linear electrical impedance tomography of brain function[END_REF]. Splitting algorithms have few parameters that need to be tuned. For the case of split Bregman algorithm, previous work provided some guidelines for manual parameter selection [START_REF] Goldstein | The Split Bregman Method for L1-Regularized Problems[END_REF] [19] [START_REF] Abascal | Comparison of total variation with a motion estimation based compressed sensing approach for self-gated cardiac cine MRI in small animal studies[END_REF] as well for automatic selection [START_REF] He | Fast total-variation image deconvolution with adaptive parameter estimation via split Bregman method[END_REF] for some of these parameters. Tikhonov Regularisation has also received attention on parameter selection, both automatic [START_REF] Xu | An adaptive Tikhonov regularization parameter choice method for electrical resistance tomography[END_REF] and manual, such as L-Curve method [START_REF] Hansen | The L-curve and its use in the numerical treatment of inverse problems[END_REF].

The aim of this paper is to propose a method of selecting parameters of the image reconstruction algorithms automatically and to assess it on planar array ECT data. The proposed method is applied to a 'Hybrid Tiknovov' with NOSER regularisation [START_REF] Cheney | NOSER: An algorithm for solving the inverse conductivity problem[END_REF] and split Bregman based TV methods [START_REF] Goldstein | The Split Bregman Method for L1-Regularized Problems[END_REF]. The algorithm proposed here will attempt to automate selection of multiple parameters.

Manual reconstruction parameters selection involves an expert who knows how each parameter affects the reconstruction. This is often done through a method of trial and improvement which relies on looking at the result and changing the parameters accordingly. This is usually very time consuming and requires the expert knowledge on the algorithm and application. However a method of automatic parameter selection, which can select reconstruction parameters based only on the measured data, would remove the need for the expert and thus potentially speed up reconstructions and add consistency to results. The proposed method of automatic parameter selection involves finding a set of optimal parameters using a Genetic Algorithm (GA). GAs have been successfully used for parameter tuning in several applications [START_REF] Wu | A Genetic Algorithm Approach for Selecting Tikhonov[END_REF] [START_REF] Serinagaoglu Dogrusoz | Genetic algorithm-based regularization parameter estimation for the inverse electrocardiography problem using multiple constraints[END_REF].

Previous work in electrical tomography which is investigating distance detection is often vague on quantifying how a distance was calculated [START_REF] Zhang | Near subsurface 3D imaging using planar array : EIT, ECT, MIT[END_REF]. New methods of distance detection and inclusion scoring from 3D reconstructions is proposed.

II. ELECTRICAL CAPACITANCE TOMOGRAPHY

ECT is the method of reconstructing a permittivity distribution from a set of capacitance measurements taken between different combinations of electrode pairs. The raw capacitance data measured can be used to infer something about the viewing region in front of the sensor head.

A. Forward Problem

Before image reconstruction can be performed some precalculations need to be made about the sensor head and the viewing region. This is often referred to as the forward problem and consists of modelling of the environment to understand how a change of permittivity in the viewing region will affect the capacitance measured by the electrodes on the sensor head. To model the relationship between electric potential from the electrodes and the permittivity of the viewing region a combination of Gauss's law is used with Poisson's equation and as there is no charge escaping the viewing region:

∇ • (ε∇φ) = 0 (1)
where ε is the permittivity distribution, φ is the electric potential distribution and ∇• is the divergence operator. This is the Partial Differential Equation (PDE) that can be used to calculate electric potential distribution with a fixed permittivity of the viewing area. Capacitance is proportional to the charge on a surface so this can be calculated for one electrode on the sensor head by finding the electric flux density over the surface area of the electrode which can be found by taking the surface integral:

C = 1 V ∮ Ω ε 0 (x, y, z)E(x, y, z) dω ( 2
)
where C is the inter electrode capacitance, V is the electric potential on the excitation electrode, Ω is the surface area and E is the electric field distribution. From this it can be seen that capacitance is a non-linear function of permittivity which can be linearised as follows:

∆C = J∆ε (3)
where J, the Jacobian, is essentially a conversion matrix that maps a change in permittivity into a change in capacitance values. The number of voxels in the viewing region is defined as n v and the number of unique capacitance measurements, m, is dictated by the following equation:

m = N v (N v -1) 2 (4)
where N v is the number of electrodes. This makes the Jacobian size equal to m × n v . Most commonly n >> m depending on the image resolution selected which defines the number of nodes.

B. Inverse Problem

Equation 3 shows that a permittivity is linearly related to a set of capacitance values. This is a linear ill posed problem, so regularization is needed to obtain a unique solution. Tikhonov regularisation [START_REF] Neubauer | Tikhonov regularisation for non-linear ill-posed problems: Optimal convergence rates and finite-dimensional approximation[END_REF] is the most common and simple choice of regularization.

Improvement on standard Tikhonov Regularisation can be achieved by adding NOSER regularisation [START_REF] Cheney | NOSER: An algorithm for solving the inverse conductivity problem[END_REF]. Tikhonov regularisation performs very well with ill-posed problems by reducing noise but it tends to over-smooth the output image as it assumes the solution to be smooth. The NOSER method is very good at position reconstruction but performs worse with noise which affects the solution more when the object is at larger distances. However the combination of Tikhonov and NOSER perform well. This leads to a 'Hybrid Tikhonov' which is defined by the following equation:

∆ε = J T J + α 1 Γ + α 2 diag(J T J) -1 J T ∆C (5)
where Γ, the regularisation matrix, is the identity and α 1 and α 2 are the scaling parameters of Tikhonov and NOSER respectively. The problem with this solution however is that boundaries gradients between different permittivity levels will be smooth and so they won't show sharply contrasting boundaries. A buried foreign object has very distinct boundaries with the ground around it. For this reason L-2 regularization is not an ideal prior for object detection.

III. TOTAL VARIATION ALGORITHM

Total variation methods have been shown to outerperform L2regularization methods in terms of image quality, as TV functional achieves piecewise-constant reconstructions with sharp edges [START_REF] Vogel | Computational methods for inverse problems[END_REF]. This is a reasonable assumption in ECT where target objects can be accurately represented with piecewiseconstant functions and sharp edges would improve the characterization of the object's shape and size.

The unconstrained formulation of the TV image reconstruction problem is given by

min ∆ 1 2 J∆ -∆C 2 2 + α ∇∆ 1 , (6)
where ∇∆ 1 is the TV functional, i.e. the L1-norm of the gradient of the reconstructed image, and α is a regularization parameter that can be selected using the L-curve or similar method [START_REF] Hansen | The Use of the L-Curve in the Regularization of Discrete Ill-Posed Problems[END_REF].

The TV functional is an optimal choice for many applications; however, it suffers from non-differentiability and a loss of contrast [START_REF] Burger | A guide to the TV zoo[END_REF]. A common approach is to use a smooth approximation of the TV functional [29] [31], but this can compromise effective regularization or slow down convergence. An alternative that has been proposed to solve these issues is the use of the split Bregman formulation [START_REF] Goldstein | The Split Bregman Method for L1-Regularized Problems[END_REF], which is an efficient formulation to solve L1-regularization approaches. Using this formulation, one aims at solving the constrained formulation of the TV image reconstruction problem

min ∆ ∇∆ 1 st. 1 2 J∆ -∆C 2 2 ≤ σ 2 (7)
where σ 2 accounts for noisy data. The constrained formulation 6 enjoys further benefits with respect to the unconstrained formulation 7: it does not need to estimate the regularization parameter using the L-curve and the data constraint is imposed iteratively, which has the potential to achieve better reconstruction.

In the Split Bregman formulation constraints are imposed iteratively using the Bregman iteration as explained below.

In addition, introducing auxiliary variables allows separating L2-and L1-norm functionals in such a way that they can be solved analytically in two alternating steps, avoiding nondifferentiability issues that appear in standard approaches. To allow for splitting, we include new variables, d x , d y , d z , and formulate a new problem equivalent to equation 7, as follows:

min ∆ ,d x ,d y ,d z (d x , d y ) 1 + d z 1 st. 1 2 J∆ -∆C 2 2 ≤ σ 2 , d x = ∇ x ∆ , d y = ∇ y ∆ , d z = ∇ z ∆ (8)
Using the Bregman iteration, the constrained problem 8 can be easily handled by using an equivalent unconstrained formulation for which constraints are imposed by adding a Bregman iteration b i for each constraint. Thus, equation 8 becomes

min ∆ ,d x ,d y ,d z (d x , d y ) 1 + d z 1 + µ 2 J∆ -∆C k 2 2 + λ 2 d x -∇ x ∆ -b k x 2 2 + λ 2 d y -∇ y ∆ -b k y 2 2 + λ 2 d z -∇ z ∆ -b k z 2 2 (9)
where k is the iteration number and the Bregman iterations are given by

b k+1 x = b k x + ∇ x ∆ k+1 -d k+1 x b k+1 y = b k y + ∇ y ∆ k+1 -d k+1 y b k+1 z = b k z + ∇ z ∆ k+1 -d k+1 z ∆C k+1 = ∆C k + ∆C -J∆ k+1 (10)
The Bregman iterations impose constraints iteratively by adding the error back into the respective functionals [START_REF] Osher | An Iterative Regularization Method for Total Variation-Based Image Restoration[END_REF] [15], thus making the solution of the problem 9 to converge to the solution of the constrained problem 8. The last line in equation 10 enforces the data constraint, producing a sequence of solutions such that the solution error norm and the data fidelity term decrease monotonically.

To solve 9 we note that reconstructed image and auxiliary variables are independent of each other, so equation 9 can be split into several problems, one for each variable, and be solved sequentially. The permittivity change is solved analytically by solving the following linear system:

(µJ T J + λ i=x,y,z ∇ T i ∇ i + γI)∆ = µ∆C k + λ i=x,y,z ∇ T i (d i -b i ) + γ∆ k (11)
where the last term on both the LHS and RHS is added for stability. Auxiliary variables are given by the following shrinkage formulae [START_REF] Goldstein | The Split Bregman Method for L1-Regularized Problems[END_REF]]:

d k+1 x = max(s k -1/λ, 0) ∇ x ∆ k+1 + b k x s k , ( 12
)
d k+1 y = max(s k -1/λ, 0) ∇ y ∆ k+1 + b k y s k , ( 13
)
s k = i=x,y,z |∇ i ∆ k+1 + b k i | 2 , ( 14
)
d k+1 z = max(|∇ z ∆ k+1 + b k z | -1/λ, 0) ∇ z ∆ k+1 + b k z |∇ z ∆ k+1 + b k z | . (15)

IV. EXPERIMENTAL PROCEDURE

The TV algorithm parameters, P TV , that are going to be optimised in order to produce the best result are µ, λ and γ.

The algorithm also has a Bregman iteration count, n b , which controls how many iterations are used in the reconstruction. The higher this is, the more the data constraint is imposed. Decreasing µ (weight of the data fidelity term) and increasing lambda (weight of the gradient constraints) leads to lower n b , requiring fewer iterations to converge. Optimal selection of these parameters can reduce computational time significantly [START_REF] Abascal | Fluorescence diffuse optical tomography using the split Bregman method[END_REF]. Tikhonov, equation 5, also requires input parameters, P tikhonov , but only 2, and these are α 1 and α 2 .

ECT is greatly affected by the distance of the object to the sensor [START_REF] Hu | Planar capacitive sensors â Ȃ Ş designs and applications[END_REF] which means that objects imaged with ECT will have a much more accurately reconstructed front surface (the closest surface to the ECT sensor) than the other surfaces further away. For this reason, the closest surface is used to localise the object. This surface is referred to as the front surface. The other sides of the object are likely to be reconstructed with less accuracy.

The software used in these experiments is DeTECT [START_REF] Tholin-Chittenden | Planar Array Capacitive Imaging Sensor Design Optimization[END_REF], a MATLAB based toolbox used for ECT simulation, reconstruction and visualisation. It is specifically designed for use with planar array ECT systems. It uses a finite difference method (FDM) for the forward problem modelling.

A test in this paper is defined as reconstructing a 3D permittivity distribution for a particular set of parameters and detecting the distance of the front surface of the reconstructed object. The detected distance, d d , is then compared to the expected distance, d e , and used to compute the difference, d di f f . A 2D slice is also taken from the original permittivity distribution and an inclusion is extracted from it through segmentation. This inclusion is then compared to the expected inclusion and a score is calculated to show how accurate it is. This results in 'distance difference' and 'reconstruction score' qualifiers for all tests.

All tests will use both simulated capacitance data as well as real captured capacitance data and the same Jacobian to reconstruct the permittivity. The simulated capacitance data is obtained by simulating a block with DeTECT , suspended at 3cm incrementing heights above the sensor head which is a 3 x 4, 12 electrode sensor as shown in figure 1 with external dimensions of 20cm by 20cm. The real captured capacitance data is measured with the same distances using a wooden block as the inclusion. The Jacobian is computed with DeTECT at a resolution of 21x21x21 in the x, y and z directions respectively and then used with the capacitance data to reconstruct the 3D permittivity distribution. The sensor used is the same as that referred to as 'original sensor' in previous work on planar array ECT [START_REF] Tholin-Chittenden | Planar Array Capacitive Imaging Sensor Design Optimization[END_REF].

A. Front Surface Distance Detection (FSDD)

The distance d d to the front surface is calculated in several stages. As the reconstruction is not perfect, the front surface may not be smooth which means that if the distance was measured from a single point on the surface it might not be representative of the actual distance of object. For this reason the distance detection is performed on multiple points and an average distance detected is taken from this. The algorithm 1 explains how the distance is detected and a visualisation of this can be seen in figure 2.

Algorithm

T otsu = OtsuT hreshold(ε 1D ) z = 1 while ε 1D (z) < T otsu do z = z + 1 end while ∆ε = ε 1D (z) -ε 1D (z -1) ∇ε = n-(n-1) ∆ε ∆d = ∇ε T otsu -ε 1D (z -1) d = z -1 + ∆d d cor e (x,

B. Cross-section Reconstruction Score (CSRS)

The distance d d from FSDD is only useful if the inclusion that was reconstructed is close in shape to that of the actual inclusion otherwise the detected distance could be coincidental and not representative of a good parameter set. For this reason every test also checks a cross-section of the permittivity distribution to check on the shape of the reconstructed inclusion. Algorithm 2 explains how this score is calculated.

C. Parameter Optimisation

Parameter optimisation is a hard problem. Several methods have been proposed to optimise the regularisation parameter for Tikhonov [START_REF] Vogel | Computational methods for inverse problems[END_REF]. Performance of these methods depends on the application and data dependent, so methods are generally compared for a new application [START_REF] Abascal | Comparison of methods for optimal choice of the regularization parameter for linear electrical impedance tomography of brain function[END_REF]. Automatic selection of other hyperparameters, such as those in the split Bregman formulation (they are not regularization parameters), is still an open issue, where some guidelines have been previously proposed [START_REF] Goldstein | The Split Bregman Method for L1-Regularized Problems[END_REF] [19] [START_REF] Abascal | A novel prior-and motion-based compressed sensing method for smallanimal respiratory gated CT[END_REF]. For optimising TV and Tikhonov parameters three methods were tried. 1) Brute Force: A 'brute force' method of evenly searching all combinations of parameters in a given size range was attempted. It was found however that if the number of parameter combinations was too large then the optimisation took too long and if too small then the gaps between parameter values was too large. 2) Neighbour Search: This method used a starting set of parameters and then varied them to try improving the results. If the reconstruction scored badly, the parameters step size was increased and then decreased as the reconstruction score improved thus aiming to converge on an optimal solution. The starting parameter set used was that of the optimal parameter set of the 'neighbouring' reconstruction distance. This method produced good results but not consistently enough. 3) Genetic Algorithm Optimisation (GAO): The GA was implemented using the MATLAB Optimization Toolbox [START_REF] Coleman | Matlab Optimization Toolbox[END_REF]. It uses a multi-start global search method which means that it has many start points and searches for the best parameter set regardless of parameter values.

It is attempting to find the global minima of difference between FSDD and expected distance by searching for local minima and selecting the best one. The GA is given 2 stopping criteria to prevent it over-searching. It is given a maximum number of local minima, L M max , found before it then assigns the lowest one as the solution.

It also stops if it finds a local minima lower than a chosen threshold , T di f f , as this is deemed a suitable solution. However if a solution has a CSRS lower than the threshold, T csr s , then this solution is not counted as a valid solution.

V. EXPERIMENTAL RESULTS

For training of reconstruction parameters, real capacitance data was collected at 30mm, 60mm, 90mm and 120mm as described in the experimental procedure. Simulated data was then created for the same distances. These two sets are referred to as Sample 1.

For both Sample 1 data, GAO was used to find optimal parameters at each distance, d e , of the wooden block. GAO used the settings, T di f f = 0.05d e , T csr s = 50 and L M max = 10. These values were selected as they gave the best balance between testing time and testing accuracy.

The TV algorithm used a n b value of 10 to allow for a balance between quick reconstruction (≈ 2s) and reconstruction accuracy. Given the relation between µ and n b [START_REF] Abascal | Fluorescence diffuse optical tomography using the split Bregman method[END_REF], one can fix one of these parameters and modify the other one.

Figure 4 shows a reconstruction by the TV algorithm of an inclusion at 90mm from the sensor head. In this reconstruction and all subsequent reconstructions shown the sensor is located at the top of the image as the images are mimicking a landmine detection scenario where the viewing direction would be down through the ground. The red square in the image shows the expected location of the inclusion.

A. Capacitance Measurements

The raw capacitance measurements of Sample 1 can be seen to show an interesting relationship between capacitance and distance of object. As the distance of the object increased from the sensor head, the euclidean norm of the raw capacitance measurements proportionally changed too. This can be seen in figure 5.

This Capacitance-Distance relationship allows an 'estimated' distance to be found by simply taking a capacitance measurement, before any reconstruction is performed. Using this estimated distance, appropriate parameters can be chosen by using the Distance-Parameter relation found through Parameter Optimisation.

This relationship is very specific to this experimental setup, a higher permittivity object observed at different distances in a lower permittivity surrounding. However for other situations the relationship can be found through experimental calibration in that environment which will give a unique Capacitance-Distance relationship.

B. Parameter Optimisation Comparison

The GAO method found 'optimal' parameters for both TV and Tikhonov reconstructions for each distance tested. This data now forms Distance-Parameter relationship curves which can be used for Automatic Parameter Selection.

A brief look at the individual parameters found by the optimisation showed that the TV parameters were a lot less affected by their individual magnitudes and more affected by the ratio to each other. For example using parameters of 1, 1 and 1 for µ, γ and λ respectively was almost identical to using 1e -10 , 1e -10 and 1e -10 respectively. However for Tikhonov the parameters magnitude was far more important as they effectively act as weights to decide how much regularisation is applied. Therefore they are not as strongly linked to each other but more to their own values.

C. Distance Detection Comparison

The TV algorithm was much more consistent when detecting distance than Tikhonov which can be seen in figure 6. The TV reconstructions of both real and simulated Sample 1 data were not accurate beyond 60mm but the error was consistent. If the distance detection error is consistent it means that a scaling factor could be used to get more accurate detection measurements. In these results when the inclusion was at 120mm the detected distance was ≈80mm. Therefore in practice a detected 80mm reconstruction could be scaled to give a 120mm reading as an actual 80mm inclusion would result in a ≈70mm reading as shown by figure 6.

The Tikhonov results were far more inconsistent. It performed poorly at 120mm and was not consistent with results at 90mm, therefore the same scaling theory used for TV would not work.

Both algorithms performed extremely well at 30mm and 60mm when optimised with real data but with simulated data the results are not so good.

Overall it shows that both TV and Tikhonov show an ability to detect up to 90mm with limited error and promise of being able to potentially detect up to 120mm. This not only proves the ability of the algorithms and DeTECT software but also the hardware used to collect the data [START_REF] Tholin-Chittenden | Planar Array Capacitive Imaging Sensor Design Optimization[END_REF].

D. Automatic Parameter Selection

Automatic Parameter Selection (APS) uses the Parameter-Distance relationship curves for both TV and Tikhonov combined with the Capacitance-Distance relationship curve, to automatically select the optimal parameter set for a given capacitance reading. Algorithm 3 describes how this is performed.

In order to assess the ability of algorithm 3, new capacitance data sets, Sample 2, were collected with the same setup as before, but with the inclusions at varying distances and with training data. For this reason interpolated parameters are used instead of exact optimal parameters. Using the exact optimal parameters found through training on Sample 1 would be expected to get the same results. If interpolated parameters from algorithm 3 were used, then this should result in a parameter set which is yet untested, and if this is able to reconstruct the inclusion accurately, then this helps to support a relationship between parameters and distance.

Figure 8 shows the detected distances, d d , using parameter sets from algorithm 3 as well as the CSRSs of these reconstructions. Looking at detected distance, d d shows that all the optimal parameters sets perform very similarly. There appears to be 2 distinct regions on the graph where the behaviour differs. The first region is up to 60mm which is roughly the typical burial depths of landmines [START_REF] Keeley | Understanding landmines and mine action[END_REF]. The reconstruction algorithms behave similarly and are on average accurate to within 7.2mm. Tikhonov reconstruction was able to detect objects with an average error of 5.7mm between distances 30mm -60mm and TV reconstruction had an average error of 8.7mm. Above 60mm is quite deep for a landmine and here there seems to be an constant shift in detected distance from expected distance. If this is consistent in future experiments it could mean that the values could be scaled such that for example a detected distance of 100mm actually represents an object at 120mm. The percentage error from expected distance to detected distance is quite consistent at ≈30%.

Using APS both algorithms are still able to produce similar results to the Sample 1 results 6, and Tikhonov is even able to reconstruct up to 130mm with only ≈4% error. It could be that the new data sets, in Sample 2, are better quality with less noise and fewer background signals. But the consistency of the distance detection for both reconstruction methods and optimisation parameter sets shows that for this application there is some relationship between parameter values and front surface distance of an object.

Both TV and Tikhonov have similar FSDDs and therefore it could be a one-off result meaning that Tikhonov might perform better in some tests and TV in other tests. However when comparing CSRSs, TV performs much better on average. In figure 8 the TV CSRSs are far better at almost all distances. This means that TV is able to reconstruct the inclusions with a much better overall shape, not just the front surface and therefore the TV reconstructions are more reliable as the inclusion can be located with more certainty.

Figure 9 shows reconstructions of Sample 2 data using optimal parameters trained from real data. The red lines indicate the expected location of the top surface. Both reconstruction methods produce similar images and none of them perfectly find the top surface.

The TV reconstructions have slightly better defined edges than the Tikhonov reconstructions. This is to be expected due to the nature of the TV algorithm. It means that the overall object edges can be defined with more certainty whereas with Tikhonov the exact location of the inclusion boundary is estimated using Otsu thresholding select edges from the blurred inclusion.

Landmine detection is far safer if objects can be accurately reconstructed and located. The TV reconstructions show the inclusion location more accurately and the shape can be assumed with more certainty than Tikhonov reconstructions.

The aim of automatic parameter selection was not to produce the best images but to remove the need for an expert to select the reconstruction parameters. The images produced show that it is possible to select parameters automatically. Through some manual parameter selection during experimentation is was observed that changing the parameters can drastically change the appearance of the reconstructed image. Thus, automatic parameter selection can lead to more consistent distance detection.

VI. CONCLUSION

In order to use ECT for landmine detection, distance detection of planar array ECT needs to be consistent and reliable. Planar array ECT suffers from poor reconstruction of inclusions further away from the sensor. A way of improving this is through optimisation of the image reconstruction parameters. However selecting appropriate parameters is often done by The development of the APS method has removed the need for expert calibration of image reconstruction parameters for TV and Tikhonov. The initial training of the APS method showed that there was not much difference between training with simulated data or real data of Sample 1 and both reconstruction algorithms performed similarly. However when Sample 2 data was used to test the automatic parameter selection algorithm Tikhonov was more consistent in distance detection than TV although TV was still producing good results at smaller distances. These results could be explained by the fact that selecting Tikhonov parameters is more robust due to the noniterative nature and fewer parameters of the algorithm whereas TV relies more on iteration count and the larger number of parameters. It could be that better TV optimal parameters can be found using different training methods instead the GA used in this work.

Fig. 1 :

 1 Fig. 1: Sensor head layout and experimental setup for training data (sample 1).

Fig. 2 :

 2 Fig. 2: Distance detection method showing interpolation between voxel levels around the Otsu Threshold value. GREEN -Smaller than threshold, RED -Bigger than threshold.

Fig. 3 :

 3 Fig.3: 2D slice taken from centre of 'core' and then the inclusion cut-out based on threshold values from the FSDD method.

Fig. 4 :

 4 Fig. 4: TV reconstruction of real data captured with object at 90mm from sensor head. Red box shows expected location of object.

Fig. 5 :

 5 Fig. 5: Capacitance-Distance relationship curve.

Fig. 6 :

 6 Fig. 6: Distance detected from Sample 1 reconstructions using optimal parameters.

Algorithm 3

 3 Get optimal parameter set, P for current capacitance reading, C Input: Capacitance Measurement, C Output: Optimal Parameter Set, P 1: Load capacitance-distance relationship, distances d r , norm capacitances C r 2: n = 1 3: C nor m = C 2 4: while C r (n) > C nor m do 5: n = n + 1 6: end while 7: ∆C = C r (n) -C r (n -1) 8: ∇C = d r (n)-d r (n-1) ∆C 9: ∆d = ∇C C nor m -C r (n -1) 10: d = d r (n -1) + ∆d d is the 'estimated' distance 11: Load optimal parameter sets, P combined , and associated distances, d p 12: m = 1 13: for all p ∈ P combined do 14: n = 1 15: while d p (n) < d do 16: n = n + 1 17: end while 18: ∆d = d(n)d(n -1) 19: ∇d = n-(n-1)

∆d 20 :

 20 ∆p = ∇d dd(n -1) 21: P(m) = p(n -1) + ∆p 22: m = m + 1 23: end for

Fig. 7 :

 7 Fig. 7: Experimental setup of Sample 2 gathered for testing of Automatic Parameter Selection.

Fig. 8 :

 8 Fig. 8: Detected distances d d against the expected distance d e for TV and Tikhonov reconstructions on Sample 2 data.

Fig. 9 :

 9 Fig. 9: Detected distances d d against the expected distance d e for TV and Tikhonov reconstructions on Sample 2 data.

1

  Calculate front surface distance, d d

	Input: Permittivity distribution, ε
	Output: Detected distance, d d
	while ε MedianFilter(ε) do
	ε = MedianFilter(ε)
	end while
	Extract 3x3 core, ε cor e , from centre of ε
	Setup empty cores distances variable, d cor e
	for x = [1 2 3] do
	for y = [1 2 3] do
	ε 1D = ε cor e (x, y)

  Algorithm 2 Calculate cross section correlation, score csr s Input: Permittivity distribution, ε Output: Cross section correlation score, score csr s Extract 3x3 core, ε core , from centre of ε Setup empty ε aver agefor x = [1 2 3] do ε 2D = ε(x) T otsu = OtsuT hreshold(ε 2D ) ε 2D (ε 2D < T otsu ) =0 ε aver age = ε aver age + ε 2D end for ε aver age (ε aver age 0) = 1 Load expected inclusion ε ex pected Initialise correct voxel count, cv = 0 Initialise incorrect voxel count, iv = 0 Calculate total area of cross section, t = 21 * 21 for x = 1 to 21 do for y = 1 to 21 do if ε aver age (x, y) = ε ex pected (x, y) then

	cv = cv + 1
	else
	iv = iv + 1
	end if
	end for
	end for
	score csr s = cv-iv t

The main conclusion of this paper is the ability to automatically select parameters and produce results which are consistent and quick without any expert calibration. Also for future comparisons to be made with this work, the clear explanation of the FSDD and CSRS scoring has created quantifiers for accurate distance scoring whilst making sure the reconstruction is showing expected inclusion shape.

VII. ACKNOWLEDGEMENT

This work is sponsored by Find A Better Way, a landmine detection and clearance charity and the University of Bath Alumni Fund. JFPJ Abascal has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 701915.