# Welcome!

You’ve reached the blog of Sébastien Brisard (follow the About me link to read my CV). My most recent posts are listed below. A full list of posts can be found in the Archives page. These posts frequently refer to work published by myself or others; the full bibliographic references are provided in the References page.

While you’re at it, you can also visit my other pages

- https://sbrisard.github.io/janus: the documentation of the
`Janus`

library, - https://sbrisard.github.io/pw85: the documentation of the
`PW85`

library.

## 2018-03-26 – On the periodic-plus-smooth decomposition of an image, part 7: improved implementation of Moisan’s algorithm

In the previous instalment of this series, we implemented Moisan’s (2011) efficient algorithm to compute the periodic-plus-smooth decomposition of an image. This algorithm relies heavily on the discrete Fourier transform, and already improves quite a lot over our previous conjugate gradient-based implementation. In the present post, we will show that performance of the implementation can be slightly improved with very little effort.

## 2018-03-19 – On the periodic-plus-smooth decomposition of an image, part 6: minimizing the energy, the clever way

In the previous instalment of this series, we computed Moisan’s (2011) periodic-plus-smooth decomposition of an image by means of the conjugate gradient method. This worked like a charm, but was fairly inefficient, owing to the iterative nature of the method. Moisan actually showed that the whole decomposition could be computed explicitly in Fourier space.

## 2018-03-12 – On the periodic-plus-smooth decomposition of an image, part 5: minimizing the energy, the clumsy way

In this post, we will compute Moisan’s (2011) periodic-plus-smooth decomposition of an image by direct minimization of the energy introduced in the second instalment of this series. More precisely, u being a m×n image, we will minimize the function F(v, u) over the space of m×n images v. The minimizer, s, is the smooth component of u, while its complement p=u-s is the periodic component of u.

## 2018-03-05 – On the periodic-plus-smooth decomposition of an image, part 4: implementing the linear operators

In the previous instalment of this series, we introduced the linear operators Q₁ and Q that allow to define Moisan’s (2011) periodic-plus-smooth decomposition. In theory, the linear operators Q₁ and Q are matrices. However, for m×n images, these matrices would be unnecessarily large: mn×mn! We will therefore adopt here a matrix-free approach for the Python implementation of these operators.

## 2018-02-26 – On the periodic-plus-smooth decomposition of an image, part 3: the energy as a quadratic form

In the previous instalment of this series, we introduced the periodic-plus-smooth decomposition of an image as a pair of images which minimizes an energy functional. Observing that this energy is a quadratic form, the purpose of this post is to derive closed form expressions of the underlying linear operators. These expressions will then be combined in the next instalments to a conjugate gradient algorithm in order to minimize the energy of the periodic-plus-smooth decomposition.

## 2018-02-19 – On the periodic-plus-smooth decomposition of an image, part 2: defining the decomposition

In the previous instalment of this series, we discussed the need for periodic
images. Although not all images are periodic, some image analysis techniques
are best performed in Fourier space (using the fast Fourier
transform). Applying Fourier-based techniques to images that are *not* periodic
(as is often the case) generates artifacts. In order to reduce these artifacts,
Moisan (2011) proposed to construct a periodic image that is as close as
possible to the original image. For reasons that will become clearer in the
remainder of this post, he called the resulting construction the
“periodic-plus-smooth decomposition”.

## 2018-02-12 – On the periodic-plus-smooth decomposition of an image, part 1: introduction

In this new series, we will explore the so-called *periodic-plus-smooth*
decomposition of an image, introduced by Moisan in 2011. This series is largely
based on Moisan’s paper, called Periodic plus Smooth Image Decomposition (the
author’s version can be found on HAL). Besides introducing a quite smart
technique (that I do use for real materials science applications), this series
will be an opportunity to discuss an important topic: how to test the
implementation of an algorithm, and be confident that the test actually
validates the implementation.

## 2017-11-17 – Announcing the defense of my “Habilitation à Diriger des Recherches”

On monday 20^{th}, november 2017 (at 13:30), I will be defending my
“Habilitation à Diriger des Recherches” at École des Ponts ParisTech.

## 2017-08-07 – New conference paper on variational estimates of poroelastic coefficients

I recently had the pleasure to give a talk on “Variational estimates
of the poroelastic coefficients” at the 6^{th} Biot Conference on
Poromechanics that took place at École des Ponts ParisTech on July
9-13, 2017. This is joint work with my colleague Siavash GHABEZLOO. It
has been ongoing for years now, but for lack of time, the full paper
is still not out! (To tell the truth, I need to find and correct a
mistake in the illustrative application…).

Meanwhile, the preprint of the conference paper is now online on HAL; you can also find the slides of my talk there.

## 2017-05-31 – Scrapy-ing the NIST X-ray Attenuation Databases

I am currently preparing with two other colleagues a review paper on X-ray tomography of cementitious materials, for which I need to retrieve tabulated values of the X-ray mass attenuation coefficients of all elements. NIST provides such data (see X-Ray Mass Attenuation Coefficients by J. H. Hubbell and S. M. Seltzer). However, it does not come in the form of simple files that can be downloaded. Rather, the values are presented in nicely formatted HTML tables (see here for an example).

In this post, I set out to extract this data as a HDF5 table. Of course, this looks like a job for Scrapy!

## 2017-04-28 – New paper on one-dimensional compression of granular materials

My student Mohamed Hassan Khalili has recently had one of his papers
published in *Physical Review E*. The paper is called “A numerical
study of one-dimensional compression of granular materials”. I have
only been very marginally involved in this DEM study, which is divided
in two parts.

## 2017-04-26 – New paper on Discrete Digital Projections Correlation

Our paper “Discrete Digital Projections Correlation: a
reconstruction-free method to quantify local kinematics in granular
media by X-ray tomography” has recently been published in
*Experimental Mechanics*.

## 2017-02-23 – New paper out!

My paper “Towards improved Hashin–Shtrikman bounds on the effective moduli of random composites” has recently been published in *Mechanics & Industry*.

## 2016-06-23 – Orientation correlations among rice grains, part 8: estimating the correlations

In the previous instalment of this series, we have analyzed the morphology of the rice grains. In particular, we have defined their *orientation* as that of the major axis of inertia. We are now in a position to quantify the statistics of the orientations. We will first discuss one-point statistics (each grain is considered individually), then two-point statistics (mutual orientations of pairs of grains are considered). Finally, we will try to quantify how the wall of the sample container influences the orientation of the rice grains.

## 2016-06-13 – New paper

My paper “Reconstructing displacements from the solution to the periodic Lippmann–Schwinger equation discretized on a uniform grid” has recently been published in *International Journal of Numerical Methods in Engineering*.

## 2016-02-19 – Orientation correlations among rice grains, part 7: analysis of the shape of the grains

In the previous instalment of this series, we have segmented the 3D image of the assembly of rice grains. In other words, each voxel of the image is attributed the label of the grain to which it belongs. Remember the initial goal of this series: we want to quantify orientation *correlations* between grains. To do so, we must analyse the orientation of each individual grain. This is the topic of the present post.

## 2015-12-08 – New paper published

My student Vinh Phuc Tran has recently had one of his paper published in *Mechanics of Materials*. The paper is called *Stochastic modeling of mesoscopic elasticity random field*.

## 2015-09-30 – Orientation correlations among rice grains, part 6: segmentation

In the previous instalment of this series, I showed that a convincing binary image could be produced from the gray level 3D reconstruction of the assembly of rice grains, using Otsu’s threshold. However, I intend to carry out statistical analyses of the *grains* themselves in the subsequent instalments. Therefore, instead of a binary image of the rice grains, what is really needed is a *labelled* image, where all voxels which are thought to belong to the same rice grain are tagged with the same label. This is called *segmentation*, which is the topic of the present post. I will first show that the most basic segmentation technique (namely, detecting connected components in the image) fails in the present case. This calls for a more elaborate strategy, based on the widely popular watershed method. However, blind application of the standard watershed strategy leads to over-segmentation. This post will therefore close on a problem-dependent strategy better suited to the present case.

## 2015-07-09 – Orientation correlations among rice grains, part 5: thresholding

In the previous instalment of this series, we used the circle Hough transform to find the boundary of the sample and define the circular ROI. Within this ROI, we now need to segment the rice grains. In other words, starting from a gray-level image (Fig. 1, left), we want to produce a binary image, where all pixels that we believe belong to rice grains are white, and all background pixels are black (Fig. 1, right). This is the topic of the present post, where we will use Otsu’s automated threshold selection. I will first discuss Otsu’s method, and propose what I believe is a new interpretation of this rather old technique. Then, I will apply this method to the 3D image of rice grains, using scikit-image.

## 2015-07-06 – When a thin shell is not so thin, part 3: the thick shell solution

Finding the stress resultant and stress couple in a spherical pressure vessel seems easy enough. Well, this apparently simple problem allows us to highlight subtle thickness effects within shells. It should be emphasized again that these thickness effects have nothing to do with shear corrections. Indeed, shear stresses are null at any point of the spherical pressure vessel. Rather, the thickness effects we are discussing in this series are due to curvature.

In the first instalment of this series, I showed that using Koiter’s thin shell solution leads to a non-null stress couple, which came as quite a surprise. In the second instalment, I used the exact, 3D solution to derive reference values of the stress couple and stress resultant through integration over the thickness of the shell. In particular, I showed that thickness corrections *had* to be incorporated.

This still does not settle the matter, since the results obtained within the framework of Koiter’s thin shell theory (see first instalment) are inconsistent with the reference, 3D elasticity values (see second instalment). To reconcile both approaches (3D elasticity vs. shell theory), we need to use a *thick* shell theory, as illustrated below.

## 2015-07-01 – When a thin shell is not so thin, part 2: the 3D, exact solution

In the previous instalment of this series, we analysed a spherical pressure vessel by means of Koiter’s linear theory of thin shells. We found the somewhat unexpected result that the stress couple was not null. Besides, we also found that the stress resultant was slightly different from the well-known value N=pR/2. So what was wrong with the previous analysis? To explore this question, we will make use of the exact solution in 3D elasticity.

## 2015-07-01 – This blog is now licensed!

As indicated at the bottom of each page of this website, this blog is now licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. This means that you are free to share and/or adapt the contents of this blog, as long as you give appropriate credit to the author. You may not make profit out of this website, though.

## 2015-06-08 – When a thin shell is not so thin, part 1: Koiter’s linear theory

In structural analysis, thick beams (resp. plates) usually refer to shear deformability, and the Timoshenko beam theory (resp. Mindlin–Reissner plate theory). With curved elements however (e.g. curved beams or shells), the situation is more subtle, as thickness corrections may be necessary even in shells *where the shear stress is null at any point*. In this series, this is illustrated with a spherical pressure vessel, for which the stress resultants and couples are studied. “Easy enough”, you probably think: N=pR/2, M=0 and that’s the end of it (Wikipedia). Well, maybe…

## 2015-05-29 – Orientation correlations among rice grains, part 4: defining the ROI

In the previous instalment of this series, we obtained binned slices of the sample. Fig. 1 is a typical example of these binned slices. We now want to segment the rice grains. However, the analysis (in particular, Otsu thresholding) might be perturbed by the fact that the walls of the sample container are visible on the 3D image. In this post, I will show how we can locate these walls. Then, any subsequent analysis will be performed within the Region Of Interest (ROI) thus defined.

## 2015-03-30 – Orientation correlations among rice grains, part 3: intermezzo – Binning images

The full reconstructed image resulting from the tomography experiment described in the second instalment of this series is a 1747×1751×688 stack. The voxel size is about 0.030 mm. This is far too much for the purpose of the present study, since all we are interested in is the determination of the location (coordinates of the centroid) and orientation (principal axes of inertia) of the rice grains. In order to reduce the computation time, the images will first be *binned*, that is each set of (say) 4×4×4 voxels will be replaced with a unique voxel, with average gray value. 3D binning would usually require three uggly nested loops. There is, however, a much more pythonic way. This is the topic of the present post.

## 2015-03-10 – Orientation correlations among rice grains, part 2: acquisition of tomography images

In the first instalment of this series, I briefly introduced orientational order in assemblies of anisotropic particles, and proposed an illustration based on synthetic microstructures. But what I really intend to do in this series is to show how orientational order can be quantified in a “real-life” sample, namely a packing of rice grains. This of course requires a 3D image of the packing. Fortunately, my lab (Laboratoire Navier) is the proud host of a collaborative microtomography platform and my former colleague Nicolas Lenoir –who was in charge of this instrument– gave me the opportunity to scan my toy sample in december 2013.

## 2015-02-23 – Orientation correlations among rice grains, part 1: introduction

In this series, I will explore the notion of orientational order in random packings of anisotropic (flat or elongated), hard particles. By orientational order, I mean that particles which are close to each other tend to adopt the same orientation. This leads to strong *local* anisotropy, while the packing may well be *globally* isotropic; in particular, all orientations of single grains are equiprobable. Local orientational order is stronger when the volume fraction of particles, or their aspect ratio increases.

## 2014-12-19 – New Blog!

My blog was previously hosted by Blogger. I have now moved to GitHub Pages, and a statically generated blog. This new website is entirely generated with Emacs’ Org mode (the source files of my blog can be found here; see in particular the config file). The stylesheet is inspired by the Hello Worg website, the simplicity of which I like. It is a rather exciting experience for me to build a whole website “from scratch”. Of course, there are many things I need to learn, and this blog should hopefully improve with time… be patient! Meanwhile, have a look to my previous posts which I have ported to this site. Happy reading!

## 2014-08-13 – Review of “Parallel Programming with Python”

I have again recently been offered by Packt Publishing to review one of their books, entitled Parallel Programming with Python (by Jan Palach).

## 2014-03-06 – The periodic Green operator for strains

The Green operator for strains is defined as the linear operator which maps –in a homogeneous material– the local prestress to the (opposite of the) local strain. Closed-form expressions of this operator can be derived in the case of periodic boundary conditions, as will be demonstrated in the present post.

## 2014-02-26 – Decomposition of transverse isotropic, fourth-rank tensors

In a previous post, I introduced the fourth-rank spherical and deviatoric projection tensors. Any isotropic fourth-rank tensor is a linear combination of these two tensors; in other words, the space of isotropic fourth-rank tensors (with minor and major symmetries) is of dimension 2. Similarly, it can be shown (Walpole, 1984) that the space of three-dimensional, *transverse isotropic* fourth-rank tensors (with minor and major symmetries) is of dimension 6. Furthermore; it is possible to produce a convenient basis of this space. This is the topic of the present post, which is mostly based on the paper by Walpole (1984).

## 2014-02-19 – On the double dot product

The double dot product of two tensors is the contraction of these tensors with respect to the last two indices of the first one, and the first two indices of the second one. Whether or not this contraction is performed on the *closest* indices is a matter of convention. In this post, I will show that this choice has some important implications.

## 2014-01-31 – The elastic acoustic tensor and its inverse

In this post, I will introduce the acoustic tensor of linearly elastic materials. Closed-form expressions of the inverse of this tensor can be derived in the case of *isotropic* materials. This will later come in handy to derive closed-form expressions of the periodic Green operator for strains.