, Explore and develop new mathematical techniques of Fourier irregular sampling for SMOS-HR. Demonstrate that the obtained configurations can be calibrated

, Explore two possibilities: a local model (from consecutive snapshots) and a global model (from clustered observations along a year on different, Earth biomes and seasons

, Perform a joint-reconstruction of the visibilities which takes into account the low dimension of the BT according to the incidence angle

, Correct the spatial aliasing in the BT map using a consecutive multi-snapshot strategy

, Reduce the influence of radiometric noise by using the low-dimension model of the BT signal

, We have an ongoing contractual collaboration with the CESBIO team at CNES. The details of this ongoing research and the first results are given in chapter 4. In particular, the research plan for the PhD of Max Dunitz is the following: ? First year 1. State of the art of the several concepts of interferometric telescopes and studies carried out for the optimization and fusion of the acquired data. The literature will be particularly analyzed around the existing SMOS satellite and telescopes (VAtacama Large Millimeter Array -ALMA, Max Dunitz is to start in October 2019 a PhD thesis supervised by Jean-Michel Morel and myself 2

, Study of the literature on the irregular sampling of the Fourier transform and methods of extension or completion of the spectrum (in particular extensions of the total variation method [137]). The PhD student will confirm the experiments proving that an optimized irregular configuration of antennas is better compared to a regular configuration

, Research on the best non-convex optimization algorithms to optimize an irregular configuration

C. Icip, . Tip, . Jmiv, . Siims, . Josa et al., Online publications and demos on IPOL allowing real-time application, and also conventional conferences and journals of image processing and mathematics applied to imaging and spatial imagery

, Mathematical analysis of the folded satellite in regular configuration: demonstration on simplified planar models that the unfolding is possible. Extension to a spherical acquisition model

, Study and optimization of separable irregular configurations, allowing to obtain a sufficient sampling in the direction orthogonal to the trace of the satellite

, Implementation of a general procedure for the calibration of antennas in irregular configuration

, Creation of a first simplified simulator showing a signal-to-noise ratio increase in the evaluation tables, and the proof of concept the proposed unfolding technique

, Parametric modeling of the angular BT and implementation of the general inversion algorithm

, Refined modeling of the BT using the vast capital of the ten years of SMOS acquisitions

, Expected impact and perspectives: a proof of usefulness of mathematics does not hurt: an advanced mathematical analysis will allow us to design a new type of satellite which is more economical and more efficient. The developments on the irregular Fourier analysis [54] and our analysis of the inversion on Earth 3 may be applicable to any interferometric instrument, vol.1

, 2. they establish a correspondence between the value of the color channel and its noise variance

, 3. they try to detect discrepancies of this correspondence, namely to find pairs of regions with similar color and different noise models

, Algorithm 11 freq2key: convert a frequency/baseline into a dictionary key 1: Input ?: frequency/baseline Output : dictionary's key of the given sampling frequency/baseline

, Algorithm 12 getBaselinesDict: given a list of antennas' positions, compute their baselines and multiplicities, vol.1

, Algorithm 13 gridBaselines: create a sparse matrix which grids the given baselines 1: Input D u : Dictionary that maps all the baselines of the instrument with the corresponding multiplicities. Input ? = 1: gridding step Output J : matrix that contains the gridded baselines, vol.2

, J

. ?-d-u, D u Set multiplicities, with DC freq. centered 5: return J Algorithm 16 separateAntennas: separate antennas which are too close 1: Input a: antenna's coordinates in ? units Outputâ: list of fused antennas 2:â ? a 3: ? ? distanceMatrix(â) For example, function spatial

, ? argmin i =j ?[i, j] 9: end while 10: returnâ Algorithm 17 isPositiveFreq: determine if the given frequency is "positive". 1: Input ?: sampling frequency Output: Boolean indicating if the given frequency is "positive

, Algorithm 18 absFreq: get the corresponding "positive" frequency 1: Input ?: sampling frequency Output: the corresponding "positive" frequency

, Algorithm 19 getBaselineDiscr: thin-grid the given baseline from the position of two antennas 1: Input a: list of antennas' positions Input i: index of the 1st antenna Input j: index of the 2nd antenna Output: thin-gridded baseline

. G-?-zeros,

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