ift.rst 6.77 KB
 Martin Reinecke committed Jan 23, 2018 1 2 3 4 5 6 7 IFT -- Information Field Theory =============================== Theoretical Background ---------------------- Torsten Ensslin committed Jan 04, 2019 8 Information Field Theory _ [1-5]_ (IFT) is information theory, the logic of reasoning under uncertainty, applied to fields. A field can be any quantity defined over some space, e.g. the air temperature over Europe, the magnetic field strength in the Milky Way, or the matter density in the Universe. IFT describes how data and knowledge can be used to infer field properties. Mathematically it is a statistical field theory and exploits many of the tools developed for such. Practically, it is a framework for signal processing and image reconstruction. Martin Reinecke committed Jan 23, 2018 9 10 11 12 13 14 15 IFT is fully Bayesian. How else could infinitely many field degrees of freedom be constrained by finite data? It can be used without the knowledge of Feynman diagrams. There is a full toolbox of methods. It reproduces many known well working algorithms. This should be reassuring. And, there were certainly previous works in a similar spirit. Anyhow, in many cases IFT provides novel rigorous ways to extract information from data. .. tip:: An *in-a-nutshell introduction to information field theory* can be found in [2]_. Torsten Ensslin committed Jan 04, 2019 16 .. [1] T. Ensslin (2019), "Information theory for fields", accepted by Annalen der Physik; arXiv:1804.03350 _ Martin Reinecke committed Jan 23, 2018 17 Torsten Ensslin committed Jan 04, 2019 18 .. [2] Wikipedia contributors (2018), "Information field theory", Wikipedia, The Free Encyclopedia. _ Martin Reinecke committed Jan 23, 2018 19 Torsten Ensslin committed Jan 04, 2019 20 21 22 23 24 .. [3] T. Ensslin (2014), "Astrophysical data analysis with information field theory", AIP Conference Proceedings, Volume 1636, Issue 1, p.49; arXiv:1405.7701 _ .. [4] T. Ensslin (2013), "Information field theory", proceedings of MaxEnt 2012 -- the 32nd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering; AIP Conference Proceedings, Volume 1553, Issue 1, p.184; arXiv:1301.2556 _ .. [5] T. Ensslin et al. (2009), "Information field theory for cosmological perturbation reconstruction and nonlinear signal analysis", PhysRevD.80.105005, 09/2009; arXiv:0806.3474 _ Martin Reinecke committed Jan 23, 2018 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 Discretized continuum --------------------- The representation of fields that are mathematically defined on a continuous space in a finite computer environment is a common necessity. The goal hereby is to preserve the continuum limit in the calculus in order to ensure a resolution independent discretization. +-----------------------------+-----------------------------+ | .. image:: images/42vs6.png | .. image:: images/42vs9.png | | :width: 100 % | :width: 100 % | +-----------------------------+-----------------------------+ Any partition of the continuous position space :math:\Omega (with volume :math:V) into a set of :math:Q disjoint, proper subsets :math:\Omega_q (with volumes :math:V_q) defines a pixelization, .. math:: \Omega &\quad=\quad \dot{\bigcup_q} \; \Omega_q \qquad \mathrm{with} \qquad q \in \{1,\dots,Q\} \subset \mathbb{N} , \\ V &\quad=\quad \int_\Omega \mathrm{d}x \quad=\quad \sum_{q=1}^Q \int_{\Omega_q} \mathrm{d}x \quad=\quad \sum_{q=1}^Q V_q . Here the number :math:Q characterizes the resolution of the pixelization and the continuum limit is described by :math:Q \rightarrow \infty and :math:V_q \rightarrow 0 for all :math:q \in \{1,\dots,Q\} simultaneously. Moreover, the above equation defines a discretization of continuous integrals, :math:\int_\Omega \mathrm{d}x \mapsto \sum_q V_q. Any valid discretization scheme for a field :math:{s} can be described by a mapping, .. math:: s(x \in \Omega_q) \quad\mapsto\quad s_q \quad=\quad \int_{\Omega_q} \mathrm{d}x \; w_q(x) \; s(x) , if the weighting function :math:w_q(x) is chosen appropriately. In order for the discretized version of the field to converge to the actual field in the continuum limit, the weighting functions need to be normalized in each subset; i.e., :math:\forall q: \int_{\Omega_q} \mathrm{d}x \; w_q(x) = 1. Choosing such a weighting function that is constant with respect to :math:x yields .. math:: s_q = \frac{\int_{\Omega_q} \mathrm{d}x \; s(x)}{\int_{\Omega_q} \mathrm{d}x} = \left< s(x) \right>_{\Omega_q} , which corresponds to a discretization of the field by spatial averaging. Another common and equally valid choice is :math:w_q(x) = \delta(x-x_q), which distinguishes some position :math:x_q \in \Omega_q, and evaluates the continuous field at this position, .. math:: s_q \quad=\quad \int_{\Omega_q} \mathrm{d}x \; \delta(x-x_q) \; s(x) \quad=\quad s(x_q) . In practice, one often makes use of the spatially averaged pixel position, :math:x_q = \left< x \right>_{\Omega_q}. If the resolution is high enough to resolve all features of the signal field :math:{s}, both of these discretization schemes approximate each other, :math:\left< s(x) \right>_{\Omega_q} \approx s(\left< x \right>_{\Omega_q}), since they approximate the continuum limit by construction. (The approximation of :math:\left< s(x) \right>_{\Omega_q} \approx s(x_q \in \Omega_q) marks a resolution threshold beyond which further refinement of the discretization reveals no new features; i.e., no new information content of the field :math:{s}.) All operations involving position integrals can be normalized in accordance with the above definitions. For example, the scalar product between two fields :math:{s} and :math:{u} is defined as .. math:: {s}^\dagger {u} \quad=\quad \int_\Omega \mathrm{d}x \; s^*(x) \; u(x) \quad\approx\quad \sum_{q=1}^Q V_q^{\phantom{*}} \; s_q^* \; u_q^{\phantom{*}} , where :math:\dagger denotes adjunction and :math:* complex conjugation. Since the above approximation becomes an equality in the continuum limit, the scalar product is independent of the pixelization scheme and resolution, if the latter is sufficiently high. The above line of argumentation analogously applies to the discretization of operators. For a linear operator :math:{A} acting on some field :math:{s} as :math:{A} {s} = \int_\Omega \mathrm{d}y \; A(x,y) \; s(y)`, a matrix representation discretized with constant weighting functions is given by .. math:: A(x \in \Omega_p, y \in \Omega_q) \quad\mapsto\quad A_{pq} \quad=\quad \frac{\iint_{\Omega_p \Omega_q} \mathrm{d}x \, \mathrm{d}y \; A(x,y)}{\iint_{\Omega_p \Omega_q} \mathrm{d}x \, \mathrm{d}y} \quad=\quad \big< \big< A(x,y) \big>_{\Omega_p} \big>_{\Omega_q} . The proper discretization of spaces, fields, and operators, as well as the normalization of position integrals, is essential for the conservation of the continuum limit. Their consistent implementation in NIFTY allows a pixelization independent coding of algorithms.