diff --git a/docs/src/background/bayesian_reconstruction.ipynb b/docs/src/background/bayesian_reconstruction.ipynb deleted file mode 100644 index 02cce5a..0000000 --- a/docs/src/background/bayesian_reconstruction.ipynb +++ /dev/null @@ -1,26 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "# Bayesian reconstruction" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [] - } - ], - "metadata": { - "language_info": { - "name": "python" - }, - "orig_nbformat": 4 - }, - "nbformat": 4, - "nbformat_minor": 2 -} diff --git a/docs/src/background/spectral_line_formation.ipynb b/docs/src/background/forward_problem.ipynb similarity index 66% rename from docs/src/background/spectral_line_formation.ipynb rename to docs/src/background/forward_problem.ipynb index 60edb93..1d72058 100644 --- a/docs/src/background/spectral_line_formation.ipynb +++ b/docs/src/background/forward_problem.ipynb @@ -4,15 +4,39 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "# Spectral line formation" + "# Forward problem\n", + "---\n", + "\n", + "The forward problem describes how the physical properties of an (astrophysical) medium give rise to a spectral line observation.\n", + "We collectively represent the physical properties of an object by a model vector, $\\boldsymbol{m}$, and we represent an observation by a vector, $\\boldsymbol{o}$.\n", + "Hence, the forward problem can be thought of as a function, $f$, that maps a model vector, $\\boldsymbol{m}$, to the corresponding observation vector,\n", + "\\begin{equation*}\n", + " \\boldsymbol{o} \\ = \\ f\\left( \\boldsymbol{m} \\right) .\n", + "\\end{equation*}\n", + "We distinguish two stages in the forward function, such that, $f\\left( \\boldsymbol{m} \\right) \\ = \\ f_{2}\\left( f_{1}\\left( \\boldsymbol{m} \\right) \\right)$.\n", + "The first part, $f_{1}$, describes the spectral line formation, i.e.\\ the radiative processes that determine the amount of electromagnetic radiation that emanates from the medium and travels towards our detectors.\n", + "This is described [here](##Spectral_line_formation).\n", + "The second part, $f_{2}$, describes the observation and includes the instrumentation effects on the observed signal. This is described in [here](##Observation)." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "vscode": { + "languageId": "plaintext" + } + }, + "source": [ + "## Spectral line formation\n", + "---" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "## Physical model\n", - "\n", + "### Physical model\n", + "---\n", "In pomme, we use Cartesian coordinates, $\\boldsymbol{x}=(x,y,z)$, and assume the line of sight to be along the positive $z$-axis, such that the plane of the sky corresponds to the $(x,y)$-plane.\n", "Furthermore, we will assume that the observer is located at $z=0$, and that the model box has a depth, $L$, along the $z$-axis.\n", "Then, in the absence of radiation scattering, considering a single ray through the model (resulting in a single pixel in the observed image), the observed intensity at $\\boldsymbol{x}=(x,y,0)$, at frequency, $\\nu$, reads,\n", @@ -30,7 +54,14 @@ "\\tau_{\\text{obs}}(\\nu; x, y, z)\n", "\\ \\equiv \\\n", "\\int_{0}^{z} \\text{d}z' \\ \\chi\\big( \\nu_{\\text{com}}(\\nu; \\, x, y, z'); \\, x, y, z' \\big) .\n", - "\\end{equation*}\n", + "\\end{equation*}" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", "Since we look along the $z$-axis, the non-relativistic Doppler-shifted frequency in the co-moving frame is given by,\n", "\\begin{equation*}\n", " \\nu_{\\text{com}}(\\nu, \\boldsymbol{x}) \\ = \\ \\left( 1 + \\frac{v_{z}(\\boldsymbol{x})}{c} \\right) \\nu ,\n", @@ -92,14 +123,16 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "## Numerical implementation" + "### Numerical implementation\n", + "---" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "### Line optical depth\n", + "#### Line optical depth\n", + "---\n", "Velocity gradients play a key role in line radiative transfer. Since spectral lines are narrowly peaked in frequency space, they are very sensitive to Doppler shifts, and thus motion (gradients), along the line-of-sight.\n", "Therefore, when numerically solving a line transfer problem, it is key to properly trace the velocity (gradient) along the line-of-sight. \n", "Since we assume to know the line profile function analytically, we can take care of this sharp frequency dependence by resolving its dependence analytically." @@ -158,7 +191,8 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "### Line radiative transfer" + "#### Line radiative transfer\n", + "---" ] }, { @@ -204,6 +238,67 @@ "The implementation of this intensity increment can be found in [src/pomme/forward.py](https://github.com/Magritte-code/pomme/blob/main/src/pomme/forward.py). For the same reaseon as with the line optical depth, we compute both cases for all values, to minimise the use of masks, and we add a small number ($10^{-30}$) to the denominator to avoid NaNs in gradients." ] }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Observation\n", + "---\n", + "In the previous section, we presented a theoretical model for spectral line formation.\n", + "In practice, however, spectral line observations are affected by instrumentation effects, such as binning and noise, the particular form of which highly depends on the way the object is observed.\n", + "\n", + "High spatial and spectral resolution observations are typically obtained using interferometry, for which the situation is even more complicated, since there are spatial scale-dependent effects which cause the smallest and largest structures to be unresolved in the resulting images.\n", + "These effects must carefully be taken into account, as they (in part) determine which information about the spatial distribution of the physical properties is encoded in the observations and which is not.\n", + "At this point, we should clarify what we mean with interferometric data and distinguish between two types.\n", + "The first type are the visibilities, i.e. the correlations between the coherent electromagnetic waves, measured in several frequency bins for the different pairs of antennas in the array.\n", + "These visibilities are often further processed into images that map the actual brightness in the plane of the sky for each frequency bin.\n", + "This is what we distinguish as the second type of interferometric data.\n", + "The conversion of the sky brightness maps from the visibilities can be achieved, for instance, with the *clean* algorithm [(Hogbom 1974)](https://ui.adsabs.harvard.edu/abs/1974A%26AS...15..417H/abstract), as implemented, for instance, in [CASA](https://casa.nrao.edu).\n", + "This conversion is in itself already an inverse problem.\n", + "Therefore, when reconstructing a model from interferometric observations, it makes sense to start directly from the visibilities.\n", + "However, since visibilities are more difficult to interpret than sky brightness maps, we consider both options.\n", + "\n", + "\n", + "The conversion of the observed intensity in the plane of the sky, $I_{\\text{obs}}(\\nu; x, y)$, to visibilities, $V(\\nu;u,v)$, can be expressed as,\n", + "\\begin{equation*}\n", + " V(\\nu; u, v) \n", + " \\ = \\\n", + " \\frac{1}{d^{2}}\n", + " \\int_{-\\infty}^{+\\infty} \\text{d}x\n", + " \\int_{-\\infty}^{+\\infty} \\text{d}y \\\n", + " \\hat{A}(x,y) \\, I_{\\text{obs}}(\\nu; x, y) \\exp\\big( -2\\pi i \\left( u x + v y \\right) \\big)\n", + "\\end{equation*}\n", + "in which $\\hat{A}(x,y) \\equiv A(x,y) / A_{0}$, is the antenna response function, normalised by, $A_{0}$, the response at the centre of the beam, $d$, is the distance between the object and the observer, and $u$ and $v$ are the components of the projected distance vectors, often referred to as baselines, between pairs of antennas in the array.\n", + "Since there are only a finite number of antennas and thus a finite number of antenna pairs, the visibilities are only known at a finite number of baseline samples, $\\{(u_{d},v_{d})\\}_{d=1}^{N_{d}}$.\n", + "This sampling, which depends on the particular antenna configuration, causes a loss of information about the spatial distribution of the source in an intricate way.\n", + "We can see that the transformation of the intensity in the plane of the sky into visibilities is essentially a 2D Fourier transform.\n", + "Although the Fourier transform is invertible, the fact that we can only sample the visibilities for a finite number of baselines, makes that the combined transformation from a sky brightness map to a finite set of visibilities is not invertible.\n", + "\n", + "In principle, we could use the exact pipeline to model the instrumentation effects in our forward model, for instance, using [CASA](https://casa.nrao.edu).\n", + "However, later, in our reconstruction algorithm,\n", + "we want the implementation of the forward function, $f$, to be automatically differentiable using the *autograd* functionality in [PyTorch](https://pytorch.org).\n", + "Therefore, we implemented this map ourselves in [PyTorch](https://pytorch.org) using a fast Fourier transform (FFT), following the [Galario](https://github.com/mtazzari/galario) package for visibility modelling [(Tazzari et al. 2018)](https://ui.adsabs.harvard.edu/abs/2018MNRAS.476.4527T/abstract).\n", + "\n", + "\n", + "Alternatively, one may want to reconstruct 3D physical models from previously obtained sky brightness maps.\n", + "In that case, it is important to consider the beam, i.e.\\ a kernel that models the spatial spread of the intensity in the plane of the sky, and the antenna response function.\n", + "\\begin{equation*}\n", + " \\tilde{I}_{\\text{obs}}(\\nu; x, y)\n", + " \\ = \\\n", + " \\frac{1}{d^{2}}\n", + " \\int_{-\\infty}^{+\\infty} \\text{d}x'\n", + " \\int_{-\\infty}^{+\\infty} \\text{d}y' \\\n", + " \\hat{A}(x,y) \\,\n", + " I_{\\text{obs}}(\\nu; x', y') \\,\n", + " B(x-x',y-y') ,\n", + "\\end{equation*}\n", + "in which $B(x, y)$ is the beam kernel, which we assume to be a 2D Gaussian, centred around $(0,0)$, and, $d$, is again the distance between the object and the observer.\n", + "\n", + "It should be emphasised that in both processes of line formation and observation information is lost.\n", + "Mathematically, this implies that the function, $f$, is not invertible.\n", + "In the next section, we describe how, with a probabilistic approach, we can circumvent this problem and obtain a probabilistic inverse." + ] + }, { "cell_type": "markdown", "metadata": {}, diff --git a/docs/src/background/index.rst b/docs/src/background/index.rst index 28fe5a1..3f649c1 100644 --- a/docs/src/background/index.rst +++ b/docs/src/background/index.rst @@ -2,10 +2,12 @@ Background ########## Some background on the physics of spectral line observations, the Bayesian reconstruction algorithm, and their numerical implementation. +This is all based `this paper`_. +Please cite `the paper`_ if you use any of this information. .. toctree:: - :maxdepth: 2 + :maxdepth: 4 :caption: Contents: - spectral_line_formation - bayesian_reconstruction \ No newline at end of file + forward_model + inverse_model \ No newline at end of file diff --git a/docs/src/background/inverse_problem.ipynb b/docs/src/background/inverse_problem.ipynb new file mode 100644 index 0000000..3545285 --- /dev/null +++ b/docs/src/background/inverse_problem.ipynb @@ -0,0 +1,283 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Inverse problem\n", + "---\n", + "The entire forward problem of spectral line formation and observation, as described above, can also be formulated from a probabilistic point of view, as determining the probability distribution, $p(\\boldsymbol{o} \\, | \\, \\boldsymbol{m})$, over all possible observations $\\boldsymbol{o}$, given a model $\\boldsymbol{m}$.\n", + "Classical numerical methods typically only consider a single solution, $\\boldsymbol{o} = f(\\boldsymbol{m})$, which corresponds to a Dirac delta distribution, $p(\\boldsymbol{o} \\, | \\, \\boldsymbol{m}) = \\delta\\big(\\boldsymbol{o} - f(\\boldsymbol{m})\\big)$.\n", + "With probabilistic numerical methods, however, the entire probability distribution can be obtained (see e.g. [De Ceuster et al. 2023](https://ui.adsabs.harvard.edu/abs/2023MNRAS.518.5536D/abstract)).\n", + "This probabilistic approach is key, for instance, to quantify uncertainties in the forward problem, but can also help to solve the (classically non-invertible) inverse problem.\n", + "Although we might not be able to invert the forward function, $f$, we can always determine the probability of the inverse situation, $p\\left(\\boldsymbol{m} \\, | \\, \\boldsymbol{o}\\right)$, with Bayes' rule,\n", + "\\begin{equation*}\n", + " p\\left(\n", + " \\boldsymbol{m} \\, | \\, \\boldsymbol{o}\n", + " \\right)\n", + " \\ = \\\n", + " \\frac{\n", + " p\\left(\\boldsymbol{o} \\, | \\, \\boldsymbol{m}\\right)\n", + " p\\left( \\boldsymbol{m} \\right)\n", + " }\n", + " {\n", + " p\\left( \\boldsymbol{o} \\right)\n", + " } .\n", + "\\end{equation*}\n", + "This allows us to determine the probability distribution over possible models, $\\boldsymbol{m}$, corresponding to an observation, $\\boldsymbol{o}$.\n", + "Since the denominator does not depend on the model, $\\boldsymbol{m}$, we can treat it as a mere normalisation constant and only concentrate on the numerator.\n", + "Here, we find the likelihood, $p\\left(\\boldsymbol{o} \\, | \\, \\boldsymbol{m}\\right)$, which is related to the forward problem, and the prior, $p(\\boldsymbol{m})$, which encodes our assumptions about the model, prior to the observation. \n", + "Just like the forward model, $f$, can be viewed as the maximum of the likelihood, $p\\left(\\boldsymbol{o} \\, | \\, \\boldsymbol{m}\\right)$, the inverse, i.e.\\ a reconstruction of a model based on an observation, can be viewed as the maximum of the posterior, $p\\left(\\boldsymbol{m} \\, | \\, \\boldsymbol{o}\\right)$.\n", + "Determining this maximum is equivalent to minimising the negative logarithm of the posterior,\n", + "\\begin{equation*}\n", + " -\\log p\\left(\n", + " \\boldsymbol{m} \\, | \\, \\boldsymbol{o}\n", + " \\right)\n", + " \\ = \\\n", + " -\\log p\\left(\\boldsymbol{o} \\, | \\, \\boldsymbol{m}\\right)\n", + " \\ - \\\n", + " \\log p\\left( \\boldsymbol{m} \\right)\n", + " \\ + \\\n", + " \\log p\\left( \\boldsymbol{o} \\right) .\n", + "\\end{equation*}\n", + "Since we want to minimise this over different models, $\\boldsymbol{m}$, but for a fixed observation, $\\boldsymbol{o}$, the last term will be constant and thus can be neglected.\n", + "In this optimisation problem, we distinguish three types of objectives or loss functions, i.e. the functions we aim to minimise,\n", + "\\begin{align*}\n", + " p\\left(\\boldsymbol{m} \\, | \\, \\boldsymbol{o}\\right)\n", + " \\ &\\equiv \\\n", + " \\exp\n", + " \\big(\n", + " -\\mathcal{L}_{\\text{tot}}\\left(\\boldsymbol{m}, \\boldsymbol{o} \\right)\n", + " \\big) , \\\\\n", + " p\\left(\\boldsymbol{o} \\, | \\, \\boldsymbol{m}\\right)\n", + " \\ &\\equiv \\\n", + " \\exp\n", + " \\big(\n", + " -\\mathcal{L}_{\\text{rep}}\\left(f(\\boldsymbol{m}), \\boldsymbol{o} \\right)\n", + " \\big) , \\\\\n", + " p\\left(\\boldsymbol{m}\\right)\n", + " \\ &\\equiv \\\n", + " \\exp\n", + " \\big(\n", + " -\\mathcal{L}_{\\text{reg}}\\left(\\boldsymbol{m} \\right)\n", + " \\big) ,\n", + "\\end{align*}\n", + "in which, $\\mathcal{L}_{\\text{rep}}$ is the reproduction loss, $\\mathcal{L}_{\\text{reg}}$ is the regularisation loss, and, $\\mathcal{L}_{\\text{tot}}$, is the total loss.\n", + "Each of these loss functions quantifies the deviation from an objective.\n", + "Hence, neglecting the last term, this can be written as,\n", + "\\begin{equation}\n", + " \\mathcal{L}_{\\text{tot}}\\left( \\boldsymbol{m}, \\boldsymbol{o} \\right)\n", + " \\ = \\\n", + " \\mathcal{L}_{\\text{rep}}\\left(f(\\boldsymbol{m}), \\boldsymbol{o} \\right)\n", + " \\ + \\\n", + " \\mathcal{L}_{\\text{reg}}\\left( \\boldsymbol{m} \\right) .\n", + "\\end{equation}\n", + "These equations allow one to translate a probabilistic problem about probability distributions into an optimisation problem with loss functions, and vice versa." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "## Reproduction loss / likelihood\n", + "---\n", + "The reproduction loss, $\\mathcal{L}_{\\text{rep}}$, is a measure on the space of observations that quantifies how badly a model fits the observation by measuring the discrepancy between a synthetic observation of a model, $f(\\boldsymbol{m})$, and the true observation, $\\boldsymbol{o}$.\n", + "In this paper, we consider a typical reproduction loss given by the mean squared error, weighted by a covariance matrix, $\\boldsymbol{\\Sigma}$, such that,\n", + "\\begin{equation*}\n", + "\\mathcal{L}_{\\text{rep}}\\big(f(\\boldsymbol{m}), \\boldsymbol{o} \\big)\n", + "\\ = \\\n", + "\\frac{1}{2} \\,\n", + "\\big(f(\\boldsymbol{m}) - \\boldsymbol{o}\\big)^{\\text{T}} \\\n", + "\\boldsymbol{\\Sigma}^{-1} \\,\n", + "\\big(f(\\boldsymbol{m}) - \\boldsymbol{o}\\big) .\n", + "\\end{equation*}\n", + "With this definition of the reproduction loss, the likelihood, up to a normalisation constant, corresponds to a multivariate Gaussian distribution.\n", + "This distribution can be used to represent the uncertainty in the observations, where the square root of the diagonal of the covariance matrix models the uncertainty per pixel or visibility.\n", + "\n", + "\n", + "For simplicity, in this paper, we omitted the noise on the observations, i.e. $\\boldsymbol{\\Sigma} = \\mathbb{1}$.\n", + "However, we did find that our reconstruction method performs significantly better by splitting the reproduction loss into a averaged and a relative part,\n", + "\\begin{equation*}\n", + " \\mathcal{L}_{\\text{rep}}\\big(f(\\boldsymbol{m}), \\boldsymbol{o} \\big)\n", + " \\ = \\\n", + " \\mathcal{L}_{\\text{rep}}\\Big( \\big\\langle f(\\boldsymbol{m}) \\big\\rangle, \\, \\left\\langle\\boldsymbol{o}\\right\\rangle \\Big)\n", + " \\ + \\\n", + " \\mathcal{L}_{\\text{rep}}\\left( \\frac{f(\\boldsymbol{m})}{\\big\\langle f(\\boldsymbol{m})\\big\\rangle}, \\, \\frac{\\boldsymbol{o}}{\\left\\langle \\boldsymbol{o}\\right\\rangle} \\right) ,\n", + "\\end{equation*}\n", + "where the brackets, $\\langle\\cdot\\rangle$, denote an arithmetic mean along an axis of the data, for instance, the frequency. For images, when considering the mean along the frequency axis, this implies that we add the loss for the frequency-averaged intensities in each pixel and the loss for the frequency-normalised intensity in each pixel. \n", + "In this way, all pixels contribute equally, at least in the relative part of the loss (i.e. the second term).\n", + "Without this, the algorithm has difficulty reconstructing dimmer regions in the observations, since their contribution to the loss is overpowered by the contributions of brighter regions." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Regularisation loss / prior\n", + "---\n", + "The regularisation loss, $\\mathcal{L}_{\\text{reg}}$, is a measure on the space of models that quantifies how well a model fits our prior assumptions.\n", + "We consider a regularisation loss, or a corresponding prior, $p(\\boldsymbol{m})$, that consists of different parts, each encoding a different assumption about our model.\n", + "Below, we will present the types of regularisation that we consider in pomme.\n", + "However, not every assumption will always be necessary.\n", + "Different combinations can be used for different reconstructions." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Often when solving inverse problems, to avoid over-fitting, one assumes a certain degree of regularity or smoothness of the solution.\n", + "In this paper, for each model parameter, $q(\\boldsymbol{x})$, with a spatial dependence, $\\boldsymbol{x}$, we use the integrated squared Euclidean norm of its gradient,\n", + "\\begin{equation}\n", + " \\mathcal{L}_{\\text{reg}}[q] \\ = \\ \\int \\text{d} \\boldsymbol{x} \\ \\| \\nabla q(\\boldsymbol{x})\\|^{2} ,\n", + "\\end{equation}\n", + "to quantify its deviation from smoothness.\n", + "The main reason for this particular choice of smoothness measure is its straightforward numerical implementation." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "For some observations it can also be useful to make assumptions about symmetries, and, in particular, about spherical symmetry. \n", + "Therefore, we implemented a loss function that can quantify deviations from spherical symmetry.\n", + "Given an origin point in the model, $\\boldsymbol{x}_{O}$, this loss quantifies the average variance within a predetermined set of spherical shells around the origin point,\n", + "\\begin{equation}\n", + " \\mathcal{L}_{\\text{sph}}[q]\n", + " \\ = \\\n", + " \\int_{0}^{\\infty} \\text{d} r \\\n", + " \\mathbb{V}\\big[q(\\boldsymbol{x}) \\, \\big| \\, \\|\\boldsymbol{x}-\\boldsymbol{x}_{O}\\|=r \\big] .\n", + "\\end{equation}" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Next, we consider a loss that can encode the physical laws that govern the variables in our model, i.e. the distributions of $\\rho$, $\\boldsymbol{v}$, and $T$.\n", + "Not every configuration of $\\rho$, $\\boldsymbol{v}$, and $T$ is equally likely to occur, since we expect any configuration to be the result of hydrodynamic evolution from some initial conditions.\n", + "We assume this hydrodynamic evolution to be governed by the conservation of mass, momentum, and energy, which can be formalised, in Eulerian form, as,\n", + "\\begin{align*}\n", + " \\frac{\\partial \\rho}{\\partial t} \\ + \\ \\nabla \\cdot \\left( \\rho \\, \\boldsymbol{v} \\right) \\ &= \\ 0 , \\\\\n", + " \\frac{\\partial \\boldsymbol{v}}{\\partial t} \\ + \\ \\left( \\boldsymbol{v} \\cdot \\nabla \\right) \\boldsymbol{v} \\ + \\ \\frac{1}{\\rho} \\, \\nabla P \\ + \\ \\nabla \\Phi \\ &= \\ 0 , \\\\\n", + " \\frac{\\partial E}{\\partial t} \\ + \\ \\nabla \\cdot \\big( \\left(E + P\\right) \\boldsymbol{v} \\big) \\ + \\ \\Lambda \\ &= \\ 0 . \n", + "\\end{align*}\n", + "Here, we defined the total internal energy,\n", + "\\begin{equation*}\n", + " E \\ = \\ \\frac{1}{2} \\, \\rho \\, \\boldsymbol{v}^{2} \\ + \\ \\rho \\, u,\n", + "\\end{equation*}\n", + "consisting of a kinetic term and a thermal internal energy, $u$, which can be related to the pressure, $P$, assuming an equation of state,\n", + "\\begin{equation*}\n", + " P \\ = \\ \\left( \\gamma - 1 \\right) \\rho \\, u ,\n", + "\\end{equation*}\n", + "where $\\gamma$ is the adiabatic index, which is related to the internal degrees of freedom in the gas.\n", + "The pressure, $P$, in turn, can be related to the temperature, $T$, through the ideal gas law,\n", + "\\begin{equation*}\n", + " P \\ = \\ \\frac{k_{\\text{B}}}{\\mu} \\, \\rho \\, T ,\n", + "\\end{equation*}\n", + "in which $k_{\\text{B}}$ is Boltzmann's constant and $\\mu$ is the mean molecular weight of the gas.\n", + "The remaining components, $\\Phi$ and $\\Lambda$, describe the gravitational potential and the cooling function respectively. \n", + "Often, there is, for instance, a central object for which we have a mass estimate, $M$, and an estimated location, $\\boldsymbol{x}_{\\text{grav}}$\\footnote{Model parameters, such as $M$ and $\\boldsymbol{x}_{\\text{grav}}$ can also be considered as free parameters that can be fitted to the observations by minimising the loss (maximising the posterior).}.\n", + "As a result, we can, for instance, assume a gravitational potential of the form,\n", + "\\begin{equation*}\n", + " \\Phi(\\boldsymbol{x}) \\ = \\ - \\frac{GM}{\\| \\boldsymbol{x} - \\boldsymbol{x_{\\text{grav}}} \\|} .\n", + "\\end{equation*}\n", + "Note that this ignores the self-gravitation of the density distribution $\\rho$.\n", + "The cooling function is often more difficult to estimate, since it depends on many other parameters of the astrophysical object.\n", + "Without additional prior knowledge, one can assume, for instance, $\\Lambda = 0$." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "The hydrodynamics equations provide five component equations that describe the time-evolution of the five components of our model variables, $\\rho$, $\\boldsymbol{v}$, and $T$.\n", + "However, we do not aim to describe the entire time evolution, but rather a snapshot in time, based on an observation.\n", + "As a result, our models lack time dependence and we cannot simply enforce the hydrodynamic equations as prior assumptions.\n", + "Instead, we want to know, at any given time, what is the most likely state to find our model in, given that its evolution is governed by the hydrodynamic equations.\n", + "One way to do this is to consider time averages of each time-dependent variable, $q(t)$, defined as,\n", + "\\begin{equation*}\n", + " \\left\\langle q \\right \\rangle_{T}\n", + " \\ \\equiv \\\n", + " \\frac{1}{T} \\int_{0}^{T} \\text{d} t \\ q(t) .\n", + "\\end{equation*}\n", + "For any bounded function, i.e.\\ there exist $q_{\\min}, q_{\\max} \\in \\mathbb{R}$, such that $q_{\\min} \\leq q(t) \\leq q_{\\max}$, for sufficiently large time intervals, i.e. $T \\rightarrow \\infty$, one can easily show that the time average of the time derivative of the function vanishes, i.e.\n", + "\\begin{equation*}\n", + " \\left|\n", + " \\lim_{T \\rightarrow \\infty}\n", + " \\left\\langle \\frac{\\partial q}{\\partial t} \\right \\rangle_{T}\n", + " \\right|\n", + " \\ = \\\n", + " \\left|\n", + " \\lim_{T \\rightarrow \\infty}\n", + " \\left(\n", + " \\frac{1}{T} \\int_{0}^{T} \\text{d} t \\ \\frac{\\partial q}{\\partial t}\n", + " \\right)\n", + " \\right|\n", + " \\ \\leq \\\n", + " \\lim_{T \\rightarrow \\infty}\n", + " \\left(\n", + " \\frac{q_{\\max} - q_{\\min}}{T}\n", + " \\right)\n", + " \\ = \\ 0 .\n", + "\\end{equation*}\n", + "As a result, one can argue that the time-averaged state of a model, which we use as an estimator for the expected state of a model at any time, is a steady-state solution, i.e.\\ a solution with vanishing time derivatives, $\\partial_{t} \\rho = 0$, $\\partial_{t} \\boldsymbol{v} = 0$, and $\\partial_{t} T = 0$.\n", + "It is also quite intuitive that it is more likely to observe a system in a state in which its time evolution is slow, since it spends comparatively more time in those states.\n", + "Assuming a steady state, the hydrodynamic equations can be rewritten in terms of our model variables, $\\rho$, $\\boldsymbol{v}$, and $T$,\n", + "\\begin{align*}\n", + " \\nabla \\cdot \\left( \\rho \\, \\boldsymbol{v} \\right) \\ &= \\ 0, \\\\\n", + " \\left( \\boldsymbol{v} \\cdot \\nabla \\right) \\boldsymbol{v} \\ + \\ \\frac{k_{\\text{B}}T}{\\mu} \\, \\nabla \\big( \\log \\rho \\, + \\, \\log T \\big) \\ + \\ \\nabla \\Phi \\ &= \\ 0, \\\\\n", + " \\rho \\, \\boldsymbol{v} \\cdot \\nabla \\left( \\frac{1}{2} \\, \\boldsymbol{v}^{2} \\ + \\ \\frac{\\gamma}{\\gamma-1} \\frac{k_{\\text{B}}T}{\\mu} \\right) \\ + \\ \\Lambda \\ &= \\ 0 .\n", + "\\end{align*}\n", + "Since these equations only contain (spatial derivatives of) our model variables, we can enforce them as an assumption on our model by defining the following loss functions,\n", + "\\begin{align*}\n", + " \\mathcal{L}_{\\rho}(\\boldsymbol{m})\n", + " \\ &\\equiv \\\n", + " \\int \\text{d}\\boldsymbol{x} \\ \\frac{1}{\\rho^{2}} \\big( \\nabla \\cdot \\left( \\rho \\, \\boldsymbol{v} \\right) \\big)^{2} , \\\\\n", + " \\mathcal{L}_{v_{i}}(\\boldsymbol{m})\n", + " \\ &\\equiv \\\n", + " \\int \\text{d}\\boldsymbol{x} \\ \\frac{1}{v_{i}^{2}} \\left( \\left( \\boldsymbol{v} \\cdot \\nabla \\right) v_{i} \\ + \\ \\frac{k_{\\text{B}}T}{\\mu} \\, \\nabla_{i} \\big( \\log \\rho \\, + \\, \\log T \\big) \\ + \\ \\nabla_{i} \\Phi \\right)^{2} , \\\\\n", + " \\mathcal{L}_{E}(\\boldsymbol{m}) \\ &\\equiv \\\n", + " \\int \\text{d}\\boldsymbol{x} \\ \\frac{1}{E^{2}} \\left( \\rho \\, \\boldsymbol{v} \\cdot \\nabla \\left( \\frac{1}{2} \\, \\boldsymbol{v}^{2} \\ + \\ \\frac{\\gamma}{\\gamma-1} \\frac{k_{\\text{B}}T}{\\mu} \\right)\n", + " \\ + \\ \\Lambda \\right)^{2} .\n", + "\\end{align*}\n", + "We included normalisation factors to ensure that all loss functions have the same unit of inverse time.\n", + "These loss functions can be combined to define a regularisation loss,\n", + "\\begin{equation*}\n", + " \\mathcal{L}_{\\text{reg}}\\left( \\boldsymbol{m} \\right)\n", + " \\ = \\\n", + " w_{\\rho} \\, \\mathcal{L}_{\\rho} \\left( \\boldsymbol{m} \\right)\n", + " \\ + \\\n", + " \\boldsymbol{w}_{\\boldsymbol{v}} \\cdot \\boldsymbol{\\mathcal{L}}_{\\boldsymbol{v}}\\left( \\boldsymbol{m} \\right)\n", + " \\ + \\\n", + " w_{E} \\, \\mathcal{L}_{E}\\left( \\boldsymbol{m} \\right) ,\n", + "\\end{equation*}\n", + "which also defines a prior distribution.\n", + "The weights, $w_{\\rho}$, $\\boldsymbol{w}_{\\boldsymbol{v}}$, and $w_{E}$, are hyper-parameters of the prior that determine the width of the distribution around each of the steady-state constraints and can, furthermore, be used to weigh their relative importance.\n", + "\n", + "We should note that the loss originating from the continuity equation, in practice, is by far the most useful, since it does not depend on external parameters, like the gravitational potential ($\\Phi$) or the cooling function ($\\Lambda$), which are often difficult to describe accurately." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "Many other types of regularisation loss can be considered.\n", + "In principle, any equation involving the model parameters, say, $a(\\boldsymbol{m})=b(\\boldsymbol{m})$, can be enforced on the model as regularisation or prior by including a loss term proportional to a monotonically increasing function of $|a(\\boldsymbol{m})-b(\\boldsymbol{m})|$.\n", + "This can be used, for instance, for non-LTE line radiative transfer, to enforce the statistical equilibrium equations on the level populations.\n", + "Since this requires an efficient way to compute the mean intensities in the line, which poses a challenge for our current implementation, we postpone this to future work and limit ourselves currently to LTE line radiative transfer. " + ] + } + ], + "metadata": { + "language_info": { + "name": "python" + }, + "orig_nbformat": 4 + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/docs/src/examples/image.html b/docs/src/examples/image.html new file mode 100644 index 0000000..05d4282 --- /dev/null +++ b/docs/src/examples/image.html @@ -0,0 +1,14 @@ + + + +
+
+ + \ No newline at end of file diff --git a/docs/src/examples/Untitled.ipynb b/docs/src/examples/multipole_regularisation.ipynb similarity index 100% rename from docs/src/examples/Untitled.ipynb rename to docs/src/examples/multipole_regularisation.ipynb diff --git a/docs/src/examples/newplot.png b/docs/src/examples/newplot.png new file mode 100644 index 0000000..ccb8614 Binary files /dev/null and b/docs/src/examples/newplot.png differ diff --git a/docs/src/examples/visualisation_plotly.ipynb b/docs/src/examples/visualisation_plotly.ipynb new file mode 100644 index 0000000..c7ea57c --- /dev/null +++ b/docs/src/examples/visualisation_plotly.ipynb @@ -0,0 +1,695174 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [], + "source": [ + "import matplotlib.pyplot as plt\n", + "import numpy as np\n", + "import torch\n", + "\n", + "from astropy import units, constants\n", + "from pomme.model import TensorModel\n", + "from pomme.haar import Haar\n", + "from pomme.plot import plot_cube_2D\n", + "from pomme.utils import planck, T_CMB\n", + "\n", + "import matplotlib.pyplot as plt # Mpl plotting\n", + "import matplotlib # Mpl\n", + "import plotly.graph_objects as go # Plotly plotting\n", + "import magritte.core as magritte # Core functionality\n", + "import numpy as np # Data structures\n", + "import os # Creating directories\n", + "import warnings # Hide warnings\n", + "warnings.filterwarnings('ignore') # especially for yt\n", + "\n", + "from matplotlib.gridspec import GridSpec # Plot layout\n", + "from plotly.subplots import make_subplots # Plotly subplots\n", + "from astropy import constants, units # Unit conversions\n", + "from scipy.interpolate import griddata # Grid interpolation\n", + "from palettable.cubehelix import cubehelix2_16 # Nice colormap\n", + "from tqdm import tqdm # Progress bars\n", + "from ipywidgets import interact # Interactive plots\n", + "from ipywidgets.embed import embed_minimal_html # Store interactive plots\n", + "from magritte.core import ImageType, ImagePointPosition # Image type, point position\n", + "from math import floor, ceil # Math helper functions\n", + "from mpl_toolkits.axes_grid1.inset_locator import inset_axes\n", + "\n", + "\n", + "# Port matplotlib colormaps to plotly\n", + "cubehelix2_16_rgb = []\n", + "for i in range(0, 255):\n", + " cubehelix2_16_rgb.append(\n", + " matplotlib.colors.colorConverter.to_rgb(\n", + " cubehelix2_16.mpl_colormap(\n", + " matplotlib.colors.Normalize(vmin=0, vmax=255)(i)\n", + " )\n", + " )\n", + " )\n", + "\n", + "\n", + "def matplotlib_to_plotly(cmap, pl_entries):\n", + " \"\"\"\n", + " Converts matplotlib cmap to plotly colorscale.\n", + " \"\"\"\n", + " h = 1.0/(pl_entries-1)\n", + " pl_colorscale = []\n", + "\n", + " for k in range(pl_entries):\n", + " C = list(map(np.uint8, np.array(cmap(k*h)[:3])*255))\n", + " pl_colorscale.append([k*h, 'rgb'+str((C[0], C[1], C[2]))])\n", + " return pl_colorscale\n", + "\n", + "\n", + "# Plotly compatible colorscale\n", + "cubehelix2_16_plotly = matplotlib_to_plotly(cubehelix2_16.mpl_colormap, 255)\n", + "\n", + "\n", + "# Plotly standard config\n", + "modeBarButtonsToRemove = ['select2d', 'lasso2d', 'zoomIn2d', 'zoomOut2d', 'autoScale2d', 'hoverClosestCartesian', 'hoverCompareCartesian', 'zoom3d', 'pan3d', 'resetCameraDefault3d', 'resetCameraLastSave3d', 'hoverClosest3d', 'orbitRotation', 'tableRotation', 'zoomInGeo', 'zoomOutGeo', 'resetGeo', 'hoverClosestGeo', 'sendDataToCloud', 'hoverClosestGl2d', 'hoverClosestPie', 'toggleHover', 'resetViews', 'toggleSpikelines', 'resetViewMapbox']" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [], + "source": [ + "model = TensorModel.load('IK_Tau/NaCl/IK_Tau_NaCl_fit.h5')\n", + "\n", + "rho = torch.exp(model['log_NaCl']).data.numpy().astype(np.float32)\n", + "\n", + "obss_2 = torch.load('IK_Tau/NaCl/data/obs.pt')\n", + "obss = torch.load('IK_Tau/NaCl/data/img.pt')\n", + "freqs = torch.load('IK_Tau/NaCl/data/freqs.pt')" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "torch.Size([35, 35, 25])" + ] + }, + "execution_count": 4, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "obss.shape" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [], + "source": [ + "x_min = -model.sizes[0]/2\n", + "x_max = +model.sizes[0]/2\n", + "y_min = -model.sizes[1]/2\n", + "y_max = +model.sizes[1]/2\n", + "\n", + "npix_x = obss.shape[0]\n", + "npix_y = obss.shape[1]\n", + "nfreqs = obss.shape[2]\n", + "\n", + "# Create image grid values\n", + "xs = np.linspace(x_min, x_max, npix_x)\n", + "ys = np.linspace(y_min, y_max, npix_y)\n", + "\n", + "x_unit = units.au\n", + "\n", + "# Convert to given units\n", + "xs = xs / (1.0 * x_unit).si.value\n", + "ys = ys / (1.0 * x_unit).si.value\n", + "\n", + "# Convert to given units\n", + "x_max = np.max(xs)\n", + "x_min = np.min(xs)\n", + "y_max = np.max(ys)\n", + "y_min = np.min(ys)" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [], + "source": [ + "def image_plotly(\n", + " v_unit = units.km/units.s,\n", + " method = 'nearest',\n", + " width = 620, # Yields approx square channel map\n", + " height = 540, # Yields approx square channel map\n", + " save_fig = True\n", + " ):\n", + " # Extract the spectral / velocity data\n", + " f_ij = freqs.mean().numpy()\n", + " # Velocity axis definition in images is flipped compared to the direction the observer is looking\n", + " velos = (f_ij - freqs.numpy()) / f_ij * constants.c.to(v_unit).value\n", + "\n", + " # Interpolate the scattered data to an image (regular grid)\n", + " Is = np.zeros((nfreqs))\n", + " zs = np.zeros((nfreqs, npix_x, npix_y))\n", + " for f in range(nfreqs):\n", + " # Nearest neighbor interpolate scattered image data\n", + " zs[f] = obss.detach().numpy()[:,:,f]\n", + " Is[f] = np.sum(zs[f])\n", + " Is = Is / np.max(Is)\n", + "\n", + " # Put zero-values to the smallest non-zero value\n", + " zs[zs<=0.0] = np.min(zs[zs>0.0])\n", + " # Put nan values to smallest positive value\n", + " zs[np.isnan(zs)] = np.min(zs[zs>0.0])\n", + "\n", + " # Get the logarithm of the data (matplotlib has a hard time handling logarithmic data.)\n", + " log_zs = np.log10(zs)\n", + " log_zs_min = np.min(log_zs)\n", + " log_zs_max = np.max(log_zs)\n", + "\n", + " # Create plotly plot\n", + " fig = make_subplots(\n", + " rows = 1,\n", + " cols = 2,\n", + " column_widths = [0.7, 0.3],\n", + " horizontal_spacing = 0.05,\n", + " subplot_titles = ['Intensity', ''],\n", + " )\n", + "\n", + " fig.add_vrect(\n", + " row = 1,\n", + " col = 1,\n", + " x0 = -1.0e+99,\n", + " x1 = +1.0e+99,\n", + " line_width = 0,\n", + " fillcolor = \"black\"\n", + " )\n", + "\n", + " # Build up plot\n", + " for f in reversed(range(nfreqs)):\n", + " fig.add_trace(\n", + " go.Heatmap(\n", + " x = xs .astype(float),\n", + " y = ys .astype(float),\n", + " z = log_zs[f].astype(float),\n", + " visible = False,\n", + " hoverinfo = 'none',\n", + " zmin = log_zs_min.astype(float),\n", + " zmax = log_zs_max.astype(float),\n", + " colorscale = cubehelix2_16_plotly,\n", + " showscale = False\n", + " ),\n", + " row = 1,\n", + " col = 1\n", + " )\n", + "\n", + " fig.add_trace(\n", + " go.Scatter(\n", + " x = (velos) .astype(float),\n", + " y = (Is/np.max(Is)).astype(float),\n", + " visible = False,\n", + " hoverinfo = 'none',\n", + " line_color = '#1f77b4',\n", + " showlegend = False\n", + " ),\n", + " row = 1,\n", + " col = 2\n", + " )\n", + "\n", + " fig.add_trace(\n", + " go.Scatter(\n", + " x = np.array([velos[f], velos[f]], dtype=float),\n", + " y = np.array([-1.0e+10, +1.0e+10], dtype=float),\n", + " visible = False,\n", + " hoverinfo = 'none',\n", + " line_color = 'red',\n", + " showlegend = False\n", + " ),\n", + " row = 1,\n", + " col = 2\n", + " )\n", + "\n", + " # Boxes around plots (more like mpl)\n", + " fig.update_xaxes(\n", + " showline=True, linewidth=2, linecolor='rgba(1,1,1,1)', mirror=True, ticks='outside'\n", + " )\n", + " fig.update_yaxes(\n", + " showline=True, linewidth=2, linecolor='rgba(1,1,1,1)', mirror=True, ticks='outside'\n", + " )\n", + "\n", + " v_min = float(min(velos)) * 1.05\n", + " v_max = float(max(velos)) * 1.05\n", + "\n", + " # Black background for channel map\n", + " fig.add_vrect(\n", + " row = 1,\n", + " col = 1,\n", + " x0 = 1000.0*x_min, # large enough so you don't see edges\n", + " x1 = 1000.0*x_max, # large enough so you don't see edges\n", + " line_width = 0,\n", + " fillcolor = \"black\",\n", + " layer=\"below\"\n", + " )\n", + "\n", + " # Plot axes\n", + " fig.update_xaxes(\n", + " row = 1,\n", + " col = 1,\n", + " title_text = f'image x [{x_unit}]',\n", + " range = [x_min, x_max],\n", + " showgrid = False,\n", + " zeroline = False\n", + " )\n", + " fig.update_yaxes(\n", + " row = 1,\n", + " col = 1,\n", + " title_text = f'image y [{x_unit}]',\n", + " scaleanchor = \"x\",\n", + " scaleratio = 1,\n", + " showgrid = False,\n", + " zeroline = False\n", + " )\n", + " fig.update_xaxes(\n", + " row = 1,\n", + " col = 2,\n", + " title_text = f'velocity [{v_unit}]',\n", + " range = [v_min, v_max]\n", + " )\n", + "\n", + "\n", + " fig.update_yaxes(\n", + " row = 1,\n", + " col = 2,\n", + " title_text = \"Relative intensity\",\n", + " side = 'right',\n", + " range = [-0.05, +1.05]\n", + " )\n", + "\n", + " # Subplot titles are annotations\n", + " fig.update_annotations(\n", + " font_size = 16,\n", + " borderpad = 7\n", + " )\n", + "\n", + " fig.update_layout(\n", + " width = width,\n", + " height = height,\n", + " plot_bgcolor = 'rgba(0,0,0,0)',\n", + " dragmode = 'pan',\n", + " font = dict(family=\"Calibri\", size=14, color='black')\n", + " )\n", + "\n", + " # Make 3 middle traces visible\n", + " fig.data[3*nfreqs//2 ].visible = True\n", + " fig.data[3*nfreqs//2+1].visible = True\n", + " fig.data[3*nfreqs//2+2].visible = True\n", + "\n", + " # Create and add slider\n", + " steps = []\n", + " for f in range(nfreqs):\n", + " step = dict(\n", + " method = \"restyle\",\n", + " args = [\n", + " {\"visible\": [False] * len(fig.data)},\n", + " {\"title\": \"Channel map: \" + str(f)}\n", + " ],\n", + " label = ''\n", + " )\n", + " # Toggle f'th trace to \"visible\"\n", + " step[\"args\"][0][\"visible\"][3*f:3*f+3] = [True, True, True]\n", + " steps.append(step)\n", + "\n", + " sliders = [\n", + " dict(\n", + " active = nfreqs//2,\n", + " pad = {\"t\": 75},\n", + " steps = steps,\n", + " tickcolor = 'white',\n", + " transition = {'duration': 0}\n", + " )\n", + " ]\n", + "\n", + " fig.update_layout(\n", + " sliders=sliders\n", + " )\n", + "\n", + " # Config for modebar buttons\n", + " config = {\n", + " \"modeBarButtonsToRemove\": modeBarButtonsToRemove,\n", + " \"scrollZoom\": True\n", + " }\n", + "\n", + " if save_fig:\n", + " # Save figure as html file\n", + " 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" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "import plotly.graph_objects as go\n", + "\n", + "X, Y, Z = model.get_coords()\n", + "values = rho\n", + "\n", + "crit = np.ones_like(values, dtype=bool) # values > 5.0 * values.min()\n", + "\n", + "X_lim = X [crit] / (1.0 * x_unit).si.value\n", + "Y_lim = Y [crit] / (1.0 * x_unit).si.value\n", + "Z_lim = Z [crit] / (1.0 * x_unit).si.value\n", + "values_lim = values[crit]\n", + "\n", + "\n", + "fig = go.Figure(data=go.Volume(\n", + " x=X_lim.flatten(),\n", + " y=Y_lim.flatten(),\n", + " z=Z_lim.flatten(),\n", + " value=values_lim.flatten(),\n", + " # isomin=0.5*rho.min(),\n", + " # isomax=rho.max(),\n", + " opacity=0.03, # needs to be small to see through all surfaces\n", + " # opacityscale=[[0, 0], [1, 0.01*rho.max()]],\n", + " surface_count=50, # needs to be a large number for good volume rendering\n", + " colorscale='magma'\n", + " ))\n", + "\n", + "fig.update_xaxes(\n", + " title_text = f'x [{x_unit}]',\n", + " range = [x_min, x_max],\n", + " showgrid = False,\n", + " zeroline = False,\n", + " tickvals=[-100, 0, +100]\n", + ")\n", + "fig.update_yaxes(\n", + " title_text = f'y [{x_unit}]',\n", + " scaleanchor = \"x\",\n", + " scaleratio = 1,\n", + " showgrid = False,\n", + " zeroline = False,\n", + ")\n", + "\n", + "fig.update_layout(\n", + " plot_bgcolor = 'rgba(0,0,0,0)',\n", + " font = dict(family=\"Calibri\", size=14, color='black'),\n", + " # plot_bgcolor = 'black',\n", + " # grid = False,\n", + " scene=dict(\n", + " xaxis_title='x [AU]',\n", + " yaxis_title='y [AU]',\n", + " zaxis_title='z [AU]',\n", + " ),\n", + ")\n", + "\n", + "fig.update_scenes(\n", + " bgcolor='rgba(0,0,0,0)',\n", + " xaxis_tickvals=[-100, 0, +100],\n", + " yaxis_tickvals=[-100, 0, +100],\n", + " zaxis_tickvals=[-100, 0, +100],\n", + " # xaxis_backgroundcolor='rgba(100, 200, 200, 0.5)',\n", + " # yaxis_backgroundcolor='rgba(200, 200, 200, 0.5)',\n", + " # zaxis_backgroundcolor='rgba(200, 200, 200, 0.5)',\n", + " xaxis_gridcolor='rgba(120, 220, 220, 0)',\n", + " yaxis_gridcolor='rgba(220, 220, 220, 0)',\n", + " zaxis_gridcolor='rgba(120, 220, 220, 0)',\n", + " yaxis_gridwidth=2,\n", + " xaxis_showbackground=False,\n", + " yaxis_showbackground=False,\n", + " zaxis_showbackground=False,\n", + ")\n", + "\n", + "\n", + "x_eye = 0\n", + "y_eye = 0\n", + "z_eye = -2\n", + "\n", + "fig.update_layout(\n", + " title='Animation Test',\n", + " width=600,\n", + " height=600,\n", + " scene_camera_eye=dict(x=x_eye, y=y_eye, z=z_eye),\n", + ")\n", + "\n", + "\n", + "# def rotate_z(x, y, z, theta):\n", + "# w = x+1j*y\n", + "# return np.real(np.exp(1j*theta)*w), np.imag(np.exp(1j*theta)*w), z\n", + "\n", + "# steps=[]\n", + "# for t in np.arange(0, 6.26, 0.1):\n", + "# xe, ye, ze = rotate_z(x_eye, y_eye, z_eye, -t)\n", + "# step = dict(\n", + "# method = \"relayout\",\n", + "# args = [\n", + "# {'scene_camera_eye': dict(x=xe, y=ye, z=ze)},\n", + "# ],\n", + "# label = '',\n", + "# )\n", + "# steps.append(step)\n", + "\n", + "\n", + "\n", + "\n", + "# sliders = [\n", + "# dict(\n", + "# active = 10,\n", + "# pad = {\"t\": 75},\n", + "# steps = steps,\n", + "# tickcolor = 'white',\n", + "# transition = {'duration': 0}\n", + "# )\n", + "# ]\n", + "\n", + "# fig.update_layout(\n", + "# sliders=sliders\n", + "# )\n", + "\n", + "\n", + "fig.show()" + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "metadata": {}, + "outputs": [ + { + "data": { + "application/vnd.plotly.v1+json": { + "config": { + "plotlyServerURL": "https://plot.ly" + }, + "data": [ + { + "colorscale": [ + [ + 0, + "#000004" + ], + [ + 0.1111111111111111, + "#180f3d" + ], + [ + 0.2222222222222222, + "#440f76" + ], + [ + 0.3333333333333333, + "#721f81" + ], + [ + 0.4444444444444444, + "#9e2f7f" + ], + [ + 0.5555555555555556, + "#cd4071" + ], + [ + 0.6666666666666666, + "#f1605d" + ], + [ + 0.7777777777777778, + "#fd9668" + ], + [ + 0.8888888888888888, + "#feca8d" + ], + [ + 1, + "#fcfdbf" + ] + ], 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" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "x_eye = 0\n", + "y_eye = 0\n", + "z_eye = -2\n", + "\n", + "fig.update_layout(\n", + " title='Animation Test',\n", + " width=600,\n", + " height=600,\n", + " scene_camera_eye=dict(x=x_eye, y=y_eye, z=z_eye),\n", + " # yaxis_tickangle=0.0,\n", + ")\n", + "\n", + "fig.update_scenes(\n", + " xaxis_tickangle=0.0,\n", + " yaxis_tickangle=0.0,\n", + " zaxis_tickangle=0.0,\n", + ")" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "magritte", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.9.13" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/docs/src/index.rst b/docs/src/index.rst index 549cc6f..6f9aec3 100644 --- a/docs/src/index.rst +++ b/docs/src/index.rst @@ -1,5 +1,5 @@ pomme -######## +##### Welcome to the *pomme* documentation! @@ -7,6 +7,7 @@ Welcome to the *pomme* documentation! :maxdepth: 1 :caption: Contents: + index background/index examples/index API/index