text:
Charles Antoine Charpy, né le  à Perpignan et mort le  à Lyon, est un général français et Gouverneur général Thrace orientale.

Biographie
Charles Antoine Charpy naît le  à Perpignan. Il étudie à l'École de Saint-Cyr et à l'École supérieure de Guerre.

En 1888, il entre dans l'armée française.
Il prend le commandement du  d'infanterie le 6 septembre 1914 au début la bataille de la Marne. Il est cité à l'ordre de l'armée pour son action lors des journées des 14, 15 et 16 septembre.
Le général Charpy est le commandant du corps d'occupation français de Constantinople. Il représente la France lors de l'armistice de Mudanya.

Références

Annexes

Bibliographie

Liens externes

Général français du XXe siècle
Grand-croix de la Légion d'honneur
Titulaire de la croix de guerre 1914-1918
Naissance en novembre 1869
Naissance à Perpignan
Décès en mars 1941
Décès à 71 ans


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text:
KLOSTERLAND e.V.
Lehnin
Topics and Offers
Prenzlau Dominican Abbey
After obtaining approval for the foundation of a abbey in 1275, the Dominicans received an area of the margravial estate on which to build the complex. The land was situated in slightly elevated position above the Uckersee lake to the south west of the old town. By 1308 the convent was one of the richest Dominican monasteries in the region. In 1519, the site was damaged by fire, weakening it economically. The abbey was dissolved during the Reformation in 1554.
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Neuste Blogeinträge
KLOSTERLAND: Bald stärker im Rheinland?
Klosterkultur und Megatrends
Kloster Stift zum Heiligengrabe ist neues Mitglied im KLOSTERLAND e.V.
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redpajama_set_name: RedPajamaCommonCrawl


text:
\section{Introduction} \label{sec:one}

The emerging field of Algebraic Vision is concerned with interactions between computer vision and algebraic geometry.
A central role in this endeavor is played by projective varieties that arise in multiview geometry~\cite{HZ}.

The set-up is as follows:
 A \emph{camera} is a linear map from the three-dimensional projective space $\mathbb{P}^3$ to the
 projective plane $\mathbb{P}^2$, both over $\mathbb{R}$.
 We represent $n$ cameras by matrices
$A_1,A_2,\ldots,A_n\in \mathbb{R}^{3\times4}$ of rank~$3$.
The kernel of $A_j$ is the {\em focal point} $f_j \in \mathbb{P}^3$.
Each image point
$u_j \in \mathbb{P}^2$ of camera $A_j$ has a line through $f_j$ as its fiber
in $\mathbb{P}^3$. This is the \emph{back-projected line}.

We assume throughout
that the focal points of the $n$ cameras are
in  {\em general position},
i.e.~all distinct, no three on a line, and no four on a plane.
Let $\beta_{jk}$ denote the line in $\mathbb{P}^3$ spanned by
the focal points $f_j$ and $f_k$.
This is the \emph{baseline} of the camera pair $A_j, A_k$.
   The image of the
focal point $f_j$ in the image plane $\mathbb{P}^2$ of the camera $A_k$ is the \emph{epipole} $e_{k\leftarrow j}$.  Note that the
baseline $\beta_{jk}$ is the back-projected line of $e_{k\leftarrow j}$ with respect to $A_j$ and also the back-projected line of $e_{j\leftarrow k}$ with respect to $A_k$.  See Figure \ref{fig:2view} for a sketch.

\begin{figure}[th]
  \includegraphics[width=\textwidth]{GeometryCamera
   \caption{Two-view geometry}
  \label{fig:2view}
\end{figure}

Fix a point $X$ in $\mathbb{P}^3$ which is not on the baseline $\beta_{jk}$,
and let $u_j$ and $u_k$ be the images of $X$ under $A_j$ and $A_k$.
 Since $X$ is not on the baseline, neither image
point is the epipole for the other camera.  The two back-projected lines of $u_j$ and $u_k$ meet in a unique
point, which is $X$.  This process of reconstructing $X$ from two images $u_j$
and $u_k$ is called \emph{triangulation} \cite[\S9.1]{HZ}.

The triangulation procedure amounts to solving the linear equations
\begin{equation}\label{eq:triangulation}
\quad  B^{jk} \begin{bmatrix}\,X\\-\lambda_j\\-\lambda_k\end{bmatrix} \, = \, 0
\qquad \hbox{where} \quad
B^{jk} \,=\, \begin{bmatrix} A_j & u_j & 0 \\ A_k & 0 & u_k\end{bmatrix} \,\in \,\mathbb{R}^{6 \times 6}.
\end{equation}
For general data we have ${\rm rank}(B^{jk}) =
{\rm rank}(B^{jk}_1) = \! \cdots \! = {\rm rank}(B^{jk}_6)  = 5$,
where $B^{jk}_i$ is obtained  from $B^{jk}$
by deleting the $i$th row.
{\em Cramer's Rule}  can be used to recover $X$.
Let $\wedge_5 B_i^{jk} \in \mathbb{R}^6$ be the column vector formed by the
signed maximal minors of $B^{jk}_i$.  Write
$\widetilde{\wedge}_5 B_i^{jk} \in \mathbb{R}^4$ for the first four coordinates
of $\wedge_5 B_i^{jk}$. These are bilinear functions of
$u_j$ and $u_k$. They yield
\begin{equation}\label{eq:X}
  X \, = \, \widetilde{\wedge}_5 B_1^{jk} \, = \, \widetilde{\wedge}_5 B_2 ^{jk} \, = \, \cdots \, = \,
  \widetilde{\wedge}_5 B_6^{jk}.
\end{equation}
We note that, in most practical applications,
the data $u_1,\ldots,u_n$ will be noisy,
in which case  triangulation
requires techniques from optimization \cite{AAT}.

The \emph{multiview variety} $V_A$ of the camera configuration $A = (A_1,\dots,A_n)$
was defined in \cite{AST} as the closure of the image of the rational map
\begin{equation}
  \label{eq:multiMap}
  \begin{matrix}
    \phi_{A}:&  \mathbb{P}^3  & \dashrightarrow &  \mathbb{P}^2 \times \mathbb{P}^2 \times \cdots \times \mathbb{P}^2, \\
&    X & \mapsto&  (A_1X, A_2X, \ldots, A_nX).
  \end{matrix}
  \end{equation}
The points $(u_1,u_2,\ldots,u_n) \in V_A$ are the
consistent views in $n$ cameras.
The prime ideal $I_A$ of $V_A$ was determined in \cite[Corollary 2.7]{AST}.
It is generated by the $\binom{n}{2}$ bilinear polynomials
${\rm det}(B^{jk})$ plus $\binom{n}{3}$ further trilinear polynomials.
See \cite{Li} for the natural generalization
of this variety to higher dimensions.

The analysis in \cite{AST} was restricted to a single world point $X \in \mathbb{P}^3$.
In this paper we study the case of two world points $X,Y \in \mathbb{P}^3$
that are linked by a distance constraint.
 Consider the hypersurface $V(Q)$ in $\mathbb{P}^3 \times \mathbb{P}^3$  defined by
 \begin{equation}
 \label{eq:biquadratic} \quad
Q \,= \,(X_0Y_3 - Y_0X_3)^2 + (X_1Y_3 - Y_1X_3)^2 + (X_2Y_3 - Y_2X_3)^2 - X_3^2Y_3^2.
\end{equation}
The affine variety $V_{\mathbb{R}}(Q) \cap \{X_3 {=} Y_3 {=}1\}$
in $\mathbb{R}^3 \times \mathbb{R}^3$ consists of
pairs of points whose Euclidean distance is $1$.
The \emph{rigid multiview map} is the rational map
\begin{equation}
\label{eq:rigidMap}
\begin{matrix}
\psi_{A}: & V(Q) & \hookrightarrow & \mathbb{P}^3 \times \mathbb{P}^3 & \dashrightarrow &
(\mathbb{P}^2)^n \times (\mathbb{P}^2)^n, \\
& (X,Y) & & & \mapsto & \bigl((A_1X, \ldots A_nX), (A_1Y, \ldots A_nY)\bigr)  .
\end{matrix}
\end{equation}
The \emph{rigid multiview variety} is the image of this map.  This is a $5$-dimensional subvariety of $(\mathbb{P}^2)^{2n}$. Its
multihomogeneous prime ideal $J_A$ lives in the polynomial ring $ \mathbb{R}[u,v] = \mathbb{R}[u_{i0},u_{i1},u_{i2},\,
v_{i0},v_{i1}, v_{i2}: \, i=1, \ldots, n]$, where $(u_{i0}{:}u_{i1}{:}u_{i2})$
and $(v_{i0}{:}v_{i1}{:}v_{i2})$ are coordinates for the $i$th factor
 $\mathbb{P}^2$ on the left respectively right in
$(\mathbb{P}^2)^n \times (\mathbb{P}^2)^n$.
Our aim is to determine the ideal $J_A$.
Knowing generators of $J_A$ has the potential
of being useful for designing optimization tools
as in \cite{AAT} for triangulation in the presence of distance constraints.

The choice of world and image coordinates
for the camera configuration $A = (A_1,\dots,A_n)$ gives our
problem the following group symmetries.
Let $N$ be an element of the \emph{Euclidean group of motions} $\text{SE}(3,\mathbb{R}) $, which is generated by rotations and translations.
We may multiply the camera configuration on the right by $N$ to obtain
$AN = (A_{1}N,\dots,A_{n}N)$.  Then $J_A = J_{AN}$
since $V(Q)$ is invariant under $\text{SE}(3,\mathbb{R})$.
For $M_{1}, \ldots, M_{n} \in \text{GL}(3,\mathbb{R})$, we may multiply $A$ on the left to obtain
$A' = (M_{1}A, \ldots, M_{n}A)$.  Then $J_{A'} = (M_1 \otimes \ldots \otimes M_n) J_A$.

This paper is organized as follows.  In Section 2 we present the explicit computation of the rigid multiview ideal for
$n=2,3,4$. Our main result, to be stated and proved in Section 3, is a system of equations that cuts out the rigid
multiview variety $V(J_A)$ for any $n$.  Section 4 is devoted to generalizations.  The general idea is to replace $V(Q)$
by arbitrary subvarieties of $(\mathbb{P}^3)^m$ that represent polynomial constraints on $m \geq 2 $ world points.  We focus on
scenarios that are of interest in applications to computer vision.

Our results in Propositions \ref{prop:n2}, \ref{prop:n3}, \ref{prop:n4} and Corollary \ref{cor:mingensH_A} are proved by computations with \macaulay \cite{M2}; for details see Appendix \ref{compu}.
Following standard practice in computational algebraic geometry, we carry out the computation on many samples in a Zariski dense set of parameters, and then conclude that it holds generically.

\section{Two, Three and Four Cameras}\label{sec:234}

In this section we offer a detailed case study of the rigid multiview variety
when the number $n$ of cameras is small.
We begin with the case~$n=2$.  The prime ideal $J_A$ lives in
the polynomial ring $\mathbb{R}[u,v]$ in $12$ variables. This is
the homogeneous coordinate ring of $(\mathbb{P}^2)^4$, so it is naturally $\mathbb{Z}^4$-graded.
The variables $u_{10},u_{11},u_{12}$ have degree $(1,0,0,0)$,
the variables $u_{20},u_{21},u_{22}$ have degree $(0,1,0,0)$,
the variables $v_{10},v_{11},v_{12}$ have degree $(0,0,1,0)$, and
the variables $v_{20},v_{21},v_{22}$ have degree $(0,0,0,1)$.
Our ideal $J_A$ is $\mathbb{Z}^4$-homogeneous.

Throughout this section we shall assume that the camera configuration $A$
is \emph{generic} in the sense of algebraic geometry. This means that $A$
lies in the complement of a certain (unknown) proper algebraic subvariety
in the affine space of all $n$-tuples of  $3 \times 4$-matrices.
All our results in Section 2 were obtained by symbolic computations with
sufficiently many random choices of $A$ (see Appendix \ref{compu} for details).
Such choices of camera matrices are generic. They will be attained with with probability~$1$.

\begin{prop} \label{prop:n2}
  For $n = 2$, the rigid multiview ideal $J_A$ is minimally generated
  by eleven $\,\mathbb{Z}^4$-homogeneous polynomials in twelve variables, one
  of degree $(1,1,0,0)$, one of degree $(0,0,1,1)$, and nine of degree
  $(2,2,2,2)$.
\end{prop}

We prove this result by sufficiently many random computations with \macaulay.
A slightly simplified version of the code is shown in Listing \ref{algo1} in Appendix \ref{compu}.

Let us look at the result in more detail.
The first two bilinear generators are the familiar $6 \times 6$-determinants
\begin{equation}
\label{eq:twobilinear}
{\rm det} \begin{bmatrix} A_1 & u_1 & 0 \\ A_2 & 0 & u_2 \end{bmatrix}
\qquad \hbox{and} \qquad
{\rm det} \begin{bmatrix} A_1 & v_1 & 0 \\ A_2 & 0 & v_2 \end{bmatrix} .
\end{equation}
These cut out two copies of the multiview threefold $V_A \subset (\mathbb{P}^2)^2$, in separate variables, for $X \mapsto u=
(u_1,u_2)$ and $Y \mapsto v= (v_1,v_2)$.  If we write the two bilinear forms in (\ref{eq:twobilinear}) as $u_1^\top F u_2$
and $v_1^\top F v_2$ then $F$ is a real $3 \times 3$-matrix of rank~$2$, known as the {\em fundamental matrix} \cite[\S
9]{HZ} of the camera pair $(A_1,A_2)$.

The rigid multiview variety $V(J_A)$ is a divisor in
$V_A \times V_A \subset (\mathbb{P}^2)^2 \times (\mathbb{P}^2)^2$.  The nine
octics that cut out this divisor can be understood as follows.  We
write $B$ and $C$ for
the  $6 \times 6$-matrices in
(\ref{eq:twobilinear}), and $B_i$ and $C_i$ for the matrices obtained
by deleting their $i$th rows. The kernels of these
$5 \times 6$-matrices are represented, via Cramer's Rule,
by $\wedge_5 B_i$ and $\wedge_5 C_i$.  We write
$\widetilde{\wedge}_5 B_i$ and $\widetilde{\wedge}_5 C_i$ for the vectors
given by their first four entries. As in
(\ref{eq:X}), these represent the two world points $X$
and $Y$ in $\mathbb{P}^3$.  Their coordinates are bilinear forms in
$(u_1,u_2)$ or $(v_1,v_2)$, where each coefficient is a $3 \times 3$-minor of $\begin{small} \begin{bmatrix} A_1 \\
    A_2 \end{bmatrix} \end{small}$.  For instance, writing $a^{jk}_i$ for the $(j,k)$ entry of $A_i$, the first
coordinate of $\widetilde{\wedge}_5 B_1$~is
\[
\begin{small}
\begin{matrix}
-(a_1^{32} a_2^{23} a_2^{34}-a_1^{32} a_2^{24} a_2^{33}-a_1^{33} a_2^{22}
a_2^{34} + a_1^{33} a_2^{24} a_2^{32}+a_1^{34} a_2^{22} a_2^{33}-
a_1^{34} a_2^{23} a_2^{32})\, u_{11} u_{20} \\
+ (a_1^{32} a_2^{13} a_2^{34}-a_1^{32} a_2^{14} a_2^{33}-a_1^{33} a_2^{12}
a_2^{34}+a_1^{33} a_2^{14} a_2^{32}+a_1^{34} a_2^{12} a_2^{33}
- a_1^{34} a_2^{13} a_2^{32}) \,u_{11} u_{21} \\
- (a_1^{32} a_2^{13} a_2^{24}-a_1^{32} a_2^{14} a_2^{23}
-a_1^{33} a_2^{12} a_2^{24}+a_1^{33} a_2^{14} a_2^{22}
+a_1^{34} a_2^{12} a_2^{23}-a_1^{34} a_2^{13} a_2^{22}) \,u_{11} u_{22} \\
+ (a_1^{22} a_2^{23} a_2^{34} - a_1^{22} a_2^{24} a_2^{33}
-a_1^{23} a_2^{22} a_2^{34} + a_1^{23} a_2^{24} a_2^{32} + a_1^{24} a_2^{22}
 a_2^{33}- a_1^{24} a_2^{23} a_2^{32}) \,u_{12} u_{20} \\
 -( a_1^{22} a_2^{13} a_2^{34} - a_1^{22} a_2^{14} a_2^{33}-a_1^{23}
 a_2^{12} a_2^{34} + a_1^{23} a_2^{14} a_2^{32} + a_1^{24} a_2^{12} a_2^{33}
- a_1^{24} a_2^{13} a_2^{32})\, u_{12} u_{21} \\ \,
+(a_1^{22} a_2^{13} a_2^{24}
-a_1^{22} a_2^{14} a_2^{23} -a_1^{23} a_2^{12} a_2^{24}
+a_1^{23} a_2^{14} a_2^{22}+a_1^{24} a_2^{12} a_2^{23}
-a_1^{24} a_2^{13} a_2^{22})\, u_{12} u_{22}.
\end{matrix}
\end{small}
\]

Recall that the two world points in $\mathbb{P}^3$ are linked by a distance
constraint~(\ref{eq:biquadratic}), expressed as a biquadratic
polynomial $Q$. We set $Q(X,Y)=T(X,X,Y,Y)$, where $T( \bullet
,\bullet,\bullet,\bullet)$ is a quadrilinear form.  We regard $T$ as a
tensor of order~$4$.  It lives in the subspace $\, \Sym_2 ( \mathbb{R}^4)
\otimes \Sym_2 ( \mathbb{R}^4) \simeq \mathbb{R}^{100}\,$ of $\,(\mathbb{R}^4)^{\otimes 4}
\simeq \mathbb{R}^{256}$.  Here $\Sym_k(\,\cdot\,)$ denotes the space of
symmetric tensors of order $k$.

We now substitute our Cramer's Rule formulas for $X$ and $Y$
into the quadrilinear form $T$.  For any choice of indices
$1 {\leq} i {\leq} j {\leq} 6$ and $1 {\leq} k {\leq} l {\leq} 6$,
  \begin{equation}
  \label{eq:zweizweizweizwei}
  T\bigl(\,\widetilde{\wedge}_5 B_i\,,\,
    \widetilde{\wedge}_5 B_j\,,\,
    \widetilde{\wedge}_5 C_k \,,\,
    \widetilde{\wedge}_5 C_l\,\bigr)
    \end{equation}
     is a multihomogeneous
    polynomial in $(u_1,u_2,v_1,v_2)$ of
    degree $(2,2,2,2)$.
    This polynomial lies in $J_A$
    but not in the ideal  $I_A(u) + I_A(v)$
    of $V_A \times V_A$, so it can
    serve as one of the nine minimal generators
    described in Proposition~\ref{prop:n2}.

The number of distinct polynomials appearing in (\ref{eq:zweizweizweizwei})
equals $\binom{7}{2}^{2}=441$.
A computation verifies that these polynomials span a real vector space
of dimension $126$. The image of that vector space modulo
the degree $(2,2,2,2)$ component of the ideal $I_A(u)+I_A(v)$
has dimension $9$.

We record three more features of the rigid multiview
with $n=2$ cameras. The first is the {\em multidegree} \cite[\S 8.5]{MS},
or, equivalently, the
cohomology class of $\,V(J_A)\,$ in
$\,H^*\bigl((\mathbb{P}^2)^4 ,\mathbb{Z}\bigr)  =
\mathbb{Z}[u_1,u_2,v_1,v_2]/\langle u_1^3,u_2^3,v_1^3,v_2^3\rangle$.
It equals
\[ \begin{matrix}
\,\, 2u_1^2v_1
+2u_1u_2v_1
+2u_2^2v_1
+2u_1^2v_2
+2u_1u_2v_2
+2u_2^2v_2 \\
+2u_1v_1^2
+2u_1v_1v_2
+2u_1v_2^2
+2u_2v_1^2
+2u_2v_1v_2
+2u_2v_2^2.
\end{matrix}
\]
This is found with the built-in command {\tt multidegree} in \macaulay.

The second is the table of the Betti numbers of the minimal free resolution of~$J_A$ in the format of \macaulay \cite{M2}. In that format, the columns
correspond to the syzygy modules, while rows denote the degrees.
For $n=2$ we obtain
\begin{small}
\begin{verbatim}
                           0  1  2  3 4 5
                    total: 1 11 25 22 8 1
                        0: 1  .  .  . . .
                        1: .  2  .  . . .
                        2: .  .  1  . . .
                        7: .  9 24 22 8 1
\end{verbatim}
\end{small}
The column labeled {\tt 1} lists the minimal generators from Proposition~\ref{prop:n2}.
Since the codimension of $V(J_A)$ is $3$, the table shows that $J_A$ is not Cohen-Macaulay. The unique $5$th syzygy has
degree $(3,3,3,3) $ in the $\mathbb{Z}^4$-grading.

The third point is an explicit choice for the nine generators of degree $(2,2,2,2) $ in Proposition \ref{prop:n2}.
Namely, we take $i=j \leq 3$ and $k=l \leq 3$ in (\ref{eq:zweizweizweizwei}).
The following corollary is also found by computation:

\begin{cor}\label{cor:mingensH_A}
The rigid multiview ideal $J_A$ for $n=2$ is
generated by $\,I_A(u) + I_A(v)$ together with the nine polynomials
  $Q\bigl(\widetilde{\wedge}_5 B_{i},\widetilde{\wedge}_5 C_{k}\bigr )$ for $\,1\leq i,k\leq 3$.
\end{cor}

We next come to the case of three cameras:

\begin{prop}
\label{prop:n3}
For $n = 3$, the rigid multiview ideal $J_A$ is minimally generated by $\,177$ polynomials in $18$ variables.
Its Betti table is given in Table~\ref{tab:n3}.
\end{prop}

\begin{table}
\begin{small}
\begin{verbatim}
          0   1    2    3     4     5     6    7    8   9  10 11
   total: 1 177 1432 5128 10584 13951 12315 7410 3018 801 126  9
       0: 1   .    .    .     .     .     .    .    .   .   .  .
       1: .   6    .    .     .     .     .    .    .   .   .  .
       2: .   2   21    6     .     .     .    .    .   .   .  .
       3: .   .    6   36    18     .     .    .    .   .   .  .
       4: .   .    1   12    42    36     9    .    .   .   .  .
       5: .   1    .    .     .     .     .    .    .   .   .  .
       6: .  24  108  166   120    42     6    .    .   .   .  .
       7: . 144 1296 4908 10404 13873 12300 7410 3018 801 126  9
\end{verbatim}
\end{small}
\smallskip
\caption{\label{tab:n3} Betti numbers for the rigid multiview ideal with $n=3$.}
\end{table}

Proposition \ref{prop:n3} is proved by computation.
The $177$ generators occur in eight symmetry classes
of multidegrees. Their numbers in these classes are
\[
\begin{matrix}
 (110000) : 1 &&    (220111):3 && (220220):9 &&    (211211):1 \\
(111000) : 1 & &    (211111):1 && (220211):3 &&  (111111):1
\end{matrix}
\]
For instance, there are nine generators in degree $(2,2,0,2,2,0)$, arising
from Proposition~\ref{prop:n2} for the first two cameras. Using various pairs
among the three cameras when forming the matrices $B_i, B_j, C_k$ and $C_l$
in (\ref{eq:zweizweizweizwei}), we can construct the generators
of degree classes $(2,2,0,2,1,1)$ and $(2,1,1,2,1,1)$.

Table \ref{tab:n3} shows the Betti table for $J_A$ in \macaulay format.
The first two entries ({\tt 6} and {\tt 2}) in the
{\tt 1}-column refer to the eight minimal generators of $I_A(u) + I_A(v)$.
These are six bilinear forms,
representing the three fundamental matrices, and two trilinear forms, representing the {\em trifocal tensor} of the
three cameras (cf.~\cite{AO}, \cite[\S 15]{HZ}).  The entry
{\tt 1} in row {\tt 5} of column~{\tt 1} marks the unique sextic
generator of $J_A$, which has $\mathbb{Z}^6$-degree $(1,1,1,1,1,1)$.

For the case of four cameras we obtain the following result.

\begin{prop}
\label{prop:n4}
For $n = 4$, the rigid multiview ideal $J_A$ is minimally
generated by $1176$ polynomials in $24$ variables.
All of them are induced from $n=3$.
Up to symmetry, the degrees of the generators in the $\mathbb{Z}^8$-grading are
\[
\begin{matrix}
 (11000000) : 1 &&    (22001110):3 && (22002200):9 &&    (21102110):1 \\
(11100000) : 1 & &    (21101110):1 && (22002110):3 &&  (11101110):1
\end{matrix}
\]
\end{prop}

We next give a brief explanation of how the rigid multiview ideals
$J_A$ were computed with \macaulay \cite{M2}. For the purpose of
efficiency, we introduce projective coordinates for the image points
and affine coordinates for the world points.  We work in the
corresponding polynomial ring
\[\mathbb{Q}[u,v][X_0,X_1,X_2,Y_0,Y_1,Y_2].\]
The rigid multiview map $\psi_A$ is thus restricted to $\mathbb{R}^3 \times
\mathbb{R}^3$.
The prime ideal of its graph is generated by the following two classes
of polynomials:
\begin{enumerate}
\item the $2\times 2$ minors of the $3\times 2$ matrices
\[\left [\begin{array}{c|c}
A_i\cdot (X_0,X_1,X_2,1)^\top&u_i\end{array}
\right ], \enspace \left [\begin{array}{c|c}
A_i\cdot (Y_0,Y_1,Y_2,1)^\top&v_i\end{array}
\right ],
\]
\item the dehomogenized distance constraint
\[Q\bigl((X_0,X_1,X_2,1)^\top,(Y_0,Y_1,Y_2,1)^\top\bigr).\]
\end{enumerate}
From this ideal we eliminate the six
 world coordinates $\{X_0,X_1,X_2,Y_0,Y_1,Y_2\}$.

For a speed up, we exploit the group actions
described in Section \ref{sec:one}.  We replace $A = (A_1, ..., A_n)$ and
$Q = Q(X,Y)$ by $A' = (M_1A_1N, ..., M_nA_nN)$ and
$Q' = Q(N^{-1}X, N^{-1}Y)$.  Here $M_i \in \GL_3(\mathbb{R})$
and $N \in \GL_4(\mathbb{R})$ are chosen so that $A'$ is sparse.
The modification to $Q$ is needed since we generally use $N \notin \text{SE}(3, \mathbb{R})$.
The elimination above now computes the ideal
$(M_1 \otimes \ldots \otimes M_n) J_A$, and it terminates much faster.  For example,
for $n=4$, the computation took two minutes for sparse $A'$
and more than one hour for non-sparse $A$.  For $n=5$, \macaulay
ran out of memory after 18 hours of CPU time for non-sparse $A$. The complete code used in this paper can be accessed via \url{http://www3.math.tu-berlin.de/combi/dmg/data/rigidMulti/}.

One last question is whether the
Gr\"obner basis property in \cite[\S 2]{AST}
extends to the rigid case.
This does not seem to be the case in general.
Only in Proposition~\ref{prop:n2}
can we choose minimal generators
 that form a Gr\"obner basis.

\begin{rem}\label{rem:octic} Let $n=2$. The reduced Gr\"obner basis of $J_A$
in the reverse lexicographic term order is a minimal generating set.
For a generic choice of cameras the initial ideal equals
\[
\begin{matrix}
{\rm in}(J_A) \, = \, \langle \,
 u_{10} u_{20},\,
 v_{10} v_{20},  \,
      u_{10}^2 u_{21}^2 v_{10}^2 v_{21}^2,\,
      u_{10}^2 u_{21}^2 v_{11}^2 v_{20} v_{21},\,
      u_{10}^2 u_{21}^2 v_{11}^2 v_{20}^2, \\ \phantom{da}
          u_{11}^2 u_{20}^2 v_{10}^2 v_{21}^2,\,\,
          u_{11}^2 u_{20} u_{21} v_{10}^2 v_{21}^2,      \,\,
          u_{11}^2 u_{20}^2 v_{11}^2 v_{20} v_{21},  \\ \phantom{dada} \quad
          u_{11}^2 u_{20}^2 v_{11}^2 v_{20}^2,\,
          u_{11}^2 u_{20} u_{21} v_{11}^2 v_{20} v_{21},\,
          u_{11}^2 u_{20} u_{21} v_{11}^2 v_{20}^2 \,\rangle.
\end{matrix}
\]
For special cameras the exact form of the initial ideal may change.
However, up to symmetry the degrees of the generators in the $\mathbb{Z}^4$-grading stay the same.
In general, a universal Gr\"obner basis for the rigid multiview ideal $J_A$ consists of  octics of degree $(2,2,2,2)$ plus the two quadrics  (\ref{eq:twobilinear}).
This was verified using the \gfan \cite{gfan} package in \macaulay.
Analogous statements do not hold for $n \geq 3$.
\end{rem}

\section{Equations for the Rigid Multiview Variety}

The computations presented in Section 2 suggest the following conjecture.

\begin{conj}
\label{conj:atmostdegree8}
{\it The rigid multiview ideal $J_A$ is minimally generated
by $\,\frac{4}{9}n^6-\frac{2}{3}n^5+ \frac{1}{36}n^4
+\frac{1}{2}n^3+\frac{1}{36}n^2-\frac{1}{3}n\,$
polynomials.
These polynomials come from two triples of cameras,
and their  number per class of degrees~is}
\[
\begin{array}{rlrl}
 (110..000..) :& \,1 \cdot 2\binom{n}{2}  &   (220..111..):&\,  3 \cdot  2 \binom{n}{2}\binom{n}{3}\\
(220..220..):&\, 9 \cdot \binom{n}{2}^2   &    (211..211..):&\, 1\cdot n^2 \binom{n-1}{2}^2 \smallskip \\
(111..000..): &\, 1 \cdot 2 \binom{n}{3} &     (211..111..):&\,  1 {\cdot} 2n \binom{n-1}{2} \binom{n}{3}   \\
(220..211..):&\, 3 {\cdot}  2 n \binom{n}{2} \binom{n-1}{2} & (111..111..):&\, 1 \cdot \binom{n}{3}^2
\end{array}
\]
\end{conj}

\setcounter{MaxMatrixCols}{12}
\begin{table}
\begin{center}
\begin{tabular}[H]{r@{\hspace{2em}}rr@{\hspace{1.5em}}rrr@{\hspace{2em}}rrrr@{\hspsace{\textcolor{red}{4em}}}rrr}
\toprule
$n\backslash$degree & 2 & 3 & 6 & 7 & 8 & total & timing (s) \\
\midrule
2 &  2 & \phantom{0} & \phantom{0} & \phantom{0} & 9 &  1 &  $<1$ \\
3 &  6 & 2 &  1 & 24 & 144 & 177 &  14\\
4 & 12 & 8 &  16 & 240 & 900 &  1176 & 130 \\
5 & 20 & 20 &  100 & 1200 & 3600 & 4940 & 24064 \\
\bottomrule
\end{tabular}
\end{center}
\caption{\label{tab:gens}
The known minimal generators of the rigid multiview ideals, listed by total degree, for up to five cameras.
There are no minimal generators of degrees $4$ or $5$.
Average timings (in seconds), using the speed up described above,
are in the last column.}
\end{table}

At the moment we have a computational proof only up to $n = 5$.
Table~\ref{tab:gens} offers a summary of the corresponding numbers of generators.

Conjecture \ref{conj:atmostdegree8}  implies that $V(J_A)$ is set-theoretically
defined by the equations coming from  triples
of cameras. It turns out that,
for the set-theoretic description,  pairs of cameras suffice.
The following  is our main result:

\begin{thm} \label{thm:six} Suppose that the $n$ focal points of $A$ are in general position
in $\mathbb{P}^3$.
The rigid multiview variety $V(J_A)$ is cut out as a subset of $V_A \times V_A $ by the $9
  \binom{n}{2}^2 $ octic generators of degree class $(220..220..)$. In other words, equations coming from any two pairs of
  cameras suffice set-theoretically.
\end{thm}

With notation as in the introduction,
the relevant octic polynomials are
\[T \bigl(\,\widetilde{\wedge}_5 B_{i_1}^{j_1k_1}\,,\,\widetilde{\wedge}_5
  B_{i_2}^{j_1k_1},\widetilde{\wedge}_5 C_{i_3}^{j_2k_2} \, ,\,
  \widetilde{\wedge}_5 C_{i_4}^{j_2k_2} \,\bigr) , \]
  for all possible choices of indices.
  Let $H_A$ denote the ideal generated by these polynomials
  in $\mathbb{R}[u,v]$, the polynomial ring in $6n$ variables.
  As before, we write $I_A(u) + I_A(v)$
  for the prime ideal  that defines the $6$-dimensional
  variety $V_A \times V_A$ in $(\mathbb{P}^2)^n \times (\mathbb{P}^2)^n$.
  It is generated by $2 \binom{n}{2} $ bilinear forms
  and $2 \binom{n}{3}$ trilinear forms, corresponding
  to fundamental matrices and trifocal tensors.
In light of Hilbert's Nullstellensatz,
Theorem \ref{thm:six} states that
the radical of $H_A + I_A(u) + I_A(v)$ is equal to $J_A$.
{To prove this, we need a lemma.

A point $u$ in the multiview variety $V_A\subset(\mathbb{P}^2)^n$ is \emph{triangulable}
 if there exists a pair of indices $(j,k)$ such that the matrix $B^{jk}$ has rank $5$.
 Equivalently, there exists a pair of cameras for which the unique
world point $X$ can be found by triangulation.  Algebraically,
 this means $X = \widetilde{\wedge}_5 B_{i}^{jk}$ for some
$i$.

\begin{lem}\label{l:nonTriangOn}
  All points in $V_A$ are triangulable except for the pair of epipoles, $(e_{1\leftarrow 2},e_{2\leftarrow 1})$, in the
  case where $n=2$.  Here, the rigid multiview variety $V(J_A)$ contains the threefolds
  $V_A(u)\times (e_{1\leftarrow 2},e_{2\leftarrow 1})\,$ and $ \,(e_{1\leftarrow 2},e_{2\leftarrow 1})\times V_A(v)$.
\end{lem}

\begin{proof}
  Let us first consider the case of $n=2$ cameras.  The first claim holds because the back-projected lines of the two
  camera images $u_1$ and $u_2$ always span a plane in $\mathbb{P}^3$ except when $u_1=e_{1\leftarrow 2}$ and
  $u_2=e_{2\leftarrow 1}$.  In that case both back-projected lines agree with the common baseline~$\beta_{12}$.
  Alternatively, we can check
  algebraically that the variety defined by the $5 \times 5$-minors of the matrix $B$ consists of the
  single point $ (e_{1\leftarrow 2},e_{2\leftarrow 1})$.

  For the second claim, fix a generic point $X$ in $\mathbb{P}^3$ and consider the surface
  \begin{equation}
  \label{eq:surface} X^Q \,\, = \,\,
  \bigl\{Y \in \mathbb{P}^3 \,:\, Q(X,Y)=0 \bigr\}.
  \end{equation}
Working over $\mathbb{C}$, the baseline $\beta_{12}$ is either
tangent to $X^Q$, or it meets that quadric in exactly two points.  Our
  assumption on the genericity of $X$ implies that no
  point in the intersection $\beta_{12}\cap X^Q$ is a focal point.
  This gives
  \begin{equation}
  \label{eq:secondclaim}
  (A_1 X, A_2 X, A_1 Y_X, A_2 Y_X) \,=\, (A_1 X, A_2 X, e_{1\leftarrow 2}, e_{2\leftarrow 1}).
  \end{equation}
  The point $(A_1 X, A_2 X)$ lies in the multiview variety $V_A(u)$.  Each generic point in $V_A(u)$ has this form for
  some $X$.  Hence (\ref{eq:secondclaim}) proves the desired inclusion $V_A(u)\times (e_{1\leftarrow 2},e_{2\leftarrow
    1}) \subset V(J_A)$.  The other inclusion $ \,(e_{1\leftarrow 2},e_{2\leftarrow 1})\times V_A(v) \subset V(J_A)$
  follows by switching the roles of $u$ and $v$.

  If there are more than two cameras then for each world point $X$, due to general position of the cameras, there is a
  pair of cameras such that $X$ avoids the pair's baseline.  This shows that each point is triangulable if $n\geq 3$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm:six}]
It follows immediately from the definition of the ideals in question  that the following inclusion
  of varieties holds in $(\mathbb{P}^2)^n \times (\mathbb{P}^2)^n $:
    \[
  V(J_A) \, \subseteq \, V\bigl(I_A(u)+I_A(v)+H_A\bigr).
  \]
  We prove the reverse inclusion.
   Let $(u,v)$ be a point in the right hand side.

   Suppose that $u$ and $v$ are both triangulable. Then $u$ has a unique
  preimage $X$ in $\mathbb{P}^3$,  determined by
  a single camera pair $\{A_{j_1},A_{k_1}\}$. Likewise, $v$ has a unique
  preimage $Y$ in $\mathbb{P}^3$, also determined by
  a single camera pair $\{A_{j_2},A_{k_2}\}$.
  There exist indices $i_1,i_2 \in \{1,2,3,4,5,6\}$ such that
  \[
  X = \widetilde{\wedge}_5 B_{i_1}^{j_1k_1} \quad \hbox{and} \quad
  Y = \widetilde{\wedge}_5 C_{i_2}^{j_2k_2} .
  \]
  Suppose that $(u,v)$ is not in $V(J_A)$.  Then $Q(X,Y) \not= 0$. This
  implies
  \[
  Q(X,Y) \,=\, T(X,X,Y,Y) \,= \,
  T \bigl(
  \widetilde{\wedge}_5 B_{i_1}^{j_1k_1},
  \widetilde{\wedge}_5 B_{i_1}^{j_1k_1},
  \widetilde{\wedge}_5 C_{i_2}^{j_2k_2},
  \widetilde{\wedge}_5 C_{i_2}^{j_2k_2} \bigr) \,\not= \, 0,
  \]
  and hence $(u,v) \not\in V(H_A)$. This is   a contradiction to our choice of $(u,v)$.

  It remains to consider the case where $v$ is not triangulable.  By
  Lemma~\ref{l:nonTriangOn}, we have $n=2$, as well as
  $v=(e_{1\leftarrow 2},e_{2\leftarrow 1})$ and $(u,v)\in V(J_A)$.
  The case where $u$ is not triangulable is symmetric, and this proves
  the theorem.
\end{proof}

The equations in Theorem~\ref{thm:six}
are fairly robust, in the sense that they work as well for
many special position scenarios.
However, when the cameras $A_1,A_2,\ldots,A_n$
are generic then the number $9 \binom{n}{2}^2$ of octics
that cut out the divisor $V(J_A)$ inside $V_A \times V_A$
can be reduced dramatically, namely to $16$.

\begin{cor} \label{cor:onlyfew}
As a subset of the $6$-dimensional ambient space $V_A \times V_A$,
the $5$-dimensional rigid multiview variety $V(J_A)$ is cut out by
$16$ polynomials of degree class $(220..220..)$.
One choice of such polynomials is given by
\[ \quad
\begin{array}{cc}
Q\bigl(\widetilde{\wedge}_5 B^{12}_{i},\widetilde{\wedge}_5 C^{12}_{k}\bigr ), &
Q\bigl(\widetilde{\wedge}_5 B^{12}_{i},\widetilde{\wedge}_5 C^{13}_{k}\bigr )\\
Q\bigl(\widetilde{\wedge}_5 B^{13}_{i},\widetilde{\wedge}_5 C^{12}_{k}\bigr ), &
Q\bigl(\widetilde{\wedge}_5 B^{13}_{i},\widetilde{\wedge}_5 C^{13}_{k}\bigr )
\end{array} \qquad
\hbox{for all $1\leq i,k\leq 2$.}
\]
\end{cor}
\begin{proof}
  First we claim that for each triangulable point $u$ at least one of
  the matrices $B^{12}$ or $B^{13}$ has rank $5$, and the same for $v$
  with $C^{12}$ or $C^{13}$.  We prove this by contradiction. By
  symmetry between $u$ and $v$, we can assume that
  $\rk(B^{12})=\rk(B^{13})=4$. Then $u_3=e_{3\leftarrow 1}$,
  $u_2=e_{2\leftarrow 1}$, and $u_1=e_{1\leftarrow 2}=e_{1\leftarrow
    3}$.  However, this last equality of the two epipoles is a
  contradiction to the hypothesis that the focal points of the cameras
  $A_1,A_2,A_3$ are not collinear.

  Next we claim that if $B^{12}$ has rank $5$ then at least one of the
  submatrices $B^{12}_{1}$ or $B^{12}_{2}$ has rank $5$, and the same
  for $B^{13}$, $C^{12}$ and $C^{13}$.  Note that the bottom $4
  {\times} 6$ submatrix of $B^{12}$ has rank $4$, since the first four
  columns are linearly independent, by genericity of $A_1$ and $A_2$.
  The claim follows.
\end{proof}

\section{Other Constraints, More Points, and No Labels}

In this section we discuss several extensions of
our results. A first observation is that there was
nothing special about the constraint $Q$ in
(\ref{eq:biquadratic}). For instance, fix positive
integers $d$ and $e$, and let
$Q(X,Y)$ be any irreducible polynomial that is bihomogeneous of degree $(d,e)$.
Its variety $V(Q)$ is a hypersurface of degree $(d,e)$ in $\mathbb{P}^3 \times \mathbb{P}^3$.
The following analogue to Theorem \ref{thm:six} holds, if we define the map $\psi_A$ as in (\ref{eq:rigidMap}).

\begin{thm} \label{thm:seven} The closure of the image of the map $\psi_A$ is cut out in $V_A \times V_A $ by $\,9
  \binom{n}{2}^2 $ polynomials of degree class $(d,d,0,\ldots,e,e,0,\ldots)$. In other words, the equations coming from
  any two pairs of cameras suffice set-theoretically.
\end{thm}

\begin{proof}
The tensor $T$  that represents $Q$ now lives in
$\Sym_d(\mathbb{R}^4) \otimes \Sym_e(\mathbb{R}^4)$.
The polynomial  $(\ref{eq:zweizweizweizwei})$ vanishes
on the image of $\psi_A$ and has degree $(d,d,e,e)$.
The proof of Theorem~\ref{thm:six} remains valid.
The surface $X^Q$  in (\ref{eq:surface})
is irreducible of degree $e$ in $\mathbb{P}^3$.
These polynomials cut out that image inside $V_A \times V_A$.
\end{proof}

\begin{rem}
In the generic case,
we can replace $9\binom{n}{2}^2$ by 16,
as in Corollary~\ref{cor:onlyfew}.
\end{rem}

Another natural generalization is to consider $m$ world points
$X_1,\ldots,X_m$ that are linked by one or several
constraints in $(\mathbb{P}^3)^m$. Taking images with $n$
cameras, we obtain a variety $V(J_A)$ which lives in $(\mathbb{P}^2)^{mn}$.
 For instance, if $m=4$ and $X_1,X_2,X_3,X_4$
are constrained to lie on a plane in $\mathbb{P}^3$,
 then $\,Q = {\rm det}(X_1,X_2,X_3,X_4)\,$ and
  $V(J_A)$ is a variety of dimension $11$ in $(\mathbb{P}^2)^{4n}$.
  Taking $ 6 {\times} 6$-matrices $B,C,D,E$ as in
(\ref{eq:triangulation}) for the four points, we then form
\begin{equation}
\label{eq:fourbyfour}
\quad  \det \bigl(\widetilde{\wedge}_5 B_i,
 \widetilde{\wedge}_5 C_j,
  \widetilde{\wedge}_5 D_k,
\widetilde{\wedge}_5 E_l \bigr)\qquad \text{for all } 1\leq i,j,k,l\leq 6.
\end{equation}
For $n=2$ we verified with \macaulay that the prime ideal $J_A$ is generated by $16$ of these determinants, along with
the four bilinear forms for ${V_A}^4$.

\begin{prop}
  The variety $V(J_A)$ is cut out in ${V_A}^4 $ by the
   $\,16 \binom{n}{2}^4$ polynomials
    from~{\rm (\ref{eq:fourbyfour})}.  In other words, the equations coming
  from any two pairs of cameras suffice set-theoretically.
\end{prop}
\begin{proof}
  Each polynomial~(\ref{eq:fourbyfour}) is in $J_A$.  The proof of Theorem~\ref{thm:six} remains valid.  The planes
  $(X_i,X_j,X_k)^Q$ intersect the baseline $\beta_{12}$ in one point each.
\end{proof}
\smallskip

To continue the theme of rigidity, we may
impose distance constraints on pairs
of points. Fixing a nonzero distance $d_{ij} $
between points  $i$ and $j$ gives
\begin{small}
\[
 Q_{ij} =     (X_{i0} X_{j3} - X_{j0} X_{i3})^2
+ (X_{i1} X_{j3} - X_{j1} X_{i3})^2
+ (X_{i2} X_{j3} - X_{j2} X_{i3})^2
- d_{ij}^2 X_{i3}^2 X_{j3}^2.
\]
\end{small}
We are interested in the image of the variety
$\mathcal{V} = V(Q_{ij}: 1 \leq i < j \leq m)$
under the multiview map $\psi_A$ that takes
 $(\mathbb{P}^3)^m$  to $(\mathbb{P}^2)^{mn}$.
For instance, for $m=3$, we consider the
variety $\mathcal{V} = V(Q_{12},Q_{13},Q_{23})$ in $(\mathbb{P}^3)^3$,
and we seek the equations for its image
under the multiview map $\psi_A$ into $(\mathbb{P}^2)^{3n}$.
Note that $\mathcal{V}$ has dimension $6$, unless
we are in the collinear case. Algebraically,
\begin{equation}
\label{eq:threecollinear} \,\,
(d_{12}+d_{13}+d_{23})
(d_{12}+d_{13}-d_{23})
(d_{12}-d_{13}+d_{23})
(-d_{12}+d_{13}+d_{23}) \,\,=\,\, 0.
\end{equation}
If this holds then ${\rm dim}(\mathcal{V}) = 5$.
The same argument as  in Theorem \ref{thm:six} yields:

\begin{cor}  The rigid multiview variety  $\overline{\psi_A(\mathcal{V})}$
has dimension six, unless (\ref{eq:threecollinear}) holds,
in which case the dimension is five.
It has real points if and only if $d_{12}, d_{13}, d_{23}$ satisfy the triangle inequality.  It is cut out in ${V_A}^3$ by $27\binom{n}{2}^2$ biquadratic equations,
coming from the $9 \binom{n}{2}^2$ equations for any two of the three points.
\end{cor}

In many computer vision applications, the $m$ world points and their
images in $\mathbb{P}^2$ will be unlabeled. To study such questions, we
propose to work with the {\em unlabeled rigid multiview variety}. This
is the image of the rigid multiview variety under the quotient map
$\bigl((\mathbb{P}^2)^m \bigr)^n \rightarrow\bigl( \Sym_m(\mathbb{P}^2)\bigr)^n$.

Indeed, while labeled configurations in the plane are points in
$(\mathbb{P}^2)^m $, unlabeled configurations are points in the {\em Chow
  variety} $\Sym_m(\mathbb{P}^2)$.  This is the variety of ternary forms that
are products of $m$ linear forms (cf.~\cite[\S 8.6]{Lan}). It is
embedded in the space $ \mathbb{P}^{\binom{m+2}{2}-1} $ of all ternary forms
of degree~$m$.

\begin{exmp}
\label{chow}
Let $m=n=2$. The Chow variety $\Sym_2(\mathbb{P}^2)$
is the hypersurface in $\mathbb{P}^5$ defined by the
determinant of a  symmetric $3 \times 3$-matrix $(a_{ij})$.
The quotient map $(\mathbb{P}^2)^2 \rightarrow \Sym_2(\mathbb{P}^2)
\subset \mathbb{P}^5$
is given by the formulas
\[
\begin{matrix}
a_{00} = 2 u_{10} v_{10}, &
a_{11} = 2 u_{11} v_{11}, &
a_{22} = 2 u_{12} v_{12}, \\
a_{01} = u_{11} v_{10}+u_{10} v_{11} , &
a_{02} = u_{12} v_{10}+u_{10} v_{12}, &
a_{12} = u_{12} v_{11}+u_{11} v_{12}. \\
\end{matrix}
\]
Similarly, for the two unlabeled images under the second camera we use
\[
\begin{matrix}
b_{00} = 2 u_{20} v_{20}, &
b_{11} = 2 u_{21} v_{21}, &
b_{22} = 2 u_{22} v_{22}, \\
b_{01} = u_{21} v_{20}+ u_{20} v_{21} , &
b_{02} = u_{22} v_{20}+ u_{20} v_{22}, &
b_{12} = u_{22} v_{21}+ u_{21} v_{22}. \\
\end{matrix}
\]
The unlabeled rigid multiview variety is the image of $V(J_A) \subset
V_A \times V_A$ under the quotient map that takes two copies of
$\,(\mathbb{P}^2)^2 \,$ to two copies of $\Sym_2(\mathbb{P})^2 \subset \mathbb{P}^5$.  This
quotient map is given by $(u_1,v_1) \mapsto a, \, (u_2,v_2) \mapsto
b$.

We first compute the image of $V_A \times V_A$
in $\mathbb{P}^5 \times \mathbb{P}^5$,
denoted $\Sym_2(V_A)$. Its ideal
has seven minimal generators,
three of degree $(1,1)$, and one each in
degrees $(3,0), (2,1), (1,2), (0,3)$.
The generators in degrees $(3,0)$ and $(0,3)$
are  ${\rm det}(a_{ij})$ and ${\rm det}(b_{ij})$.
The five others depend on the cameras $A_1, A_2$.

Now, to get equations for the unlabeled rigid multiview variety, we
intersect the ideal $J_A$ with the subring $\mathbb{R}[a,b]$ of bisymmetric
homogeneous polynomials in $\mathbb{R}[u,v]$. This results in nine new
generators which represent the distance constraint.  One of them is a
quartic of degree $(2,2)$ in $(a,b)$.  The other eight are quintics,
four of degree $(2,3)$ and four of degree $(3,2)$.
\end{exmp}

Independently of the specific constraints considered in this paper, it
is of interest to characterize the pictures of $m$ unlabeled points
using $n$ cameras.  This gives rise to the {\em unlabeled multiview
  variety} $\,\Sym_m(V_A) $ in $
{\bigl(}\mathbb{P}^{\binom{m+2}{2}-1}{\bigr)}^{n} $. It would be desirable to
know the prime ideal of $\Sym_m(V_A) $ for any $n$ and~$m$.

\bigskip
\medskip

\noindent {\bf Acknowledgements.}\ This research is carried out in the framework of Matheon supported by Einstein Foundation Berlin. Further support by Deutsche
Forschungsgemeinschaft (Priority Program 1489: ``Experimental methods
in algebra, geometry, and number theory'') and
the US National Science Foundation (grant DMS-1419018).

redpajama_set_name: RedPajamaArXiv


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Tom Boonen (Etixx-Quick Step) is planning to race in 2017 and dismissed rumours that he would retire when his contract expires at the end of the season.
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Journal of mass spectrometry : JMS. 2008 Nov; 43(11): 1458-69.
The presence and position of a single beta-alanine (betaA), gamma-aminobutyric acid (gammaABu) or epsilon-aminocaproic acid (Cap) residue has been shown to have a significant influence on the formation of b(n)+ and y(n)+ product ions from a series of model, protonated peptides. In this study, we examined the effect of the same residues on the formation of analogous [b3 - 1 + cat]+ products from metal (Li+, Na+ and Ag+)-cationized peptides. The larger amino acids suppress formation of b3+ from protonated peptides with general sequence AAXG (where X = beta-alanine, gamma-aminobutyric acid or epsilon-aminocaproic acid), presumably because of the prohibitive effect of larger cyclic intermediates in the 'oxazolone' pathway. However, abundant [b3 - 1 + cat]+ products are generated from metal-cationized versions of AAXG. Using a group of deuterium-labeled and exchanged peptides, we found that formation of [b3 - 1 + cat]+ involves transfer of either amide or alpha-carbon position H atoms, and the tendency to transfer the atom from the alpha-carbon position increases with the size of the amino acid in position X. To account for the transfer of the H atom, a mechanism involving formation of a ketene product as [b3 - 1 + cat]+ is proposed.


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Feed to a cow for an extra 5 coins when selling the milk, or still more if the milk is processed.
Feed to a Turkey for an extra 115 coins when selling black feathers, or more with white feathers.
Harvesting clovers plots enable to earn Game Clover ribbons.


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text:
Elizabeth Glaser Pediatric AIDS Foundation Awards
Remarks by First Lady Hillary Rodham Clinton
Washington Press Club
Thank you all. Thank you. Well, it is my honor to be here, once again, for one of the most important events that I think goes on in Washington every year, as we take the energy and the generosity of individuals, corporations, and foundations in a great show of philanthropic spirit and put that to work on behalf of the extraordinary, groundbreaking work that these scientists and their colleagues are doing. I am grateful to be here with all of you and to have a chance, once again, to thank those of you who support the foundation and the work that Elizabeth started so many years ago.
I want to thank my friend Paul, whom I appreciate for not embarrassing me. But more than that I want to thank him for his courage, commitment, and his support and friendship. I'm honored to be with him once again as we recognize the outstanding work that the Elizabeth Glaser Pediatric AIDS Foundation, and these scientists, are doing to push the frontiers of science even further.
Elizabeth Glazer, Susan DeLaurentis and Susie Zeegen
Founders of the Elizabeth Glazer Pediatric AIDS Foundation
I want to thank the remarkable leaders of this foundation, who have helped to build it into what it is today. I know that Suzie Zeegan is often very modest about her contributions, but she has been a stalwart, steadfast voice and heart of this organization for so many years. I'm glad that she is joined by Kate Carr and Janis Spire, and all of you who are on the board and involved in the work of the foundation.
And I want to say a personal word of appreciation and gratitude to Mary Fisher. Her tireless work, her eloquent powerful voice has meant a lot to many of us, and I know that she will not rest until we do all that we can on behalf of the work that she so well represents.
None of us would be here today—not Suzie, not Paul, not me, not Mary—none of us, were it not for the inspirational life and work of our friend Elizabeth Glaser. Her spirit animates this gathering, as it does any time we come together. No one fought harder—or more—to bring attention and resources to the issue of pediatric AIDS. No one who ever heard her speak or met her will ever forget that fabulous laugh and that very persistent effort that she put into making it clear to all she met that this was an issue that was not just confined to those who were suffering from the tragedy of HIV and AIDS—and particularly as it affected children—but for every human being. And that work that she started and that spirit that is still with us has brought hope to millions and millions of children and adults who suffer from this disease. She transformed literally the way this country responds to HIV and AIDS.
It was something that started very simply, sitting around a kitchen table with Suzie and Susan, and deciding to undertake what must, at that time, have seemed like an impossible task. But once again, one person and a small group of people have made a tremendous difference. And as Margaret Mead famously once said, "It is only a small group of people who ever make a difference in changing the direction of the world."
So here we are to celebrate the 10th anniversary, a "decade of progress, a decade of promise," and to go forward into this upcoming decade with the same level of commitment, motivation, and dedication. And there is no group that has done more than this foundation to make it possible for us to say that we are looking at a decade of promise.
Ten years ago, almost no one knew about AIDS, let alone that it could be transmitted from mother to child. There was ignorance; there was fear. And we have come a very long way indeed. We've come to the point where we can say that there is a pediatric AIDS research agenda and that we can point to the progress that all of us have seen made because of the dedication of scientists.
We can look back at the decade of breakthroughs that have dramatically reduced mother-to-child transmission and greatly improved the quality of life and life expectancy of infected children. There are also more drugs—once only available to adults—that are now benefiting children.
I want to underscore something Paul said and thank all of you who joined us in our fight to make pediatric labeling a reality. It was a long struggle, but it was one that we could not have achieved without the support of this foundation and many of you here today. Now we have a regulation in place assuring that prescription drugs are tested for safety and efficacy in children. This regulation will ensure that doctors know which drugs to prescribe and how much to give to their youngest patients. We believe this initiative has the potential for saving countless lives, and we could not have done it without you.
In the past several months, the Clinton Administration has also launched new initiatives to address AIDS in minority communities, to strengthen our efforts to find a vaccine, and to boost our international efforts to care for children orphaned by AIDS. Those are three very important initiatives, and let me just briefly address them.
As I look around this room, I see that there are not many representatives of minority communities, which are the hardest hit part of the American population. And certainly if we look abroad and we see what HIV/AIDS is doing in Africa and Asia, it makes that emphasis even more urgent.
The effort to find a vaccine, which these scientists to some extent are focusing their work on, is something that the United States is even more committed to. And working with partners around the world, we hope we will see progress.
And finally, international efforts to care for children orphaned by AIDS are extremely important. I've seen varying statistics, but I keep seeing the number: 40 million orphans from AIDS in the next ten years.
So we've come together to celebrate what we've achieved, but also to look towards the future. And we have a lot to be thankful for when we look at the health care of our children. In the last six years, because of the leadership of this President and the Administration and the supporters in Congress and elsewhere, we've seen improvements in the health of America's children. Immunization rates are at an all time high, infant mortality at an all time low. And thanks to the President's program for children's health insurance called "CHIP," millions more children are eligible for coverage, and we have to make sure that we reach the 5 million children who are eligible but not currently enrolled.
I'm also pleased to announce today that the President's fiscal year 2000 budget includes a new $40 million grant program to support graduate medical education. This is something that has worried me for quite sometime. Some of you may know that graduate medical education—how we fund the training of our physicians, our research scientists, our hospitals, and our medical centers—is funded by Medicare. And what we have found over the last several years is that the funding for those who work with children and are being trained to work with children is grossly disproportionate to the need. I can't remember the exact figures, but something like over $9,000 a year is spent out of Medicare to fund the training and work of physicians who work with adults, and about $400 a year for those physicians working with children. We have worked for quite sometime to change that, and I am very proud that the President's budget will make a start on that.
There is no way we could talk about treating children with HIV or AIDS or any chronic or acute condition if we don't have the doctors and the trained staff ready and able to treat them throughout our country. So we need to do more to train doctors who care for our children and provide vital health services for the poorest, the sickest, and the most vulnerable among our children.
Too long children's hospitals have been trying to carry this responsibility without adequate resources. I ran into Senator Kennedy as I got off the elevator. Senator Kennedy and others have introduced a bipartisan piece of legislation to make permanent the funding of those who work for children's health and I hope, in addition to the grant, we will see a permanent solution to this problem.
But we have a lot of work ahead of us. Here, and around the world especially, new HIV infections are increasing, and women of color and young people are among the most vulnerable. Every hour of every day, two more American adolescents are infected with HIV. And around the world, 1,600 children are infected every single day.
As we all know, the ravages of pediatric AIDS are the most severe in the developing countries, where over 90 percent of new infections now occur. By the end of this year, 40 million people will be living with HIV or AIDS. And by the end of the next decade, AIDS will have orphaned over 40 million children.
Now when you're confronted with statistics like these, it is sometimes difficult not to feel overwhelmed. But the founders of this foundation and this effort, and Elizabeth herself, would not accept that as an excuse at all. The legacy that Elizabeth leaves us is one that is driven by and shaped by hope, not despair; one that chooses action over inaction; one that is determined, regardless of the odds, to find effective treatments, a vaccine, and eventually a cure.
Now we know that that can only happen if we have the kind of support and dedication of the scientists that we honor today, and I am very proud to ask them to stand and receive their individual awards. These gifted individuals represent some of the brightest investigators from the international research community. While they work at separate research institutions, they all share a deep commitment to not only furthering their own work, but collaborating with those who are their partners here and around the world.
The first recipient is Dr. Robert Doms from the University of Pennsylvania. Dr. Doms, will you please come forward? Dr. Doms is investigating how a cell gets infected with HIV. And I told Dr. Doms and the other scientists that I read the summary of their work, and as a lay person, I have to confess I wasn't quite sure what CCR5 was, what it did or what it didn't do, but I know how important the work is that each of them is engaged in. Dr. Doms is attempting to discover how we can block the infection by learning how the cell gets infected in the first place—which would then lead to the development of an effective vaccine. And so on behalf of all us, and literally on behalf of millions of people who will maybe never know your name, Dr. Doms, but will benefit from your work, congratulations! [The First Lady presents the award.]
Dr. Philip Goulder from Massachusetts General Hospital is investigating the immune response of AIDS-infected children in the United States and Africa to bring us closer to the discovery of an AIDS vaccine. I want, particularly, to thank Dr. Goulder and those who work with him for partnering with scientists and physicians in Africa where the need is so acute, and where we have an opportunity to make a tremendous difference. Because, as so many of you know, the progress that we've made in extending life expectancy and quality of life for many people living with HIV/AIDS is dependent upon a very expensive drug regiment which is out of the reach of people in most of the rest of the world. So the work that is being done by Dr. Goulder and others is critical to bringing hope to people in parts of the world that now don't have an opportunity to benefit from what has already been achieved. Dr. Goulder—congratulations. [The First Lady presents the award.]
Next is Dr. Julie Overbaugh from Seattle's Fred Hutchinson Cancer Research Center. She is collaborating with other researchers to further reduce mother-child HIV transmission— particularly during breastfeeding. You know, we spent years persuading women in developing countries to breast feed. This was one of the great causes of the 1970's and early 80's, to persuade them that their breast milk was more effective for their children to build up their child's immune system, and not to be persuaded by advertisements promoting the use of formula when they didn't even have clean water that they could mix with the formula. But, of course, now we are trying to figure out what we do with that success, because the breast feeding, which is very important for many reasons, is one of the ways that HIV is transmitted. As part of the work that Dr. Overbaugh is doing, she is studying HIV-infected infants and mothers in Nairobi, Kenya. And I want to thank her also for helping to focus on the issue in Africa. Congratulations. [The First Lady presents the award.]
Dr. Paul Johnson from Harvard Medical School is researching new ways to help boost the immune system's response to HIV infection in children. And that is something else that we are so hopeful about, to learn more about what can be done to fend off HIV once it enters into a child's system, to prolong life and really do everything we can to diminish the impact of HIV and postpone or eliminate the development of AIDS. Thank you, Dr. Johnson for your important work. [The First Lady presents the award.]
You know, I feel, as I hand out these awards to these scientists, the way I felt when I first went to a doctor younger than myself. These four incredible scientists are so young, and for all of us we think of that as a very hopeful sign that the research that is going on is engaging the attention and the intelligence and expertise of our young scientists and researchers. And certainly by evidence of them today and the work that they are doing, we have every reason to be hopeful. So let me on behalf of the foundation thank all of you who have been part of this work, who have been involved in carving out this very unique contribution that the foundation is making to furthering scientific endeavor and to highlighting the work that these four scientists are doing.
It is a pleasure and an honor for me to be part of this presentation once again, and to do all that I can along with all of you to make it possible for us to move forward against this disease, and do everything we can to protect our children from it, and to help those who already suffer from it to lead lives of quality as they try to fulfill their entire God-given promise.
* Related Links *
Office of the First Lady
Pediatric AIDS Foundation


text:
Treatise on Christian Doctrine: Compiled From the Holy Scriptures Alone
Fb&c Limited, 2015 M06 26 - 765 páginas
Excerpt from Treatise on Christian Doctrine: Compiled From the Holy Scriptures Alone
To enter into a preliminary discussion of the doctrines or opinions contained in the present volume, seems, properly speaking, to be no necessary part of the Translator's duty. After stating, therefore, in the first place, the circumstances under which the original manuscript was discovered, and the reasons for considering it as the long lost theological work of Milton, it will be sufficient to subjoin, as briefly as possible, a few remarks chiefly relating to certain peculiarities in the following treatise, by which it is distinguished from the authors other compositions.
From information communicated by Robert Lemon, sen. Esq. Deputy Keeper of His Majesty's State Papers, who has lately completed from the documents under his care an entire series of the Order-Books of the Council of State during the Interregnum, it appears that Milton retired from active official employment as Secretary for Foreign Languages, about the middle of the year 1665. The following entry occurs under the date of April 17 in that year:
"The Councell resumed the debate upon the report made from the Committee of the Councell to whom it was referred to consider of the establishment of the Councell's contingencies.
"Ordered. . . That the former yearly Salary of Mr. John Milton, of Two Hundred Eighty-Eight Pounds, &c., formerly charged on the Councell's contingencies, be reduced to One Hundred and Fiftie Pounds per annum, and paid to him, during his life, out of His Highness' Exchequer."
This sum must have been intended as a retiring pension in consideration of past services, as it is evident from another entry, under the same date, that a successor was already appointed, at a reduced salary, to discharge the duties of the situation which Milton had previously occupied.
Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com
This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.
A Treatise on Christian Doctrine: Compiled from the Holy ..., Volumen1
Título Treatise on Christian Doctrine: Compiled From the Holy Scriptures Alone
Autor John Milton
Edición ilustrada
Editor Fb&c Limited, 2015


text:
Q: Adverbs for reduced relative clauses I wrote:

A caption is usually a short text enclosed by a block-level HTML element, immediately following a table or an image.

"following a table or an image" is a reduced relative clause in which I want to use the adverb "immediately". I would like to know if I used it in correct position.

A: First, the sentence isn't entirely correct, not because it's ungrammatical, but because it isn't precisely the way that HTML captions work.  I'll explain how captions work in HTML, so we can be sure that you are saying what you want to say.
The HTML
 element only works with tables, not images.  It has to be enclosed in a
 block (if you use it outside of a table block it is ignored), and defaults to being above the table (you can use CSS to reposition it below the table). The HTML
 element (HTML5 only) is used with images.  Here are two code samples, borrowed from w3schools.com.  This is a table:

Monthly savings

    Month
    Savings

    January
    $100

Here's an image:

Fig.1 - A view of the pulpit rock in Norway.

All right.  Now, your statement is saying that the caption is a short string of text positioned below the image or table.  If that is what you want to say, it is grammatically correct.  However, as I have explained, it is not factually completely accurate.
As Ustanak has observed in the comments, you may also say immediately followed by a table or image, but that has the opposite meaning: it means that the caption is on top and the table or image is underneath it.  Again, this is the default behavior with tables.  With images, it's simply a matter of whether you put the
 tag first, or the  tag first.  (In the code sample, the caption is below the image, as is typical.)
These two sentences should make the difference clear:

Joe immediately followed Pete in the parade.  (Joe was right behind Pete.)
  Joe was immediately followed by Pete in the parade.  (Pete was right behind Joe.)

redpajama_set_name: RedPajamaStackExchange


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text:
> Weddings
Jamie Lynn Spears debuts wedding ring on red carpet with husband Jamie Watson
April 07, 2014 - 15:23 BST hellomagazine.com Jamie Lynn Spears made her first red carpet appearance since becoming a married woman, showing off her wedding ring
Jamie Lynn Spears has made her first red carpet appearance since becoming a married woman.
The 23-year-old singer rocked up to the Academy Of Country Music Awards in Las Vegas, accompanied by her doting husband Jamie Watson.
Jamie Lynn looked like she was on cloud nine as she walked up the carpet in a geometric pattern, sequined dress. The blonde bombshell showed off her petite frame and toned legs in the figure-hugging number and completed her look with neon red heels.
Jamie Lynn Spears at the Academy Of Country Music Awards
All attention was focused on Jamie Lynn's latest accessory though — her beautiful wedding ring. The thick silver band sat alongside her monster engagement diamond rock which sparkled as she posed for photographs.
Fans of the Mississippi-born star first saw Jamie Lynn's engagement ring when she posted a photo on Twitter of her and Jamie announcing their happy news. "Guess what?" she captioned it, followed by another close-up snap of the impressive round-cut jewel.
Jamie Lynn's show-stopping wedding cake: get all the details here
Jamie Lynn left all attention to her wedding and engagement rings
At Sunday's event in Sin City, it was a mix of business and pleasure for Jamie Lynn as she took on the role of social media correspondent for the awards. The night seems to have been successful as she thanked her country musician Kellie Pickler on the night.
"Thank you @KelliePickler for being so sweet and guiding me thru my first #ACMs!" she tweeted, next to a photo of the pair.
Jamie Lynn's wedding photos revealed
Jamie Lynn and Jamie tied the knot last month
Speaking on the red carpet, Jamie Lynn also opened up about her debut song How Could I Want More and her pop icon sister Britney Spears.
"I worked on this project for a really long time," she said. "The single means a lot to me, and I was finally brave enough to put it out there.
"Your sister is always someone you're going to look up to. We're in the same kind of industry but different genres, so definitely it's good to take her advice."
Jamie Lynn and Jamie tied the knot in a romantic ceremony in New Orleans, surrounded by their nearest and dearest. Britney played the starring role of maid of honour, while Jamie Lynn's five-year-old daughter Maddie, from her previous relationship with Casey Aldridge, looked picture perfect as a bridesmaid.
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Psychological and Brain Sciences program
Hanover, United States
Study mode:Full-time Languages: english Duration:5 years
Deadline: Dec 15, 2021
Photos of university / #dartmouthcollege
The Graduate Program in Psychological and Brain Sciences is divided into three programmatic groups: Social and Affective Neuroscience; Cognitive and Computational Neuroscience; and Systems and Behavioral Neuroscience. Although most requirements are similar for the three groups, each group can set independent requirements beyond the minimum standards set by the Psychological and Brain Sciences Graduate Program. Regardless of the group in which the student completes his/her training, the student will receive his/her PhD in Psychology. Students who complete additional requirements can opt to receive their PhD in Cognitive Neuroscience. The Department of Psychological and Brain Sciences maintains close ties and affiliations with a number of other departments and programs, including the Dartmouth Medical School and the Neuroscience Center at Dartmouth. Coursework and research is often conducted in conjunction with these other programs.
During the first year students take Proseminar, which is taught by the entire faculty. This seminar explores the diverse areas of psychological and brain sciences represented in the department. First-year students also begin to take a series of Core Courses and conduct supervised research.
Second-year students receive training in statistics, enroll in focused seminars, carry out independent research and complete a Specialist Examination in their area before proceeding to their dissertation research in subsequent years of the program. Students serve as teaching assistants for four terms during their time in the program.
An applicant must have completed the baccalaureate degree or its equivalent prior to matriculation. The strength of an applicant's academic background and evidence of promise in graduate study is more important than the area of undergraduate study. If an individual has not been exposed to some background material, the student will be encouraged to correct these deficiencies during his or her first year in the program. Applicants must submit the following materials for completion of the application:
A completed application form, including a concise statement of previous research experience and of research interests.
An official copy of college transcripts from previous academic institutions.
Results from the General Test of the Graduate Record Examination (GRE). The GRE should be taken no later than November, and the results should be sent to the Graduate Office. The Subject Test in Psychology is encouraged but not required.
Three letters of recommendation from persons having knowledge of the training and experience of the applicant. These letters should be uploaded via the link above. Instructions are available on the page after you sign in.
A nonrefundable application fee of $50.
Dartmouth offers full financial support to all students in doctoral programs. Financial support includes a paid yearly stipend or fellowship at a minimum of $24,000. Students generally do not need to pay any tuition from their stipend as doctoral tuition is typically paid, or scholarship arrangements are made. Financial support is usually supplied for the entire duration of a student's enrollment in their doctoral program, although support terms may differ slightly by program.
Psychology (Organizational Psychology)
Minneapolis, United States
Rochester, United States
Deadline: Dec 1, 2021 StudyQA ranking: 447
Evanston, United States
Local:$ 67.2k / 1 Academic year(s) Foreign:$ 67.2k / 1 Academic year(s)
Stanford, United States
Deadline: Nov 24, 2021 StudyQA ranking: 596
South Bend, United States
Deadline: Dec 15, 2021 StudyQA ranking: 1499
Medford, United States
Foreign:$ 44.2k / 1 Academic year(s)
Deadline: Dec 15, 2021 StudyQA ranking: 535
Clinical Psychology Health
Deadline: Jan 15, 2022 StudyQA ranking: 1063
School-Clinical Child Psychology
Deadline: Feb 1, 2021 StudyQA ranking: 976
dartmouthcollege


Locating that moms day present can be very catchy, exactly as with any other event. If your mom is anything like my mine when you inquire presently she desires the regular answer is you're "I really don't need anything", thanks mommy… but that does not actually help.
It's simple when she needs something specifically (in reason of course!) But while you've got no idea you're actually in a dead end. Would you only need to have flowers for your mom? I mean blossoms are good yes, but particular and unique?
No. I believe this year I will receive my mother's something that she'll recall, I do not actually enjoy the notion of having flowers and chocolates since them kind of gifts is boring and overdone. So what other kinds of chances are there to select from?
A brand new trendy gift to provide in the past couple of years experiences times, they're essentially days outside. What's great about times out is that the surprise variable it is going to provide, will your mother be expecting you to figure outside the box and state treat her into some pampering session at a nearby spa?
She'd love a day outside like that; it'd give her something to keep in mind, something unique. Perhaps even moving a bit farther and inviting her to a weekend break someplace quiet, calm and relaxing away from town. You can also browse boxedindulgence.com.au/mothers-day-gift-baskets to know more about mother's day gift box.
Its far more special than the usual box of chocolates your mother will most likely not wind up eating since she's counting calories, or even blossoms that yes might seem fine but will probably wind up lasting a week, the lasting allure isn't really amazing.


"This story is mostly true", we're told. And while The Old Man and the Gun is ostensibly about the hunt for the Over the Hill Gang, a real-life trio of geriatric bank robbers led by Robert Redford's Forrest Tucker, it's really a stylish homage to the charming outlaw archetype enshrined by Hollywood and, of course, Redford himself. This is pure, selfreflexive cinematic pleasure.
For Forest Tucker, robbing banks isn't about making a living. It's about living. Sharply dressed in a periwinkle blue suit, he carries out his work with precision and politeness, leaving witnesses oddly admiring of his gentlemanly manner. Aided by Danny Glover's Teddy and Tom Waits' Waller, he's done countless jobs and escaped from imprisonment so many times that he can't remember when and where every jail break took place. He slips through the cops' fingers yet again when he stops to help Jewel (Sissy Spacek) with her broken down car and the police cars speed unknowingly past. While romance gently blossoms, the appropriately named Detective John Hunt (Casey Affleck) grows determined to catch him.
Laura Venning, 25 year old feminist Jedi and film and television person.


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African Women Online: How African Women are doing the internet
Women are doing a lot of awesome things on the internet. Some are tweeting, some are taking selfies, some are reading news and some are posting twerk videos. Women in Africa are doing all these things and much more including smashing all things misogynistic and patriarchal right in the face
Kagure Mugo
Here we profile online awesomeness by African women tackling everything from commerce, politics to passion. These platforms hail from all over the continent, from Ghana to Egypt to Uganda and beyond. These sites and social media platforms are looking to change the way we think about African women.
Adventures from the Bedrooms of African Women
Site: http://adventuresfrom.com/
Twitter: @adventurefrom
What are women doing between the sheets, on the kitchen counter, in international conference spaces and possibly on the bathroom floor? Are African women having good sex despite the moans that fill the night sky? Adventures will tell you all this and more.
Mon Pi Mon
Site: https://monpimon.wordpress.com/
Twitter: @Women4WomenUG
Facebook: https://www.facebook.com/monpimonuganda?fref=ts
The email that started this blog was titled 'Women and Uganda and taking over the country.' And with that Mon Pi Mon was born. Mon pi Mon translates to Women for Women in the Acholi language, and that is what this site is all about: Ugandan Women and their everyday experiences.
Chubby Vogue Divas
Site: https://charmaincarol.wordpress.com/category/chubby-vogue-divas/
Facebook: https://www.facebook.com/chubbyvoguedivas
Women with curves can make your mind go blank. Make you walk into street lights. This site, Chubby Vogue Divas, celebrates the fuller African figure not only through the art of photography but also through story telling. Alongside partaking in a photo-shoot the women profiled are allowed to speak on their struggle with body issues and the journey to love and self-acceptance.
Photo: Blend Images/Hill Street Studios
HOLAAfrica!
Twitter: @HOLAAfrica
Facebook: https://www.facebook.com/HolaAfrica
Site: www.holaafrica.org
Sometimes things are serious and sometimes you have to have a little fun online, this dichotomy is exactly what HOLAA! is about. Started in 2012, HOLAA! explores all things related to female sexuality from eating pussy like a champ to tackling motherhood and body positivity. From handling homophobia in the workplace to handling your sex toys.
Sister Speak237
Twitter: @ComfortMussa
Site: http://sisterspeak237.com/
Facebook: https://www.facebook.com/pages/Sisterspeak237/274295492721998?fref=ts
In 2014, celebrated journalist Comfort Mussa set up Sister237 after noticing that women's issues were only discussed in women's groups and in whispers around kitchen tables. So she wanted to open up a space were these issues could be spoken about openly. Sisterspeak237 is a supportive online space were Cameroonian women can speak about what's important to them.
Ms Afropolitan
Twitter: @Msafropolitan
Site: http://www.msafropolitan.com/
Facebook: https://www.facebook.com/MsAfropolitan?fref=ts
This is an award-winning pan-African blog started in 2010. Written by the intoxicatingly beautiful Minna Salami, this blog has been described as a 'game changer' and tackles everything from sexual revolutions, to literature and politics. It looks to give readers "sharp, witty commentary about modern African society and popular culture from a feminist perspective."
Inkanyiso
Twitter: @InkanyisoN
Site: www.inkanyiso.org
Facebook: https://www.facebook.com/Inkanyiso
Inkanyiso(registered in 2009) is the brain child of visual activist and photography guru Zanele Muholi. It was started as a response to the lack of visual histories and skills training produced by and for LGBTI persons, especially artists. This platform plays with photography, film, visual arts and multimedia and has a lot of talented contributors.
Photo: Grant Thornton
Thursday's Space
Twitter: @ThursdaysSpace
Site: www.ThursdaysSpace.wordpress.com
Thursday's Space looks at issues that affect women but are often kept on the hush by mainstream media and within general conversational spaces. The platform takes on everything from religion to culture to street harassment. This site is one were the voices of African women, especially Kenyan women, can grow and be amplified.
The Feminist UnFairytales of a Young African Mother
Site: http://theloulette.blogspot.com/
Twitter: @TheLoulette
When a woman is taking care of a child, managing to hold down a job and blogging about the existence of African women all at the same time we should stop and take notice. The woman in question is a number of things: (young, unmarried) mom, writer, feminist, amateur photographer, A-religious and Yoruba. All of these things form part of what makes her and will appear on this blog, so hold on tight.
Ghana feminisms
Twitter: @obaa_boni
Site: www.ghanafeminism.wordpress.com
Described as a site that houses 'modernized Indigenous feminism from the Ghanaian soul,' founder Adwoa wanted to see Ghanaian feminism tackled by a Ghanaian woman. The 'Ghaminism' movement (Ghanaian feminism) wants to liberate Ghanaians from constricting gender oppression and this site is adding to this movement, with posts such as Vaginas in Ghana: The Elixir of Bliss.
Spectra Speaks
Twitter: @spectraspeaks
Facebook: http://www.facebook.com/spectraspeaksalot
Site: www.spectraspeaks.com
A spin off from keeping a journal and voicing the feelings of being misunderstood this online platform has evolved into a strong voice tackling issues of sexuality, helping ground Spectra in the idea of being African and Queer. It's an online space that is, arguably, one of most vocal in terms of representing the LGBTI African experience and truth be told, she has such a way with words and a killer smile.
FeminstsSA
Twitter: @Feministssa
Facebook: https://www.facebook.com/FeministsSA
Site: http://feministssa.com/
The site was started in 2011 by Jen Thorpe because she wanted to see more feminist writing tailored for a South African audience. Along with a number of other projects including My First Time (an archive of women's first times of absolutely everything) this was a platform for young women and more specifically young feminists to contribute ideas about the world around them on issues that mattered to them.
9jaFeminista
Twitter: @9jafeminista
Site: www.9jafeminista.wordpress.com
Started in September 2014 to challenge the way women were portrayed within the Nigerian media, the site seeks to be an advocacy tool to show how inequality brings more harm than good while also wanting to start a sexual revolution in some respects.
Uncultured Sisterhood
Site: www.unculturedsisterhood.wordpress.com
Twitter: @EstellaMZ
These are "the scribblings of a Ugandan feminist" named Estella. She found herself frustrated by the constant misogyny in the press, parliament and even just walking down the streets. In reaction to this she started this space to celebrate women but also to vent. Because sometimes we just need to scream into cyberspace.
Arabic Literature in English
Site: Arablit.org
Twitter: @arablit
Facebook: https://www.facebook.com/pages/Arabic-Literature-in-English/276130032405652
This site was started by Marcia Lynx Qualey, a writer and reader based in Cairo. Initially a 'solo effort' the site has grown to include over dozen contributors from around the region. Hoping to deepen people's understanding of Arabic literature the site also has a female spin as it looks at Arab feminism and even hosts #ReadWomen2015 in which Arab women novelists recommend texts written by women. Although not necessarily geared to the average reader (it interacts mainly with Arabic literature specialists, translators and the like) this amazing site will open you up a whole new world of literature to you.
KinnaReads
Twitter:@kinnareads
Site: www.kinnareads.com
On this site you will find a good mix of commentary and reviews on novels, some short stories, some poetry and other such magical literary things. The site also focuses on African women writing, showcasing their talents and accomplishments.
Her Zimbabwe
Twitter: @HerZimbabwe
Facebook: https://www.facebook.com/herzimbabwe
Site: www.herzimbabwe.co.zw
The Tag line for this is 'Her Voice, Her revolution.' That about says it all. Her Zimbabwe documents the untold stories of Zimbabwean women's lives and their histories. It is about showing that it is not all just about economic and political woes. The site also brings the world to Zimbabwe with 'Her Africa', showing the overlaps and parallels in our various existences across the continent.
African Feminist Forum
Twitter: @AFF001
Site: www.africanfeministforum.com
Facebook: https://www.facebook.com/AfricanFeministForum
The site is the home of the African Feminist Forum biennial conference that brings together African feminist activists to deliberate on issues that are important to the movement. It is a space for conversation and dialogue on the challenges facing the African women's movement. This site publishes pieces of writing, provides resources and information on upcoming events.
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