This room has an area of 18 m² and it is rich of natural light thanks to the private terrace. It also has a panoramic view of the city, including a spectacular view of the lake and the swimming pool.
Room facilities: heating, parquet or wooden floor, clothes rack, private bathroom, Air Conditioners, hairdryer, free Wi-Fi.


SAPPERS from 3 Armoured Engineer Squadron, part of 22 Engineer Regiment, based in Perham Down, Tidworth are working in Cumbria to build a new temporary footbridge spanning the River Derwent in Workington.
The logistic support bridge will provide a vital connection between the north and south sides of the town which have been cut off after the Workington Bridge was destroyed and the Calva Bridge suffered significant structural damage from the floods and remains unstable and dangerous.
It is hoped the footbridge will be open to the public by the weekend.


text:
Community-Minded Enterprises COVID-19 Updates
by Char Ladyman | Nov 16, 2020 | News | 0 comments
5pm – Monday, November 16, 2020
In light of new COVID-19 guidelines from the State of Washington, Community-Minded Enterprises is making some changes in services effective from midnight on Monday, November 16 through Monday, December 14. In some cases, there is no change. Here is a complete list.
Recovery Café will provide virtual services. Food service is suspended.
Recovery Community Services will provide virtual assistance to clients.
ECEAP at both our South Hill and Division locations is not impacted by the new guidelines and will remain open. Click here to read a letter from our ECEAP Director.
Early Learning services based in Spokane and Pasco will continue with staff serving clients virtually.
Plan to Work will continue to serve clients over the phone.
CMTV14 will serve those already scheduled to use the studio or have video services scheduled.
Our goal is to provide a healthy and safe place for clients and staff. We appreciate your understanding during this difficult time.
4:00pm – Friday, July 24, 2020
Following Governor Jay Inslee's "Stay at Home" Proclamation on Monday, March 23, 2020, Community-Minded Enterprises closed our locations to the public. All programs and staff moved to work remotely to serve our members and clients. CME is excited to announce that each of our five locations has begun the process and planning to re-open. All plans to re-open began with our staff and will then advance to include public access to offices.
Recovery Cafe Spokane is once again open to members. Health and safety guidelines are in place including; limited capacity in the building and health checks for all staff and members who enter. Recovery Circles and School for Recovery classes have resumed with in-person sessions but are also being offered virtually for those unable to attend in person. Recovery Care Associates from the SOR/SAS program are also seeing clients both virtually and in-person at our Division St location.
CMTV14 Studios are now open by appointment only. Our production team has put safety plans into place for both in house and on-location shoots.
Early Achievers coaches in Spokane and Pasco have moved to online and virtual coaching for all providers in the program. Professional Development training has also moved exclusively online. The Resource Center for Early Learning has plans in place to re-open to staff following health and safety guidelines. Our Pasco office remains closed to staff until Benton/Franklin counties advance in the WA State Safe Start Phases.
Plan to Work and Child Care Assistance Programs will continue services to clients remotely whenever possible but will be able to meet and serve clients in person when needed.
Early Connections Play & Learn sessions are still on hold. Our Play & Learn facilitators have begun distributing craft and activity kits to families at both the Recovery Café and Community-Minded ECEAP.
Community-Minded ECEAP will expand services for the 2020-21 school year, offering both full and part-day sessions. Our ECEAP staff will continue to stay in contact with families and children throughout the summer months. ECEAP teachers have also begun hosting weekly Play & Learn sessions at neighborhood parks.
8:00am – Tuesday, March 31, 2020
In response to the growing crisis among child care providers who are remaining open during the statewide "Stay at Home" order, Child Care Aware of Eastern Washington will be collecting and distributing much needed supplies to area providers. More information is available HERE.
4:30pm – Tuesday, March 24, 2020
Following Governer Jay Inslee's "Stay at Home" Proclamation on Monday March 23, 2020, Community-Minded Enterprises will continue to close our locations to the public. Additionally, all staff will be working remotely until further notice.
Recovery Cafe Spokane will still be able to provide sack lunches to members from 12:00 – 1:00pm Tuesday thru Fridays. Cafe staff has also begun providing virtual recovery circles for members.
Early Achievers coaches in Spokane and Pasco have suspended all on-site coaching visits with early learning providers. All Child Care Aware of Washington Professional Development trainings are postponed. Our team is working with Child Care Aware of Washington and the Department of Emergency Management to assess the immediate needs of child care providers during this ongoing crisis.
CMTV14 Studios remain closed.
Plan to Work and Child Care Assistance Programs will be able to continue services to clients via telephone.
Early Connections Play & Learn sessions have been suspended.
Southwest Community Center ECEAP program will remain closed through April 24, 2020 in alignment with Spokane Public Schools.
11:30AM- Monday, March 16, 2020
With the increasing public health crisis facing our country, Community-Minded Enterprises continues to value the safety and long-term well-being of our staff, their families and our community.
Following this morning's declaration from Gov. Inslee that temporarily closes all restaurants, bars and recreational/entertainment facilities, our Executive Team has decided to suspend public access at all CME locations.
Beginning Tuesday, March 17 we will be closing Recovery Café Spokane to the public and our members/volunteers. All our Recovery Services teams will be looking at ways to meet with members and clients virtually or via phone.
Our Early Achievers coaches in Spokane and Pasco will suspend all on-site coaching visits with early learning providers. All Child Care Aware of Washington Professional Development trainings will be postponed.
We are closing the CMTV14 studios to the public.
Our Plan to Work and Child Care Assistance Programs will be able to continue services to clients via telephone.
The Early Connections Play & Learn sessions have been suspended.
Our ECEAP program operated at Southwest Community Center will close beginning tomorrow, 03/17 until at least March 24. This may be updated upon further instruction from Washington State Department of Children, Youth & Families.
We have plans in place for employees to work remotely when possible. We are encouraging our staff to practice social distancing in order to stem the spread of the virus. This includes holding meetings online or rescheduling when appropriate.
This health concern is a fluid and rapidly changing situation and we will likely update protocols as new information becomes available. We will continue to follow leadership of Spokane Regional Heath District, Benton-Franklin Health District as well as State and Federal governments.
4:00pm – Friday, March 13, 2020
With the emerging and ongoing public health crisis facing our country, Community-Minded Enterprises values the safety and long-term well-being of our staff, their families and our community above all else.
Currently all CME offices will remain open, including the Recovery Café Spokane. Our Early Connections Play & Learn groups held at the Maple Street Community Service Office will be suspended until further notice. At this time the status of our ECEAP program at Southwest Community Center is still being determined.
We are currently making plans and putting protocols into place for employees to work remotely when needed and when possible. We are encouraging our staff to practice social distancing in order to stem the spread of the virus. This includes holding meetings and trainings online or rescheduling when appropriate.
This health concern is a rapidly changing situation and we will likely update protocols as new information becomes available. We will continue to follow leadership of Spokane Regional Heath, Benton-Franklin Health as well as State and Federal governments.

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: Find the names of recipes where at least 3 oz. (quantity) of every ingredient are used INGREDIENT(ingredient-id,name,price-ounce)
RECIPE(recipe-id,name,country,time)
USES(rid,iid,quantity)

rid is a foreign key to recipe-id and iid is a foreign key ingredient_id
Find the names of recipes where at least 3 oz. (quantity) of every ingredient are used. NOTE: this
means that, however many ingredients are used in a recipe, at least 3 oz. of each ingredient is used.
I've tried this code, but not sure if its correct.
SELECT  name
FROM    recipe
WHERE   recipe_ID IN(SELECT rid
            FROM    USES
            WHERE   quantity>=3);

I don't get an error but not sure if its getting the correct results.
Would this code get the correct result:
enter image description here
EDIT
The code from the image...
SELECT name
FROM   recipe
WHERE  recipe_id IN (SELECT rid
                     FROM   uses
                     WHERE  rid NOT IN (SELECT rid
                                        FROM   uses
                                        WHERE  quantity < 3))

A: The subquery will match any "Uses" row that has quantity >= 3. So, you will end with the list of recipes that has at least one ingredient with quantity >= 3.
What you want is that none of Uses should be less than 3 in order to match a recipe.
I would do some "double negative" here (NOT IN, and use "<" instead of ">=") :
SELECT  name
FROM    recipe
WHERE   recipe_ID NOT IN(SELECT rid
            FROM    USES
            WHERE   quantity<3);

With this you will get every recipe that doesn't have any ingredients with quantity less than 3, which is equivalent to having all ingredients with quantity greater than 3.

redpajama_set_name: RedPajamaStackExchange


Home / Pattern Reviews / My #smyly2018 dress – another Pattydoo Chloe dress!
My #smyly2018 dress – another Pattydoo Chloe dress!
O.K, I know. Another Pattydoo Chloe dress. I just can't help it, I love it so much!
I recently made this as part of a contest entry over on the Pattern Review website and decided to mention it here as part of the #smyly2018 challenge.
The idea of the #smyly2018 challenge is to make a garment that makes you feel good and helps the sewing community talk about positive body image and mental health issues.
Whilst I have been incredibly touched with reading and listening to so many situations that other sewists have found themselves dealing with, I feel that it is still too difficult to share with you my struggles with overwhelming shyness and other issues that affect my mental health.
I will say however that I agree with everyone that has taken part, in that sewing definitely makes a huge difference to me and I am constantly amazed by the support I have from fellow sewists, and that which I see shown to others. It is always important to remember that we all have struggles that we are dealing with, often unseen, and these affect us all to varying extents, but kindness costs nothing and patience with others is always something to aim for as we can never truly know what others are dealing with.
I have talked before about how I love the style and comfort of this dress, and if you would like to read a couple of my previous reviews on this dress you can do so here and here. I have made others, but not blogged them.
I have used a moss green scuba fabric which I picked up from The Rag Market in Birmingham last year. It has a bold floral design with the addition of berries, butterflies and bees.
… and pockets and pleats. Yay!
Previous post: Another Cocowawa Chestnut Sweater, with added bows!
Next post: The Mandy Boat Tee from Tessuti Patterns.
I, too, shy away from talking about personal issues on my blog, but I really appreciate the support shown by the sewing community. I enjoy reading other people's stories and it's truly amazing to see all the love and support they receive from fellow sewists.
You've made yourself another beautiful dress. It's such a wonderful pattern, no wonder you've made it several times. And the fabric is so pretty.
That's a lovely dress and if you find a pattern that works for you then you can never have too many of them. I have issues around my middle and have found two patterns that suits me and I keep making them again and again.
Very true and when you are petite it's difficult to buy RTW dresses.
I hope we meet again one day – I have not made a new sewing friend via Secret a Valentine this year!


According to the material provided by American Association for the Advancement of Science, Scientists here have created engineered proteins that brought down body weight, bloodstream insulin, and cholesterol levels in obese mice, rats and primates. These findings, to some extent, pave the way for earnestly required alternatives to bariatric surgery for treating obesity in humans-the rates of which have almost tripled worldwide since 1975.
In view of the perception that obese mice, rats, and humans all had raised serum concentrations of a protein named GDF15 in contrast with learn controls, Yumei Xiong and partners set out to create treatments got from the molecule. In various mouse models of diet-induced and genetic obesity, conveyance of the GDF15 gene reduced body weights, food intake, and serum insulin levels in the animals.
Since GDF15 has a short plasma half-life and is hard to deliver in substantial quantities, the researchers produced two distinctive fusion proteins that were more stable in the circulation and prompted higher yields. Both fusion proteins adequately diminished body weights for obese mice and cynomolgus monkeys.
What's more interesting is that, Xiong, together with her colleagues, additionally demonstrated that the GDF15 regimen changed food preferences in mice—driving the animals to pick lower calorie chow when offered a decision between standard food and an additional condensed-milk diet.
The authors established that GDF15 enacted a population of nerve cells called AP neurons that make up a portion of the gut-brain axis, yet take not of that further investigations to identify the protein's cellular receptor are required as potential therapeutics advance toward the clinic.


This 14k yellow gold 16in lasered titan curb chain has a lobster clasp. Chain width: 1mm. This chain is manufactured using a special, hard, gold alloy that is extremely firm and is specifically designed to enhance the laser welding process. This produces the strongest pendant chain available. It has a higher tensile strength and is up to four times more stretch resistant than ever before. This allows you to use lighter weight chains with heavier weight pendants. Higher tensile strength and laser welded joints guarantee that you will have less breakage. Weighs 0.55 grams. Please note: weight is approximate, actual weight may vary.


Infosys Self-Service BI introduces a paradigm shift in how end-users consume data. A shift we call Pervasive BI. A shift that removes dependency on your IT teams and empowers end-users. A shift that allows your organization to take full advantage of your data. Watch this video to learn more.


text:
\section{Introduction}
Statistics on Lie groups have became more and more popular in the last decade. Applications can be found in shape statistics \cite{fletcher2003gaussian,fletcher2004principal}, medical imaging \cite{fletcher2003statistics}, multiple scattering processes \cite{le2009nonparametric,Said2010}, crystallography \cite{Hielscher2013}, robotics and mechanics \cite{chirikjian2000engineering,Chirikjian2012} and many others. In directional statistics, the abundance of datasets taking values on spheres and homogeneous spaces has also motivated the study of random variables and processes on Lie groups \cite{Mardia}. Amongst all the matrix Lie groups, the most popular one is the rotation group in ${\mathbb R}^3$, {\em i.e} the Special Orthogonal group $SO(3)$. This is due to its predominent use in engineering problems \cite{chirikjian2000engineering}.

Even though the number of engineering challenges including random rotations has grown dramatically in the last years, the problem of probability density estimation for such variables was only considered in details recently in \cite{Hielscher2013}. In parallel, the concept of wavelets on manifolds was transposed to the case of the rotation group, due to its relation to the 2-sphere which has attracted a lot of work since the 90s \cite{schroder1995spherical,antoine1999wavelets,starck2006wavelets}. In \cite{Ebert2011}, authors introduce diffusive wavelets on manifolds. The definition of a wavelet transform on a surface/manifold is conditioned by the possible definition of two operations on the manifold: scaling and translation. While translation is easily defined on Lie groups as it is based on the group action, the definition of scaling is less obvious. In \cite{Ebert2011}, authors have chosen an intrinsic definition for scaling, whereas some extrinsic approaches had been proposed previously in \cite{bogdanova2005stereographic}. The difference in these definitions resides in the way the mother wavelet is scaled: it is either firstly projected in the tangent plane before scaling and back projection (extrinsic); or scaled on the manifold directly (intrinsic). In this paper, we will make use of the intrinsic approach and study the ability of diffusive wavelets to define interesting estimators for densities on $SO(3)$.

The specificity of diffusive wavelets is that they are based on the heat kernel. As this kernel can be defined on manifolds, the diffusive wavelet approach overcomes the problem of ``scaling'' on manifold. Note that this definition problem was already pointed out by many authors \cite{bogdanova2005stereographic,	antoine1999wavelets} when defining wavelets on the 2-sphere for example.

In this paper, we present an estimation technique for densities on $SO(3)$ based on the diffusive wavelet transform. Using wavelet estimators (linear or thresholded) is well known in non parametric estimation \cite{Tsybakov2009}. We propose the use of the linear\footnote{Linear wavelet estimation refers to the standard estimation through wavelet coefficients estimation, as opposed to thresholded wavelet coefficients estimator sometimes refered as nonlinear \cite{Tsybakov2009}} wavelet estimator to the case of $SO(3)$-valued random variables and give some of its properties. In particular, the wavelet-based estimator is a characteristic kernel estimator. A comparison with other types of estimators (characteristic function and kernel) is provided.

The rest of the paper is organized as follows: Section \ref{sec:CharaF_SO3} is dedicated to the presentation of harmonic analysis on $SO(3)$ and various concepts about random variables on this Lie group. In Section \ref{sec:DiffWavSO3}, the diffuse wavelet formalism is introduced for the case of the rotation group. It is then used in Section \ref{sec:estim} to propose a probability density function estimator. Finally, Section \ref{sec:simu} presents simulation results and comparison of the wavelet estimator with two other estimators.

\section{Characteristic functions for random variables on $SO(3)$}
\label{sec:CharaF_SO3}
We first review some basic concepts on the rotation group $SO(3)$ and random variables taking values on this well known Lie group. The concept of characteristic function for $SO(3)$-valued random variables is of importance as it can be identified as the ``Fourier transform'' of the probability density of the random variable. It will also be of use in Section \ref{sec:estim} to provide a probability density estimator. The results presented here can be found in various textbooks, for example \cite{Altmann2005,Dym1972}.

\subsection{The rotation group $SO(3)$}
The set of rotations in the 3D space forms a compact Lie group denoted $SO(3)$. An element $x \in SO(3)$ can be parametrized in several ways \cite{Altmann2005}. Using the so-called ZYZ convention of Euler angles parametrization, any element $x \in SO(3)$ can be associated to a matrix $R^x=R^x(\varphi,\theta,\psi)$ with $0 \leq \varphi,\psi < 2\pi$ and $0 \leq \theta \leq \pi$. With this convention, $R^x(\varphi,\theta,\psi)$ takes the form:
\begin{align*}
R^x(\varphi,\theta,\psi) = &
\left[
\begin{array}{ccc}
\cos \varphi & -\sin \varphi & 0 \\
\sin \varphi & \cos\varphi & 0 \\
0 & 0 & 1
\end{array}
\right]
\left[
\begin{array}{ccc}
\cos \theta & 0 & \sin \theta \\
0 & 1 & 0 \\
-\sin \theta & 0 & \cos\theta
\end{array}
\right]
\\
& \times
\left[
\begin{array}{ccc}
\cos \psi & -\sin\psi & 0 \\
\sin \psi & \cos\psi & 0 \\
0 & 0 & 1
\end{array}
\right]
\end{align*}

It is also possible to parametrize an element of $SO(3)$ in terms of its \emph{rotation axis} and its \emph{rotation angle} \cite[Chap. 3]{Altmann2005}. The rotation axis $\eta$ is a unit vector in ${\mathbb R}^3$, \emph{i.e.} $\eta \in {\cal S}^2$. The rotation angle, denoted $\omega(x)$, is given by:
\begin{equation}
\cos \omega(x)= \frac{Tr(R^x) - 1}{2}
\label{eq:rotangleS03}
\end{equation}
Note that in this parametrization, the angle takes values in: $-\pi < \omega(x) \leq \pi$. It is also possible to express this angle $\omega(x)$ in terms of the Euler angles:
\begin{equation}
\omega(x)= 2 \arccos \left( \cos \frac{\theta}{2} \cos \frac{\varphi+\psi}{2}\right)
\end{equation}
The \emph{rotation angle} is of particular interest as it is a metric on $SO(3)$. In fact, one can define the distance between $x \in SO(3)$ and $y \in SO(3)$, denoted $d(x,y)$, as:
\begin{equation}
d(x,y)=|\omega(yx^{-1})| = \arccos \left(\frac{Tr(R^y(R^{x})^{-1}) - 1)}{2}\right)
\label{eq:geodistSO3}
\end{equation}
In the sequel, we will make use of the notation ${\bm R(\omega(x),\eta)}$ for an element $x \in SO(3)$, keeping in mind that:
\begin{equation}
{\bm R(\omega(x),\eta)} = \exp(\omega(x){\bm M}) \quad \text{with} \quad {\bm M}=\left(\begin{matrix} 0 & -\eta_3 & \eta_2 \\ \eta_3 & 0 & -\eta_1 \\ -\eta_2 & \eta_1 & 0 	\end{matrix}\right)
\end{equation}
where $\eta_1,\eta_2,\eta_3$ are the components of the vector $\eta$, {\em i.e.} $\eta=\left[\eta_1,\eta_2,\eta_3\right]^T$ and $\exp(.)$ the matrix exponential \cite{Altmann2005}.

\subsection{Fourier series on $SO(3)$}
We now consider the Fourier series expansion of functions taking values on the rotation group. Consider the set of square integrable functions $L^2(SO(3),{\mathbb R})$. By the Peter-Weyl theorem \cite{Dym1972}, a function $f \in L^2(SO(3),{\mathbb R})$ can be expressed as:
\begin{equation}
f(x)= \sum_{\ell\geq 0}\sum_{n = -\ell}^{+\ell}\sum_{m = -\ell}^{+\ell} (2\ell+1) \hat{f}_{nm}^{\ell} D_{nm}^{\ell}(x)
\label{eq:FourierSeriesSO3}
\end{equation}
for $x \in SO(3)$ and where $D_{nm}^{\ell}(x)$ are the Wigner-D functions \cite{Barut1986} evaluated at position $x$. This infinite series expansion over $\ell$ has matrix coefficients $\hat{f}^{\ell}$ of dimension $(2\ell+1)\times(2\ell+1)$ with elements $\hat{f}_{nm}^{\ell}$. The matrix entries $\hat{f}_{nm}^{\ell}$ are obtained by projection of $f$ on the Wigner-D functions:
\begin{equation}
\hat{f}_{nm}^{\ell} =\left<f,D_{nm}^{\ell}\right>_{\text{\tiny{$SO(3)$}}} = \int_{SO(3)} f(x) \overline{D_{nm}^{\ell}}(x)  d\mu
\end{equation}
where $\left<f,h \right>_{\text{\tiny{$SO(3)$}}}$ is the scalar product on $L^2(SO(3),{\mathbb R})$ and $d\mu$ the bi-invariant Haar measure on $SO(3)$, {\em i.e.} $d\mu=(8\pi^2)^{-1}\sin\theta d\varphi d\theta d\psi$ when using the ZYZ Euler angle parametrization for elements $x \in SO(3)$.
We also mention that the corresponding norm is:
\begin{equation}
\left\|f\right\|_2=\sqrt{\left<f,f \right>_{\text{\tiny{$SO(3)$}}}}
\end{equation}
With the chosen parametrization of $SO(3)$, the previously introduced Wigner-D functions $D_{nm}^{\ell}$ take the form:
\begin{equation}
D_{nm}^{\ell}\left(\varphi,\theta,\psi\right)= e^{-{\tt i}n\varphi} P_{nm}^{\ell}(\cos\theta) e^{-{\tt i}m\psi}
\end{equation}
where $P_{nm}^{\ell}(\cos\theta)$ are the generalized Legendre polynomials. We have used the fact that the Wigner-D functions $D_{nm}^{\ell}$ form a complete set of orthogonal functions:
\begin{equation}
\begin{array}{rcl}
\displaystyle{\left<D_{nm}^{\ell},D_{n'm'}^{\ell'}\right>_{\text{\tiny{$SO(3)$}}}}
 & =  & \displaystyle{\frac{1}{\left(2\ell + 1\right)} \delta_{nn'}\delta_{mm'}\delta_{\ell\ell'}}
\end{array}
\end{equation}
This shows clearly that the set of functions $\left\{\sqrt{2\ell+1}D^{\ell}_{nm}, \ell \geq 0, -\ell\leq n,m \leq \ell \right\}$ form an orthonormal basis for functions in $L^2(SO(3),{\mathbb R})$, allowing the decomposition of functions taking values on the rotation group using this basis. In the sequel, we may refer to $\hat{f}_{nm}^{\ell}$ (and abusively to $\hat{f}^{\ell}$) as the {\em Fourier coefficients} of $f$.

\subsection{Parseval identity}
It is also well known that the Parseval identity holds for functions $f \in L^2(SO(3),{\mathbb R})$. Using our normalization convention, the following is true:
\begin{equation}
\begin{array}{rcl}
\left\|f\right\|_2^2 & = & \displaystyle{\left<f,f\right>_{\text{\tiny{$SO(3)$}}}} \\
& = & \displaystyle{\sum_{\ell \geq 0} \sum_{n=-\ell}^{\ell} \sum_{m=-\ell}^{\ell} (2\ell +1) \left|\hat{f}^{\ell}_{nm}\right|^2}
\end{array}
\label{eq:parsevalSO3}
\end{equation}
The left hand side of equation (\ref{eq:parsevalSO3}) is commonly denoted as the {\em energy} and the Parseval identity simply states that the energy of $f$ consists in the infinite sum of its modulus squared {\em Fourier coefficients}.

\subsection{Zonal functions on $SO(3)$}
\label{sec:zonalfunctSO3}
Zonal functions, sometimes also called \emph{conjugate invariant} functions, are radially symmetric functions with center $g_0=e \in SO(3)$ where $e$ is the identity element in $SO(3)$. See for example \cite{Hielscher2007} for more details on radially symmetric fonctions on $SO(3)$. They will be at the heart of the diffusive wavelets construction in Section \ref{sec:DiffWavSO3}. A function $f:SO(3) \rightarrow {\mathbb R}$ is called \emph{zonal} iff it satisfies $\forall x,y \in SO(3)$:
$$ f(yxy^{-1})=f(x)$$
Equivalentlty, $f:SO(3)\rightarrow {\mathbb R}$ is \emph{zonal} iff $f(x)=f(x')$ $\forall x,x' \in SO(3)$ such that $\omega(x)=\omega(x')$, {\em i.e.} $x$ and $x'$ have the same {\em angle}. In fact, it simply means that $f$, evaluated at $x \in SO(3)$, only depends on the rotation angle of $x$ introduced in (\ref{eq:rotangleS03}). It is known \cite{Hielscher2007,Hielscher2013} that the subspace of zonal functions is spanned by functions $\chi^{\ell}$, $\ell \in {\mathbb N}$, called the {\em characters} of $SO(3)$, and given as:
\begin{eqnarray}
\chi^{\ell}(x) & = & \sum_{n=-\ell}^{\ell} D^{\ell}_{nn}(x) = \frac{\sin \left(\left( \ell + \frac{1}{2}\right)\omega(x)\right)}{\sin \frac{\omega(x)}{2}}\\
 & = & {\mathcal U}_{2\ell}\left(\cos\frac{\omega(x)}{2}\right)
\end{eqnarray}
where ${\mathcal U}_{2\ell}(.)$ are the Chebychev polynomials of second kind and of (even) degree $2\ell$. For these (even degree) polynomials, the following orthogonality relation stands:
\begin{equation}
 \int_{-1}^1 {\mathcal U}_{2\ell}(t) {\mathcal U}_{2\ell'}(t) \sqrt{1-t^2}dt = \frac{\pi}{2}\delta_{\ell\ell'}
\end{equation}
In terms of the characters $\chi^{\ell}$, and using the notation $\omega(x)=\omega$ for simplicity, this integral becomes:
\begin{equation}
\int_0^{2\pi}\chi^{\ell}(\omega)\chi^{\ell'}(\omega)\sin^2\left(\frac{\omega}{2}\right)d\omega = \pi \delta_{\ell \ell'}
\end{equation}
where the link with the orthogonality relation for ${\mathcal U}_{2\ell}$ comes with $t=\cos\frac{\omega}{2}$. As a consequence, the characters $\chi^{\ell}$ fulfill the following orthogonality relation:
\begin{equation}
\left< \chi^{\ell}, \chi^{\ell'} \right>_{\text{\tiny{$SO(3)$}}} =   \frac{1}{4\pi^2} \int_{{\mathcal S}^2}d\eta \int_0^{2\pi} \chi^{\ell}(\omega) \chi^{\ell'}(\omega) \sin^2\left(\frac{\omega}{2}\right)d\omega =  \delta_{\ell\ell'}
\end{equation}
where we used the expression of the Haar measure in terms of axis $\eta$ and rotation angle $\omega$ for $SO(3)$. With this convention the Haar measure takes the form $d\xi=\frac{1}{4\pi^2}d\eta \sin^2(\frac{\omega}{2})d\omega$. As a consequence, a zonal function $f \in L^2(SO(3),{\mathbb R})$ has a Fourier series expansion that can be written like:
\begin{equation}
f(x)=\sum_{\ell \geq 0} (2 \ell +1) \hat{F}^{\ell} \chi^{\ell}(x)
\label{eq:FourierZonal}
\end{equation}
where its Fourier coefficients $\hat{F}^{\ell}$ are denoted in capital to emphasize the fact that $f$ is zonal. These Fourier coefficients $\hat{F}^{\ell}$ are simply given by:
\begin{equation}
\hat{F}^{\ell} =  \frac{1}{(2 \ell +1)} \left< f,\chi^{\ell}\right>_{\text{\tiny{$SO(3)$}}} = \frac{1}{\pi} \frac{1}{(2 \ell +1)}\int_0^{2\pi} f(\omega) \chi^{\ell}(\omega) \sin^2 \left(\frac{\omega}{2}\right) d\omega
\end{equation}
The Parseval identity for a zonal function $f$ now reads:
\begin{equation}
\|f\|^2_2= \sum_{\ell \geq 0} \left(2\ell+1\right)^2\left|\hat{F}^{\ell}\right|^2
\label{eq:ParsevalZonal}
\end{equation}
Zonal functions will be used in Section \ref{sec:DiffWavSO3} in the definition of wavelets on $SO(3)$.

\subsection{Convolution}
A very important feature of Fourier transformation is its behaviour with respect to the convolution product. In the case of functions taking values on $SO(3)$, the "famous" relation still holds. First, given $f,h \in L^1(SO(3),{\mathbb R})$, their convolution product is defined as:
\begin{equation}
\left(f*h\right)(x) = \int_{\text{\tiny{$SO(3)$}}} f(g)h(g^{-1}x) d\mu
\end{equation}
where the group operation stands naturally for translation. Now, if the Fourier coefficients of $f$ and $h$ are respectively $\hat{f}^{\ell}$ and $\hat{h}^{\ell}$ (in matrix format), then the following stands:
\begin{equation}
\widehat{f*h}^{\ell}=\hat{f}^{\ell}\hat{h}^{\ell}
\end{equation}
Note that the right-hand side of the equation is a matrix product.

\subsection{Characteristic function of $SO(3)$-valued random variables}
The characteristic function of a random variable is the Fourier transform of its probability transform. This well known result extends to random variables on the rotation group, thanks to the results on Fourier series expansion introduced in the previous Section. We now give some definitions and properties for characteristic function of random variables on $SO(3)$ as it will be used to define a probability density function estimator in Section \ref{sec:estim}.

Consider the case of a random variable $X \in SO(3)$ with density $f$. Such random variables $X$ taking values on the rotation group $SO(3)$ can simply be thought of as random rotation matrices of dimension $3 \times 3$, classically parametrized by Euler angles. Alternatively, one can think of these random variables as unit quaternions from the upper hemisphere of ${\mathcal S}^3$.

\subsubsection{Definition}
Given a rotation random variable $X$ with density $f_X$, the sequence $\Phi_X=\left\{\Phi_X(\ell)\right\}_{\ell \geq 0}$ of $(2\ell + 1) \times (2\ell +1)$ matrices given by:
\begin{equation}
\Phi_X(\ell)={\mathbb E}\left[\overline{D^{\ell}}(X)\right]
\end{equation}
is the \emph{characteristic function} of $X$. The elements of the matrix $\Phi_X(\ell)$ are denoted $\Phi_{nm}^{\ell}$ and they read as:
\begin{equation}
\Phi_{nm}^{\ell}=\int_{SO(3)}f_X(x)\overline{D^{\ell}_{nm}}(x) d\mu
\end{equation}
Thus the density $f$ has the following Fourier series expansion:
\begin{equation}
f_X(x)= \sum_{\ell \geq 0}\sum_{n,m=-\ell}^{\ell} (2\ell + 1)\Phi_{nm}^{\ell} D^{\ell}_{nm}(x)
\end{equation}

One can see by comparison with Equation (\ref{eq:FourierSeriesSO3}) that the characteristic function of $f_X$ consists is the Fourier transform of its density $f_X$, \emph{i.e.} $\Phi_X^{\ell}=\hat{f}_X^{\ell}$.

\subsubsection{Basic properties}
The following properties can easily be verified:
\begin{itemize}
\item Given two rotation random variables $X$ and $Y$, then:
$$X=Y \quad \text{iff} \Phi_X=\Phi_Y$$
\item If $X$ and $Y$ are two independent rotation random variables and $Z=XY$, then:
$$\Phi_Z^{\ell}=\Phi_X^{\ell}\Phi_Y^{\ell}$$
\item A rotation random variable $U \in SO(3)$ is uniformly distributed iff:
$$\Phi_U^{\ell} = 0 \quad \forall \quad \ell > 0$$
\item Consider $n$ \emph{i.i.d.} rotation random variables $X_1,X_2,\ldots,X_n$, then the random variable consisting in the accumulated products of the $X_n$, denoted $Y=X_1X_2 \ldots X_n$ has the following characteristic function:
$$\Phi_Y^{\ell}=\left[\Phi_{X}^{\ell}\right]^n$$
where $\Phi_{X}^{\ell}$ is the characteristic function shared by the $X_n$.
\end{itemize}

\subsubsection{Zonal invariance}
	A rotation random variable $X$ with density $f_X$ is called \emph{zonal invariant} if:
	$$X \stackrel{d}{=} R X R^{-1} \quad \text{for all} \quad R \in SO(3)$$
	Now, if $X$ is \emph{zonal invariant}, then its characteristic function is:
	$$\Phi_X^{\ell} = a_{\ell} I_{\ell}$$
	where $a_{\ell} \in {\mathbb R}$ and $I_{\ell}$ is the $(2\ell+1) \times (2\ell+1)$ identity matrix. As a consequence, if $X$ is \emph{zonal invariant}, then its density $f_X$ takes the form:
	\begin{equation}
	f_X(x) = \sum_{\ell \geq 0}  \, (2\ell+1) a_{\ell} \, \chi^{\ell}(x)
	\end{equation}
	as detailed previously when considering zonal functions on $SO(3)$ in \ref{sec:zonalfunctSO3}.

This last expression shows that the characteristic function of {\em zonal invariant} random rotation variables are scalar coefficients $a_{\ell}$, as opposed to matrix coefficients for random variables with no symmetries.

\section{Diffusive wavelets on $SO(3)$}
\label{sec:DiffWavSO3}
We now introduce diffusive wavelets on the rotation group. They will be used in Section \ref{sec:estim} to propose a probability density estimator. Recently, Ebert and Wirth \cite{EbertThesis2011} introduced \emph{diffusive wavelets} on groups. We detail in the sequel the special case of the rotation group in 3D, {\em i.e.} $SO(3)$.

\subsection{Heat wavelet family on $SO(3)$}
\label{subsec:HeatWaveletSO3}
Here, we follow the construction given by Ebert \cite{EbertThesis2011}. First recall that on $SO(3)$ the heat kernel is given by:
\begin{equation}
\kappa_{\rho}(x)=\sum_{\ell \geq 0}(2\ell + 1 ) e^{- \ell (\ell +1) \rho} \chi^{\ell}(x)
\end{equation}
for $x \in SO(3)$, and where $\chi^{\ell}$ are the irreducible characters of $SO(3)$ introduced previously in Section \ref{sec:CharaF_SO3}. Note that the Fourier series expansion of $\kappa_{\rho}$ exhibits the fact that $\kappa_{\rho}$ is a zonal function with Fourier coefficients $\hat{\kappa}_{\rho}^{\ell}=e^{- \ell (\ell +1) \rho}$.

Now, we introduce some of the properties of the heat kernel. First, $\kappa_{\rho}(x)$ is a \emph{diffusive approximate identity} as it fullfils the following properties:

\begin{itemize}
\item $\left\|\hat{\kappa}_{\rho}^{\ell}\right\| \leq C \quad \forall \ell \quad \rho \in {\mathbb R}^+$
\item  $\lim_{\rho \rightarrow 0} \hat{\kappa}^{\ell}_{\rho}= Id \quad \forall \ell$
\item $\lim_{\rho \rightarrow \infty} \hat{\kappa}^{\ell}_{\rho}= 0 \quad \forall \ell > 0$
\item $-\frac{\partial}{\partial \rho} \hat{\kappa}^{\ell}_{\rho}$ is a positive matrix for all $\rho > 0$ and $\ell \geq 0$
\end{itemize}
where $Id$ denotes the identity operator. In fact, $\left\{\kappa_{\rho},\rho > 0\right\}$ defines a family of kernels associated to an approximate identity (\emph{i.e.} an admissible semi-group\footnote{A continuous family $\left\{D_\rho,\rho>0\right\}$ of operators on $L^2(SO(3))$ is called an admissible semi-group if:
\begin{itemize}
\item $D_\rho$ is a bounded operator independent of $\rho$
\item $\lim_{\rho \rightarrow 0} = Id$
\item $D_\rho$ is postive $\forall \rho$
\item $D_{\rho_1}D_{\rho_2} = D_{\rho_1+\rho_2}$
\end{itemize}
Now, an admissible semi-group that can be expressed as a convolution operator, \emph{i.e.} for the semi-group $\left\{D_\rho,\rho>0\right\}$ there exits a family of kernels $\left\{\kappa_\rho, \rho>0\right\}$ such that $D_\rho(f)=f*\kappa_\rho$. Such operator is also called \emph{approximate identity with kernel $\kappa_\rho$}.} with kernel $\kappa_{\rho}$). It means that:
$$
\kappa_{\rho} * h \underset{\rho \longrightarrow 0}{\longrightarrow} h \quad \forall h \in L^2(SO(3))
$$

As detailed in \cite{EbertThesis2011}, a family of wavelets corresponding to the heat kernel on $SO(3)$ is consequently of the form:
\begin{equation}\label{eq:wavheatso3}
\displaystyle{\Psi_\rho(x)=\frac{1}{\sqrt{\alpha(\rho)}} \sum_{\ell \geq 0} (2 \ell +1) \sqrt{\ell (\ell +1)} e^{- \frac{\ell (\ell +1)}{2} \rho} \chi^{\ell}(x)
}
\end{equation}
with $\alpha(\rho)$ a normalizing factor to be detailed below. Rephrasing this equation in terms of Fourier transform, noticing that the heat wavelet family is zonal, one can write that:
\begin{equation}
\Psi_\rho(x) = \sum_{\ell \geq 0} (2\ell +1) \hat{\Psi}_\rho \chi^{\ell}(x)
\end{equation}
where the Fourier coefficients of $\Psi_\rho$ are:
\begin{equation}
\hat{\Psi}_\rho = \frac{1}{\sqrt{\alpha(\rho)}} \sqrt{\ell (\ell +1)} e^{- \frac{\ell (\ell +1)}{2} \rho}
\end{equation}
As can be seen in Equation (\ref{eq:wavheatso3}), the choice for the normalization coefficients $\alpha(\rho)$ has to be made. In order to have a normalized/unitary wavelet family, we can use the Parseval identity to choose $\alpha(\rho)$. In fact, by (\ref{eq:ParsevalZonal}), we have that:
\begin{equation}
\|\Psi_\rho\|^2_2 = \sum_{\ell \geq 0} (2\ell + 1)^2 \left|\hat{\Psi}_\rho\right|^2
\end{equation}
and imposing that $\|\Psi_\rho\|^2_2=1$ $\forall \rho$, one obtains that:
\begin{equation}
\displaystyle{\alpha(\rho)= \sum_{\ell \geq 0} (2\ell + 1)^2 \ell(\ell+1) e^{-\ell(\ell+1)\rho}}
\end{equation}
This choice for $\alpha(\rho)$ will be made throughout the rest of the paper. Notice also that the heat wavelet family  $\Psi_\rho \in L^2(SO(3))$ is a \emph{diffusive wavelet family} as it satisfies the \emph{admissibility condition}:
\begin{equation}
\kappa_\rho(x)  =  \displaystyle{\int_{\rho}^{+\infty} \left(\check{\Psi}_{t} * \Psi_{t}\right)(x) \alpha(t)dt}  =  \displaystyle{\int_{\rho}^{+\infty} \int_{SO(3)} \check{\Psi}_{t}(g) \Psi_{t}(g^{-1}x) d\mu(g) \alpha(t)dt}
\label{eq:heatkernelSO3}
\end{equation}
in which $\kappa_{\rho}(x)$ is an diffusive approximate convolution identity and where we used the notation $\check{\Psi}_{\rho}(g)=\overline{\Psi_\rho(g^{-1})}$. The fact that $\kappa_{\rho}(x)$ is an approximate convolution identity ensures that the wavelet transform can be inverted. Note that the approximate convolution identity can be expressed using the Fourier coefficients. In the case of $\kappa_\rho$, it means that:
$$\lim_{\rho \rightarrow 0} \hat{\kappa}_{\rho}^{\ell}=1 \quad \forall \ell $$
This means that an approximate convolution identity is characterized by constant Fourier coefficients in the limit.

With the diffusive wavelets introduced, we can now introduce the wavelet transform for functions on the rotation group.

\subsection{Wavelet transform on $SO(3)$}
\label{subsec:WTSO3}
Recall that we are interested in estimating probability density functions of $SO(3)$-valued random variables using the diffusive wavelet transform. A probability density function $f$ belongs to $L^2(SO(3),{\mathbb R}) \cap L^1(SO(3),{\mathbb R})$ with the aditionnal condition that $\displaystyle{\int_{SO(3)} f d\mu}=1$ and that it is non-negative. The diffusive wavelet transform for such densities is as follows. Given a diffusive wavelet family $\Psi_{\rho} \in L^1(SO(3))$ as defined in \ref{subsec:HeatWaveletSO3}, then the Wavelet Transform (WT) of $f \in L^2(SO(3),{\mathbb R})\cap L^1(SO(3),{\mathbb R})$ is:
$$
\begin{array}{rccc}
WT: & f & \longrightarrow & WT_f \\
 & SO(3) & & {\mathbb R}^+ \times SO(3)
\end{array}
$$
with the following expression:
\begin{equation}
WT_f(\rho,g)= \left( f * \check{\Psi}_{\rho} \right) (g) = \int_{SO(3)} f(x)\check{\Psi}_{\rho}(x^{-1}g) d\mu(x)
\end{equation}
and where we made use of the notation $\check{\Psi}(x)=\overline{\Psi(x^{-1})}$. Using scalar product on the rotation group, this expression can be written like:
\begin{equation}
WT_f(\rho,g)= \int_{SO(3)} f(x) \overline{\Psi_{\rho}(g^{-1}x)} d\mu(x) = \left<f, T^*_g \Psi_\rho \right>_{\text{\tiny{$SO(3)$}}}
\label{eq:wavelso3}
\end{equation}
with $T^*_g$ the following operator: $T^*_g:\Psi \rightarrow \Psi(g^{-1}.)$. Equation (\ref{eq:wavelso3}) is a very general definition of wavelet transform on $SO(3)$. In the sequel, we will only make use of the heat wavelet family given in (\ref{eq:wavheatso3}) to define the heat wavelet transform. One of the interesting properties of the wavelet transform is that it is invertible. The density $f$ can thus be reconstructed in the following way:
\begin{equation}
\begin{array}{rcl}
f(x) & = & \displaystyle{\int_{{\mathbb R}^+}\int_{SO(3)} WT_f(t,g) \Psi_{t}(g^{-1}x) d\mu(g) \alpha(t)dt} \\
 & = & \displaystyle{\int_{{\mathbb R}^+}\int_{SO(3)} \left( f * \check{\Psi}_{t}\right)(g) \Psi_{t}(g^{-1}x) d\mu(g) \alpha(t)dt}\\
 & = & \displaystyle{\int_{{\mathbb R}^+} \left(f * \check{\Psi}_{t} * \Psi_{t} \right)(x) \alpha(t)dt} \\
 & = & \displaystyle{ f * \int_{{\mathbb R}^+} \left( \check{\Psi}_{t} * \Psi_{t} \right)(x) \alpha(t)dt} \\
 & = & \displaystyle{ f * \int_{\rho \rightarrow 0}^{+\infty} \left( \check{\Psi}_{t} * \Psi_{t} \right)(x) \alpha(t)dt} \\
 & = & \displaystyle{ \lim_{\rho \rightarrow 0} \left(f*\kappa_{\rho}\right)(x) }\\
 & = & \displaystyle{f(x)}
\end{array}
\label{eq:invWTSO3}
\end{equation}
where the last equality is obtained thanks to the fact that $\kappa_t(x)$ is a convolution identity. This can be verified in the Fourier domain, when denoting $\hat{F}^{\ell}$ the Fourier coefficients of $f$, by:
\begin{equation}
\begin{array}{rcl}
 \displaystyle{\lim_{\rho \rightarrow 0} \left( f*\kappa_{\rho} \right)(x)} & = & \displaystyle{\lim_{\rho \rightarrow 0} \sum_{\ell \geq 0} (2\ell +1) e^{-\ell(\ell+1)\rho} \hat{F}^{\ell} \chi^{\ell}(x)}\\
  & = & \displaystyle{\sum_{\ell \geq 0} (2\ell +1) \hat{F}^{\ell} \chi^{\ell}(x)}\\
  & = & \displaystyle{f(x)}
\end{array}
\end{equation}

An interesting property of the heat wavelet transform as defined above is that it is {\em unitary}. This property reads
\begin{equation}
\left<WT_{f_1}(\rho,g),WT_{f_2}(\rho,g)\right>_{\text{\tiny{${\mathbb R}^{+} \times SO(3)$}}} = \left< f_1,f_2\right>_{\text{\tiny{$SO(3)$}}}
\end{equation}
where we use the notation $\left<.,.\right>_{\text{\tiny{${\mathbb R}^{+} \times SO(3)$}}}$ for the scalar product between wavelet transforms. This unitary condition makes possible to compare density in the wavelet domain for example.

We have introduced the Heat wavelet transform (also called diffusive wavelet transform) for probability densities of rotation random variables. In the next Section, we will make use of this transform to define an estimator for densities on $SO(3)$.

\section{Estimation}
\label{sec:estim}
In this section, we consider the problem of estimating the probability density function of a $SO(3)$-valued random variable given an independent sample of size $K$: $\left\{X_1,X_2,\ldots,X_K\right\}$. After presenting kernel estimators as defined in \cite{Hielscher2013}, we show that the characteristic function estimator and the heat wavelet estimator are actually kernel estimators. We provide the MISE of each of them and explain their differences.

\subsection{Kernel estimators on $SO(3)$}
First, we recall some of the results given in \cite{Hielscher2013} for kernel estimators of probability density functions on $SO(3)$.

\subsubsection{Definition}
A Kernel estimator, with Kernel $\Xi \in L^2(SO(3))$ has the following expression:
\begin{equation}
\zeta_K (x) = \frac{1}{K}\sum_{k=1}^K \Xi(X_k^{-1}x)
\end{equation}
This definition is the extension of the classical kernel estimator known for densities of random variables on the line (see \cite{Tsybakov2009} for details). Once again, we emphasize that translation is made through the group action. Before introducing different types of kernels $\Xi(.)$, we introduce how to study their estimation performances.

\subsubsection{Bias and variance}
\label{subsec:BVK}
In order to characterize the Kernel estimator, we provide here the expression of its Mean Integrated Square Error (MISE). This expression is general for kernel estimators and will be of use later to analyze the behaviour of the characteristic function and the wavelet estimators.

In the sequel, we will make use of the following notation in order to distinguish the different density estimator. Every estimator, based on a simple sample of size $K$, will be denoted $\zeta^{\Delta}_K(.)$ with the superscript $\Delta$ taking the following values: $\Delta=\alpha$ for the ``general'' kernel estimator, $\Delta=\beta$ for the characteristic function estimator and $\Delta=\gamma$ for the Heat wavelet estimator. Associate quantities will exhibit the $\alpha,\beta,\gamma$ values when needed.

As known in the classical case \cite{Tsybakov2009} and as given for the $SO(3)$ case in \cite{Hielscher2013}, the MISE of the Kernel estimator $\zeta^{\alpha}_K (x)$ is made of a bias and a variance term:
\begin{equation}
MISE\left(\zeta^{\alpha}_K (x)\right) = \left\|f - {\mathbb E}\left[\zeta^{\alpha}_K\right]\right\|_2^2 + {\mathbb E}\left[ \left\| \zeta^{\alpha}_K (x) - {\mathbb E}\left[\zeta^{\alpha}_K (x)\right] \right\|_2^2 \right]
\end{equation}
with the following fact:
\begin{equation}
\displaystyle{ {\mathbb E}\left[\zeta^{\alpha}_K (x)\right]} = \displaystyle{{\mathbb E}\left[ \frac{1}{K}\sum_{k=1}^K \Xi(X_k^{-1}x) \right]} =  \displaystyle{ \frac{1}{K}\sum_{k=1}^K {\mathbb E}\left[\Xi(X_k^{-1}x) \right] } = \displaystyle{ \frac{1}{K}\sum_{k=1}^K \int_{SO(3)} \Xi(y^{-1}x)f(y)d\mu(y) } = \displaystyle{ \left(\Xi * f \right)(x)}
\end{equation}
This means that the mean value of the Kernel consists in the convolution of the density with the kernel. One can also express the bias $b_{\alpha}$ and the variance $\sigma^2_{\alpha}$ as follows:
\begin{equation}
b^2_{\alpha}=\left\|f - {\mathbb E}\left[\zeta^{\alpha}_K\right]\right\|_2^2 = \left\|f - f * \Xi \right\|_2^2
\end{equation}
and
\begin{equation}
\sigma^2_{\alpha}={\mathbb E}\left[ \left\| \zeta^{\alpha}_K (x) - {\mathbb E}\left[\zeta^{\alpha}_K (x)\right] \right\|_2^2 \right] = \frac{1}{K} \left( \left\| \Xi \right\|_2^2 - \left\| \Xi * f \right\|_2^2\right)
\end{equation}
Also, it is interesting to note that the following property holds:
\begin{equation}
\lim_{K \to \infty} \zeta^{\alpha}_K (x) = \left(\Xi * f\right)(x)
\end{equation}
As a consequence, the MISE for the kernel estimator is:
\begin{equation}
MISE(\zeta^{\alpha}_K) = \left\|f - f * \Xi \right\|_2^2 + \frac{1}{K} \left( \left\| \Xi \right\|_2^2 - \left\| \Xi * f \right\|_2^2\right)
\end{equation}

This general expression is useful for the study of kernel estimator. Using results from representation theory (Fourier series expansion), it is also possible to write the MISE in terms of the Fourier coefficients of $\Xi$ and $f$. Remembering the Parseval identity and the convolution property of the Fourier series expansion on $SO(3)$, one gets, just like in \cite{Hielscher2013} but with a slight $(2\ell+1)$ factor due to our normalization choice, the following representation:
\begin{equation}
\displaystyle{MISE(\zeta^{\alpha}_K) = \sum_{\ell \geq 1} \left( (2\ell +1) \left.\hat{f}^{l}\right.^2  (1 - \hat{\Xi}^{\ell})^2  + \frac{(2\ell +1)}{K} \left.\hat{\Xi}^{\ell}\right.^2 \left(1 - \left.\hat{f}^{l}\right.^2\right)\right)}
\label{eq:MISE_K}
\end{equation}
where the following notation was used for the Fourier coefficients of the probability density function $f$:
\begin{equation}
\left.\hat{f}^{l}\right.^2 = (2\ell +1)^{-1} \sum_{n,m=-\ell}^{\ell}\left|\hat{f}^{\ell}_{n,m}\right|^2
\end{equation}
This expression of the MISE of a kernel estimator on $SO(3)$ will be used with a specific kernel, namely the De La Vall\'ee Poussin kernel, in Section \ref{sec:simu} for comparison purposes with the characteristic function estimator and the wavelet estimator.

\subsection{Characteristic function estimator on $SO(3)$}

The characteristic function estimator presented here was used for example in \cite{Said2010} to provide a non-parametric estimation of the density of a compound Poisson process on the rotation group $SO(3)$. Recall that our notation is $\zeta^{\beta}_K (.)$ for this estimator. It is shown here that this estimator is in fact a kernel estimator and use can be made of the MISE expressions given in Section \ref{subsec:BVK}. Note that we consider the truncated version of the estimator, with $\ell_{max}=L$, given by:
\begin{equation}
\zeta^{\beta}_K (x) =  \sum_{\ell = 0}^{L}\sum_{n=-\ell}^{\ell}\sum_{m=-\ell}^{\ell} \left(2\ell+1\right) \widetilde{\hat{f}^{\ell}_{nm}} D^{\ell}_{nm}(x)
\end{equation}
 and where:
\begin{equation}
\widetilde{\hat{f}^{\ell}_{nm}}=\frac{1}{K}\sum_{k=1}^K \overline{D^{\ell}_{nm}}(X_k)
\end{equation}
The use of truncated version of the Fourier expansion is necessary for obvious computational reasons. Such a truncation has effects on the performance of the estimator as it only converges to a low-resolution version of the density. However, it is well-adapted to naturally band-limited functions. In the simulation in Section \ref{sec:simu}, we will investigate the influence of the bandwidth on the estimation performances.

The definition of the characteristic kernel estimator leads by simple calculation to:
\begin{equation}\label{eq:charfunctest}
\zeta^{\beta}_K (x) = \frac{1}{K} \sum_{k=1}^K \sum_{\ell \geq 0}(2\ell+1) \chi^{\ell}(X_k^{-1}x)
\end{equation}

This can be obtained thanks to the following property (see \cite{Hielscher2010} for example):
\begin{equation}
\sum_{n=-\ell}^{\ell} \sum_{m=-\ell}^{\ell} \overline{D^{\ell}_{nm}(x)} D^{\ell}_{nm}(y) = \chi^{\ell}(x^{-1}y)
\end{equation}
Note that this estimator indirectly estimates the density, as it is designed to estimate its Fourier coefficients. The estimator $\widetilde{\hat{f}^{\ell}_{nm}}$ is the characteristic function estimator. From equation (\ref{eq:charfunctest}), one can see that the characteristic function estimator is a Kernel estimator, with the specific kernel:

$$\Xi (.) = \sum_{\ell \geq 0}^{\ell_{max}=L} (2\ell+1) \chi^{\ell}(.)$$

From the expression of the MISE for a kernel estimator given in Equation (\ref{eq:MISE_K}), one can directly deduce the MISE for the characteristic function estimator. It suffices to replace the Fourier coefficients $\hat{\Xi}^{\ell}$, remembering the linearity property of the Fourier expansion, by the sum of the unit coefficients up to $\ell_{max}=L$. This make the characteristic function estimator a very ``simple'' kernel with constant Fourier coefficients. Illustration of its behaviour compared to other estimators will be presented in Section \ref{sec:simu}.

\subsection{Linear diffusive wavelet estimator}

Using the diffusive wavelet introduced in Section \ref{subsec:WTSO3}, it is possible to build an estimator of the density $f$ through its wavelet expansion. Recall that with our notation, the wavelet estimator is denoted $\zeta^{\gamma}_K (.,.)$. Note also that an extra scalar parameter is introduced when using the wavelet estimator. This is the scaling parameter of the wavelet transform. The estimator based on the wavelet transform consists in replacing the wavelet transform $WT_f(\rho,x)$ in the inversion formula given in Equation (\ref{eq:invWTSO3}) by its empirical estimate, obtained from the data sample. The wavelet coefficients are thus estimated using:
\begin{equation}
\widetilde{WT}_f(\rho,x) = \frac{1}{K} \sum_{k=1}^K \check{\Psi}_{\rho}(X_k^{-1} x)
\end{equation}
The estimated density $\zeta^{\gamma}_K (x)$ takes the following expression when plugging the estimated coefficients in the inversion formula:
\begin{equation}
\zeta^{\gamma}_K (x,t) = \frac{1}{K}\sum_{k=1}^K \int_{t}^{+\infty}\int_{SO(3)} \check{\Psi}_{\rho}(X_k^{-1} y) \Psi_{\rho}(y^{-1} x) d\mu(y)\alpha(\rho)d\rho
\end{equation}
The 'scaling' coefficient $\rho$ is a parameter for this estimator. Ideal range should be $0$ to $+\infty$. Obviously it is not possible to reach the upper limit nad one can only numerically tend to very high values. As can be seen in the Simulation section, it is not a limitation for the use of the estimator. In the sequel, we will only keep the $t\geq 0$ constraint and keep in mind the upper limit issue.

It can easily be shown that the wavelet estimator can be expressed using the kernel $\kappa_t$ previsouly introduced in Equation (\ref{eq:heatkernelSO3}).
The following equality thus stands:
\begin{equation}
\begin{array}{rcl}
\displaystyle{\int_{t}^{+\infty}\int_{SO(3)} \check{\Psi}_{\rho}(X_k^{-1}g) \Psi_{\rho}(g^{-1} x) d\mu(g)} & = & \displaystyle{ \int_{t}^{+\infty}\int_{SO(3)} \check{\Psi}_{\rho}(g') \Psi_{\rho}(g^{'-1} X_k^{-1}x) d\mu(g') }\\
& = & \displaystyle{ \int_{t}^{+\infty} \left( \check{\Psi}_{\rho} * \Psi_{\rho}\right) (X_k^{-1}x) \alpha(\rho)d\rho} \\
& = & \displaystyle{\kappa_t(X_k^{-1}x)}
\end{array}
\end{equation}
which is obtained by simple change of variable $g'=X_k^{-1}g$ and using the fact that the Haar measure is bi-invariant, inducing that $d\mu(X_k g')=d\mu(g')$. As a consequence, the linear wavelet estimator is:

\begin{equation}
\displaystyle{\zeta^{\gamma}_K (x,t) = \frac{1}{K}\sum_{k=1}^K \kappa_t(X_k^{-1}x)}
 \end{equation}
where one can obviously see that this estimator is a kernel estimator. In order to study this estimator, one can look at its MISE with the bias and variance term. The bias term is:

\begin{equation}
\begin{array}{rcl}
\displaystyle{ {\mathbb E}\left[\zeta^{\gamma}_K (x,t)\right]} & = & \displaystyle {{\mathbb E}\left[\frac{1}{K}\sum_{k=1}^K \int_{t}^{+\infty}\int_{SO(3)} \check{\Psi}_{\rho}(X_k^{-1} y) \Psi_{\rho}(y^{-1} x) d\mu(y)\alpha(\rho)d\rho\right]} \\
 & = & \displaystyle{ \frac{1}{K}\sum_{k=1}^K \int_{t}^{+\infty}\int_{SO(3)} \int_{SO(3)} f(z) \check{\Psi}_{\rho}(z^{-1} y) \Psi_{\rho}(y^{-1} x)d\mu(z) d\mu(y)\alpha(\rho)d\rho} \\
  & = & \displaystyle{ f * \int_{t}^{+\infty} \left( \check{\Psi}_{\rho}*\Psi_{\rho} \right) (x) \alpha(\rho) d\rho } \\
  & = & \displaystyle{\left(f*\kappa_t\right)(x)}
\end{array}
\end{equation}
Note that we have:
\begin{equation}
\lim_{t \rightarrow 0} {\mathbb E}\left[\zeta^{\gamma}_K (x,t)\right] = f(x)
\end{equation}
Note that this is due to the fact that we used an approximate convolution identity. Thus, the bias of the linear wavelet estimator takes the following form:
\begin{equation}
b^2_{\gamma,t}=\left\|f - {\mathbb E}\left[\zeta^{\gamma}_K(t)\right]\right\|_2^2 = \left\|f - f*\kappa_t \right\|_2^2
\end{equation}
It is noticeable that, asymptotically with $t$, the bias term vanishes:
\begin{equation}
\lim_{t \rightarrow 0} b^2_{\gamma,t} = 0
\end{equation}
thanks to the definition of $\kappa_t$. This makes the wavelet estimator an unbiased estimator when $t$ reaches $0$. Now, for the variance term, notice first that:
\begin{equation}
\begin{array}{rcl}
\displaystyle{{\mathbb E}\left[ \zeta^{\gamma}_K(x,t) - {\mathbb E}\left[\zeta^{\gamma}_K(x,t)\right]\right]^2} & = & \displaystyle{var\left[\zeta^{\gamma}_K(x,t)\right]} \\
& = & \displaystyle{\frac{1}{K^2}  \sum_{k=1}^K \operatorname{var} \left[ \ \kappa_t(X_k^{-1}x) \right]} \\

 \end{array}
 \end{equation}

It comes then naturally that the variance term takes the form:
\begin{equation}
\begin{array}{rcl}
\displaystyle{\sigma^2_{\gamma,t}} & = & \displaystyle{\int_{SO(3)}var\left[\zeta^{\gamma}_K(x,t)\right] d\mu(x)}\\
 & = & \displaystyle{\frac{1}{K^2}\sum_{k=1}^K\int_{SO(3)} \left({\mathbb E}\left[\kappa_t^2(X_k^{-1}x)\right] - {\mathbb E}^2\left[\kappa_t(X_k^{-1}x)\right] \right) d\mu(x)} \\
  & = & \displaystyle{ \frac{1}{K^2}\sum_{k=1}^K\int_{SO(3)} \int_{SO(3)} \kappa_t^2(z^{-1}
  x)f(z)d\mu(z)d\mu(x) - \frac{1}{K^2}\sum_{k=1}^K \int_{SO(3)} \left(f*\kappa_t\right)^2(X_k)d\mu(X_k)}
 \end{array}
\end{equation}
where we made use of the bi-invariance of the Haar measure and of the independence between the samples $X_k$. Finally, it is possible to give the expression of the variance as:
\begin{equation}
\sigma^2_{\gamma,t}={\mathbb E}\left[ \left\| \zeta^{\gamma}_K (x,t) - {\mathbb E}\left[\zeta^{\gamma}_K (x,t)\right] \right\|_2^2 \right] = \frac{1}{K} \left( \left\| \kappa_t \right\|_2^2 - \left\| f*\kappa_t \right\|_2^2\right)
\end{equation}

As was noticed earlier, the wavelet estimator is a kernel estimator. Using the general formula for the MISE of kernel estimators and the expressions of the bias and variance given above, one gets the following expression for the MISE:

\begin{equation}
MISE(\zeta^{\gamma}_K,t)=\sum_{\ell \geq 1} \left( (2\ell +1) \left.\hat{f}^{l}\right.^2  (1 - e^{-\ell(\ell+1)t})^2  + \frac{(2\ell +1)}{K} e^{-2\ell(\ell+1)t} \left(1 - \left.\hat{f}^{l}\right.^2\right)\right)
\end{equation}

It is then possible to give a first order approsimation of this expression. This takes the form:
\begin{equation}
MISE(\zeta^{\gamma}_K,t)=\sum_{\ell \geq 1} \left[\frac{(2\ell +1)(1-2\ell(\ell+1)t)}{K} +  \left.\hat{f}^{l}\right.^2  \frac{(2\ell+1)}{K} \left( 2\ell\left(\ell+1\right)t\left(1+K\right)-1\right) \right]
\end{equation}
This expression will be used in the Simulation Section. It is of interest to note that the behaviour of the MISE with parameter $t$ is as follows:
\begin{equation}
\lim_{t \rightarrow 0} MISE \left(\zeta^{\gamma}_K,t \right) = \sum_{\ell \geq 1}\frac{(2\ell + 1 )}{K}(1-\left.\hat{f}^{l}\right.^2)
\end{equation}

Also notice that as $f$ is a probability density, then $f(x) \geq 0$ almost everywhere and we have that $\hat{f}^{0}=1$ because $\hat{f}^{0}=\int_{SO(3)}f(x)d\mu(x)$.

We have presented the MISE for the kernel estimator and for the characteristic function and linear wavelet estimators. The three studied estimators belong to the kernel estimator family, but they do have differences. The given expressions will allow allow their behaviours to be compared. The following section highlights these differences.

\section{Simulations}
\label{sec:simu}
In this section, we present some simulations that illustrate the differences between the three considered estimators studied in this paper. The comparison is performed in terms of the respective MISE computed in the context of estimation of a mixture of densities on the rotation group $SO(3)$.

\subsection{Definition of a test function}
 First, we have to define a test function $f$ on the rotation group, which we take similar to the one studied in \cite{Hielscher2013}. This mixture takes the form:
\begin{equation}
f(\bm x) = 0.2 + 0.7\psi_{\text{VP}}^{30}(\bm R(\bm e_1,\pi/6)\cdot \bm x) + 0.1\psi_{\text{VP}}^{45}(\bm R(\bm e_2,4\pi/9)\cdot \bm x),
\end{equation}
where $\bm R(\eta, \omega)$ denotes the rotation of angle $\omega$ and axis $\eta$, and $\psi_{\text{VP}}^\kappa$ is the de La Vallee Poussin kernel. This kernel has a closed form expression, which reads
\begin{equation}
\psi_{\text{VP}}^\kappa(x)= \frac{(2\kappa+1)2^{2\kappa}}{\binom{2\kappa+1}{\kappa}}\cos^2\left(\frac{\omega(x)}{2}\right) = \binom{2\kappa+1}{\kappa}^{-1}\sum_{\ell=0}^\kappa(2\ell+1)\binom{2\kappa+1}{\kappa-\ell}\chi^\ell(x).
\end{equation}
The parameter $\kappa$ can be understood as the analog of the $\rho$ parameter in the heat kernel wavelet approah, that is it plays the equivalent role of a bandwidth.
\subsection{Computing the MISE}
We recall that the MISE can be expressed in the Fourier domain for a kernel function $\Xi$ by
\begin{equation}
MISE(\Xi) = \sum_{\ell \geq 1} \left( (2\ell +1) \left.\hat{f}^{\ell}\right.^2  (1 - \left.\hat{\Xi}^{\ell}\right.)^2  + \frac{(2\ell +1)}{K} \left.\hat{\Xi}^{\ell}\right.^2 \left(1 - \left.\hat{f}^{\ell}\right.^2\right)\right).
\end{equation}
The last expression of the MISE  in terms of the Fourier coefficients of both the test function $f$ and the chosen kernel $\Xi$ allows us to compute the MISE in a simple way. The Fourier coefficients of the kernel are in general known, as it is the case here for the kernel used (de la Vallée Poussin, Heat kernel, and characteristic function kernel). However the coefficients of $f$ have to be computed with a numerical implementation of the Fourier transform on the rotation group.

We used in our simulation an implementation of the FFT on $SO(3)$ as proposed by Kostelec and Rockmore \cite{kostelec2008ffts}. This FFT is based on a equiangular sampling of the Euler angles. Since our test function $f$ is a linear combination of de la Vall\'ee Poussin kernels, it is bandlimited by the largest $\kappa$ value chosen, that is in our case $\kappa = 45$. The FFT was thus performed up to degree $L = 49$, leading to 166650 complex valued Fourier coefficients $\hat{f}_{lmn}$. The energy per degree $\hat{f}^{\ell^2}$ can be computed like:
\begin{equation}
\hat{f}^{\ell^2} = \frac{1}{(2\ell+1)}\sum_{m,n = -\ell}^\ell\vert f_{lmn}\vert^2.
\end{equation}
This expression, up to the maximum value of $\ell$ is then plugged into the MISE expression.
\subsection{Results}
In figure \ref{fig:kernels} we have plotted the kernels used in our simulations. We consider here only three kernel types, the de La Vall\'ee Poussin kernel, the Heat kernel (sometimes called the Gauss-Weierstrass kernel) and the characteristic function kernel. For each of these kernels we looked at different bandwidth parameters, respectively $\kappa$, $\rho$, and $L$, the latter being the truncation order in the characteristic function estimator. The plots in figure \ref{fig:kernels} emphasize the role of the respective bandwidth, as the concentration of the kernel functions increases with larger bandwidths. We also note that the de La Vall\'ee Poussin and heat kernels are nonnegative kernels, whereas the characteristic function kernel exhibits negative values. Such behaviour has drawbacks, especially when the density to estimate exhibits narrow modes.
\begin{figure}
\centering
\includegraphics[scale=.5]{kernels}\caption{Kernels used in the simulations. From left to right: Heat kernel, La Vall\'ee Poussin kernel and characteristic function kernel. Kernels are displayed for different values of the ``bandwidth'' parameter.}\label{fig:kernels}
\end{figure}

Before stepping into the MISE figures, we recall an important result from Hielscher \cite[theorem 3]{Hielscher2013}, which gives the MISE optimal bound for the function $f$. It reads explicitly as
\begin{equation}\label{eq:MISE_optimal_bound}
MISE_\text{opt} = \sum_{\ell=1}^\infty (2\ell+1)\frac{\hat{f}^{\ell^2}(1-\hat{f}^{\ell^2}}{(K-1)\hat{f}^{\ell^2}+1}
\end{equation}
where we have adapted the prefactor in $(2\ell +1)$ due to our normalization choice. This optimal MISE is displayed on figures \ref{fig:MISE1}, \ref{fig:MISE2} and \ref{fig:MISE3} for comparison with the studied estimators.

We investigate now the behavior of each kernel and its influence on the MISE for different values of bandwidth parameters. It is important to note that in each experiment, bandwidths were fixed, as opposed to what was done in \cite{Hielscher2013} were the bandwidths were chosen as optimal with respect to the test function and the number of observations. Our approach is indeed motivated by a multiresolution approach, with the minimum amount of information about the function $f$ being incorporated in the kernels.

In figure \ref{fig:MISE1}  we have the MISE evaluated for de la Vall\'ee Poussin kernel with bandwidth parameter ranging for $\kappa  = 1,8,17,22,29,36,43$ along the theoretical lower bound. In figure \ref{fig:MISE2}  we have the MISE evaluated for the heat kernel with bandwidth parameter ranging from $\rho = 2^{0}$ to $\rho = 2^{-9}$, (which corresponds to a dyadic scaling) along the theoretical lower bound. Finally, in figure \ref{fig:MISE3} we have the MISE evaluated for the characteristic function kernel with bandwidth parameter ranging from $L=1$ to $L=9$, along the theoretical lower bound.

As expected with a multiresolution approach, for a given bandwidth the MISE reaches a lower threshold after a sufficient number of observations $K$. Moreover as the bandwidth increases the threshold is lowered. Comparing the three kernel types, the de La Vall\'ee Poussin kernel seems to converge slower than the heat or characteristic function kernel, as the threshold lowers slowly. Therefore, the de La Vall\'ee Poussin kernel seems less appropriate for a multiresolution analysis.
The heat kernel and characteristic function kernel are performing better, with a slight advantage to the characteristic function kernel as it comes closest to the MISE optimal bound. We note however that the characteristic function kernel does not have the nonnegative property of the heat kernel, which may be critical when the density to estimate possesses sharpened modes. It is also noticeable that for small sizes of samples (from $10^1$ to $10^2$) the wavelet estimator gets closer to the optimal bound. This suggests that an adaptive choice of the parameter $\rho$ could lead to very good estimates in most of the situations.
\begin{figure}
\centering
\includegraphics[scale=.5]{MISE1}\caption{MISE computed for a de la Vallee Poussin kernel type, for bandwidth values $\kappa  = 1,8,17,22,29,36;43$ (from light yellow to dark red). The optimal MISE bound given in Equation (\ref{eq:MISE_optimal_bound}) is displayed in black for comparison. }\label{fig:MISE1}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=.5]{MISE2}\caption{MISE computed for the heat kernel, for bandwidth values $\rho  = 2^{-j}$ with $j=1,2,\ldots,9$ (from light yellow to dark red). The optimal MISE bound given in Equation (\ref{eq:MISE_optimal_bound}) is displayed in black for comparison. }\label{fig:MISE2}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=.5]{MISE3}\caption{MISE computed for the characteristic function kernel, for bandwidth values $L=1,2,\ldots,9$ (from light yellow to dark red).The optimal MISE bound given in Equation (\ref{eq:MISE_optimal_bound}) is displayed in black for comparison.	}\label{fig:MISE3}
\end{figure}

\section{Conclusion}

We have demonstrated that the characteristic function estimator and the linear Heat wavelet estimator on $SO(3)$ both belong to the larger family of kernel estimators for densities on $SO(3)$. The characteristic function estimator consists in a kernel estimator with constant Fourier coefficients up to a maximum bandwidth, while the wavelet estimator leads to a Heat kernel with coefficients driven by the scaling parameter of the heat wavelet family. The MISEs of the introduced estimator have been presented and illustration of the differences between heat kernel, characteristic function and De La Vall\'ee Poussin kernels investigated. The diffusive wavelet based estimator combines the nice property of converging faster than the De La Vall\'ee Poussin kernel and of being strictly positive (as opposed to the characteristic function kernel), allowing good performances in many configurations. The results presented demonstrate the advantage of using the heat wavelet transform for probability density estimation on the rotation group $SO(3)$. Future work should consist in studying the nonlinear (thresholded) version of the heat wavelet estimator which is known to perform better than linear wavelet estimator on the real line. Validation on real datasets should also be investigated to completely validate the proposed estimator.

\bibliographystyle{IEEEtran}

redpajama_set_name: RedPajamaArXiv


The World Is A Neat Place, Usually: Adopt Barbara Boxer!
Wow, they couldn't just wait for the end of the month to shoot themselves in the foot?
Barbra Boxer is complaining that since Condolizza Rice doesn't have children (whom will "pay the price" for her decisions), she doesn't have a right to make important decisions about the military of the nation.
Wow. Once again, the Demcrats prove that we need a drug plan, so I can GET the drugs these people are using. I want to be that disconnected from reality. Sadly, to get that way, it seems I need to read more Marx, DailyKOS, and consider Michael Moore a "serious" film maker. I'm not that insane yet.
However, I think the perfect solution has been hit upon. Condolizza Rice should adopt Barbra Boxer. It would solve everything!


text:
Matt Dorado
Serial Killers and Supper
Bonsoir, mortals! Your friend, the Drunken Devil, is delving into a different type of territory with a brand-new, theatrical dining experience: To Live and Di(n)e in LA. This special event will be part of Think Tank Gallery's latest art show, Drinkin', Smokin', and West Coastin' – billed as a "group love/hate letter to the city," with a creative take by local event producers each night.
To Live and Di(n)e in LA is an intimate, 45-person supper club, where guests will sit down for dinner with the ghosts of the most infamous figures in Los Angeles' true crime history, as they navigate the city's sordid past through cuisine, cocktails, and cabaret-type performances.
"While we've had such great success with our past parties, Bloody Gras and Curse of the Jungle Drums, I want to return to my roots by providing something much smaller and much more sinister than anything we've produced as a company," says Matt Dorado, the Creator and Executive Producer of Drunken Devil. "I've been craving the opportunity to produce a dining experience, and I think this will be a phenomenal first entry for the company."
In To Live and Di(n)e in LA, guests will sit down for dinner with the dead, as they come face-to-face with Elizabeth Short, better known as the Black Dahlia; Harvey Glatman, infamously nicknamed the "Glamour Girl Slayer" for his harrowing methods of killing aspiring young models in the late 1950's; Susan Atkins, one of the most prominent members of the Manson Family (although Charles will not appear, his ominous presence will be noticeable throughout the event); and Richard Ramirez, also known as the "Night Stalker," who terrorized suburban Los Angeles for years.
The menu will be created by Felix Barron, executive chef of KTCHNLA. Each course will be based off of a different figure and different era, with thematically appropriate cocktails paired to each dish. "For example, as you sip cocktails and enjoy hors d'oeuvres inspired by the Black Dahlia and frenzy of the city circa 1947, you'll be mingling with Elizabeth Short and a host of characters from that time period. Then, you'll sit down for the first course and find yourself in 1958, as Harvey Glatman's story unfolds before you," says Dorado. "The food interweaves with the theatrical and interactive elements, creating an entirely unique narrative."
In between courses, guests will experience dark burlesque, performance art and more that will help guide the the story. "Theatrical, cabaret-type performances are integral to the Drunken Devil experience, and we have a number of eye-popping acts that we're excited to bring to the event," Dorado says.
To Live and Di(n)e in LA is the first in what Dorado hopes to be a recurring series. "My goal is to have this show serve as a launching point for us to produce more intimate, terror-tinged experiences, bringing the distinct horror edge of Drunken Devil to the dinner table."
Guests will be required to sign a waiver of consent, as some of the interactive moments may involve contact with actors. Audience members should also note that To Live and Di(n)e in LA explores mature subject matter including violence and disturbing sexual content.
The event will take place at Think Tank Gallery, 939 S Maple Ave, Los Angeles, 90014, on Thursday, September 21, 2017. The show begins with a cocktail hour at 7:00pm. Tickets are $165.00 each and include the four-course meal, cocktails, and performances. Only 45 tickets are available. 21+.
Tagged: dinner, supper club, serial killer, horror, burlesque, downtown la


text:
Q: SQL to return NULL rows and corresponding next rows I have a mysql table where a column can have NULL values. Can someone help me with a SQL to return all the rows with the column is null, and as well as the next non-null rows?
Eg:
row 1, NULL
row 2, foo 1
row 3, foo 2
row 4, NULL
row 5, foo 3
row 6, foo 4
row 7, foo 5
row 8, NULL
row 9, foo 6

SQL will return:
row 1, NULL
row 2, foo 1
row 4, NULL
row 5, foo 3
row 8, NULL
row 9, foo 6

A: I would suggest something like
select * from table t1
where t1.col2 = null
or (t1.col2 != null and exists (select 1 from table t2 where t1.col1 - 1 = t2.col1 and t2.col2 = null))

We return rows whos col2 is null, or if the col2 is null, we run a subquery to see if the row one before it (col1 - 1) has null in col2, if so we return that too.

A: Here's one way to do it using user defined variables:
select col1, col2
from (
  select *,
    @showRow:=IF(@prevCol2 is NULL,1,0) showRow,
    @prevCol2:=col2
  from yourtable
    join (SELECT @showRow:= 0, @prevCol2:= NULL) t
  ) t
where showRow = 1 or col2 is null
order by col1, col2

SQL Fiddle Demo

A: Basing on your comment, update the answer. Actually for your case, you don't need any order by in the last, you need the native order of data I think. Probably I'm wrong, if so, please add order by as you need.
I borrow @sgeddes's solution, actually his solution fit your case best as far as I know. Only a tiny change is the second variable intialization, instead of NULL, give a non-null contanst value, then it can handle the first fews rows of data are not null scenario.

select col1, col2
from (
  select *,
    @showRow:=IF(@prevCol2 is NULL,1,0) showRow,
    @prevCol2:=col2
  from yourtable
    join (SELECT @showRow:= 0, @prevCol2:= 'foo') t
  ) t
where showRow = 1 or col2 is null

SQL FIDDLE DEMO


text:
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Mephitsu
Review : Doctor Who, The Jungles of Mechanus Dalek Set, Character Exclusive
In August 2020, Character Options announced an exclusive Dalek set based on the 1965 story, The Chase. The set was sold exclusively online via the Character website and contained two of the "movie" Daleks that were loaned to the BBC and used as background Daleks on The Chase. The Jungles of Mechanus set was released just two weeks after the announcement, arriving with collectors in early September 2020.
Packaging 4.5
Our Jungles of Mechanus set is delivered in an outer box brown sleeve with the packed figure within. Interestingly there is no packing or protection and should the outer box be damaged in anyway in transit the inner box will also suffer.
The design principle is the same as the one established in 2019 and used on the B&M exclusive sets. It is however interesting that while the Jungles set uses the principles, it is not the same as the boxes used for the B&M Dalek packs. To start with the box has been reversed, with the TARDIS and logo panel now sitting on the right and not the left. The box has also been extended by about an inch in length and an inch in depth allowing the Daleks to face more forward in the box. While the window is now bigger and displays the Daleks much better than the smaller footprint B&M box, the lack of overall consistency may both MIB collectors a little.
The same dark blue trim and yellow tramline retain as does the red flash in the window which confirms the set as coming from The Chase (1965). There is no foil "limited" or "exclusive" sticker and instead a circular gold gut out has been added to the box that confirms this as a Character Online Exclusive. There is no mention of the 5.5 inch collectors line as seen on most other packaging of the last few years.
Like the other Dalek Sets in the past couple of years, Character have provided a very comprehensive overview on the back of the box - both for the story, The Chase, as well as the background for the Dalek props themselves.
The text is probably too extensive, giving you the full story for The Chase rather than tease the story and encourage people to go and watch it. There are also the usual grammatical errors in the text.
The Chase (June 1965)
The Doctor and his companions Ian, Barbara and Vicki are forced to flee the TARDIS when they learn from a Time/Space Visualiser, taken from a museum, that a group of Daleks equipped with their own Time Ship are on their trail with orders to exterminate them. Starting on the sandy Saggaro desert of the Planet Aridius there follows an epic 'Chase' through time and space which includes some diverse and bizarre locations including the Empire State Building, the Marie Celeste, and an old haunted house populated by classic fantasy creations. The fictional characters stalk the Doctor and his companions but also attack the Dalek squad sent to kill them. Escaping again, the next stop the TARDIS makes is the Planet Mechanus, where Daleks arrive shortly afterwards and unleash a duplicate of the Doctor in order to destroy him. The planet is covered in thick dense and aggressive jungles surrounding a strange metallic city suspended high in the sky. The Daleks' duplicate Doctor is discovered and destroyed but the Daleks continue to pursue the Doctor and his friends through the jungle. Eventually the 'Chase' leads to a final confrontation between the Mechanoids and the Daleks. Both sides are utterly destroyed and the travellers persuade the Doctor to show them how to use the now abandoned Dalek Time Ship to return Ian and Barbara to Earth.
This story, broadcast in 1965, is special for many reasons. It's the final adventure for the original TARDIS team, and features the first and last appearance of metallic creatures at one point considered to replace the Daleks, the Mechanoids. It also marks the appearance of some rather special 'Guest Daleks'. At the time of filming the BBC were short of Dalek props and borrowed two Daleks from the upcoming Dalek Movie. These are seen in altered form guarding the elevator ramp to the Dalek control centre but also in their full glory towering above their BBC counterparts in the jungle chase sequence (along with a BBC camera at one point) and stand, to this day, as the very first appearance of the Dalek props built for that production.
The set is opened by the top flap that is taped in three places. The inner tray slides out as a whole with the clear plastic Dalek tray taped into the cardboard backing section. You will need to untape the Daleks to release their tray and then cut the two string ties that hold them into position.
The cardboard piece features a raised plinth base and a decor in the style of the Jungle of Mechanus, and this is done in a semi-realistic illustration that is sensitive to the sort of artwork we saw in Doctor Who Annuals of the era. A corner piece is created to cut across the base in the shape of the bottom two thirds of the TARDIS doors.
The TARDIS itself is a little out of scale to the Daleks, and the whole thing is only really designed for the packaging and not for use out of the box. The two sides are taped heavily and plain cardboard on the side. With some work there is a good set piece here for display if you wish - and of course Al from Character has included some Easter Eggs in the artwork...
Paint & Sculpt 4.5
Since their inception, the Dalek figures from Character have been fairly modular and to create this "Movie" version the base 1960's Dalek has been planted onto the base that was used on the Planet Supreme - which itself was a Movie Dalek Prop. The two stage base as used on the Dalek Invasion of Earth Daleks is not present. The plunger and eye stalk are also 60's Dalek era while the gun may be new. It doesn't have the ridges we see on the 60's Dalek and is also not scooped in the middle like the later 70's and 80's Dalek. The ear pieces are also new and the extended translucent red with ridges built in to the cast as seen on the 60's movie Daleks.
Both Daleks are identical and feature a very crisp paint job with a duck egg silver body and deep blue metallic dome and hemispheres trimmed with gold around the central panels. The pale blue base is cast in the base colour, which is a touch too light vs the actual prop. There was a slight burr on the side of one of my Dalek skirts leaving a white scar. All the other elements appear well painted with no obvious errors on either Dalek in the set I am reviewing here.
This is the same colour scheme used on the big screen - but of course was not seen in The Chase as it was filmed in black and white. The whole look is fantastic and save the Claw Arm attachment, which obviously was outside of Characters licence and remit, these are as good a stand in for the Technicolour Movie Daleks as you will get.
Accessories N/A
The Daleks have no included accessories.
Articulation 4.0
Our Daleks are articulated essentially the same as their onscreen counterparts. The eye stalk is on a pivot and can be raised up and down. The head can also be rotated on its axis a full 360 degrees. The gun and plunger are ball joints so can rotate and move fully around the central panel to the point they meet the outer casing.
The Dalek is then fitted with three wheels under the enlarged skirt piece. There are two rear fixed wheels and a forward caster that rotates. This allows the Dalek to be rolled around and gives the impression of them sitting just off the ground.
At £24.99 the Character Exclusive set is £5 more expensive than the B&M sets - but for this you get a much nicer box and in my opinion a crisper paint job. The two Daleks are identical, but this is fine and representative of the two 'Guests" we see in The Chase. It also allows the opportunity for Army Builders to pick up more sets for a larger squad.
We were never going to get the claw for the set, that would probably tread too far into a licence that we assume Character do not own. The inclusion of the diorama base is nice for display in the box, out of the packaging it would need a bit of work to be aesthetically pleasing on all sides.
The set delivers what it sets out to do and more Daleks from Character will never be an issue for Who fans, and these are brand new to the line and not a repack of earlier releases. I score this impressive, well executed set a very strong 4.0 overall
Order your Doctor Who Jungles of Mechanus Dalek Set... while stocks last
Visit our Doctor Who Database for more Doctor Who figures
About Me : As a child of the 70's and 80's I grew up in a golden age for action figures and in my youth bought and sold myself through collections of Star Wars, G.I. Joe (Action Force) and M.A.S.K. while also dabbling in He-Man, Transformers and Ghostbusters. Roll forward and I am now reliving that Youth with the action figures of today and am a collector and fan of the larger 6-8 inch figures from my favourite movie and TV licences - including the ones mentioned above, but also the Marvel Cinematic Universe, Doctor Who and the Aliens. I launched The Mephitsu Archives in 2015 with a view of creating a UK focused site or these figures where fans can pick up the latest action figure news, read reviews and get information on where to buy their figures and what is currently on store shelves. I hope I am delivering that to you guys...
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Review : Star Wars Black Series Commander Pyre, Resistance
Review : Doctor Who Friends of the 13th Doctor Set (B&M Exclusive)
New Super7 Ultimates Conan the Barbarian figure
Mattel Jurassic World Amber Collection figures incoming of Ray Arnold, Ian Malcolm & Tiger Raptor
NECA Ultimate Greta the Gremlin, packaging shots
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text:
Volunteers in Hull
Changing the lives of volunteers in Hull UK City of Culture 2017
Did you know that by playing The National Lottery you are helping to recruit and train volunteers across the UK to help deliver a huge range of activities?
Volunteers are at the heart of events across the UK, from national and international sporting events, individual community projects and even becoming ambassadors for high profile events such as Hull UK City of Culture 2017, which has been in the spotlight this year.
Throughout 2017, National Lottery funding has helped recruit over 2,500 volunteers for events happening in Hull. These volunteers were then given extensive training in communication, interpersonal, history, customer service and artistic skills. But these skills go far beyond just helping the city, they have also built confidence and friendships within the volunteers.
How volunteer Iris became a lioness
Iris, a volunteer from Hull shares what the experience has meant to her,
"I hadn't planned on becoming a volunteer but I got talking to a lady one day last year about her experience as a Hull 2017 volunteer and it sounded really interesting. I decided to put an application in but I didn't expect to get through.
"I have dyslexic problems, eczema on my face and really lacked in confidence so I thought all this would hold me back. But I was made to feel really welcome and I got through.
"I ended up being involved in 7 Alleys, the Land of Green Ginger event in East Park. I was completely pushed out of my comfort zone as I'd never done acting or barrier carrying, but it was the best thing I could have done.
"As I was driving home after the first performance, I was in my car and 'I am so alive' by the Goo Goo Dolls came on and that summed up exactly how I was feeling. My experience woke me up and has made me realise there is no such thing as a comfort zone, I can do anything. I'm no longer the quiet person who was unsure of herself, now I'm really confident; I've gone from mouse-like to a lioness who wants to roar!"
There are plenty more events happening in Hull as part of the UK City of Culture celebration for the rest of the year, so why not pop along and ask how you can get involved?


text:
Home > Resources > The 4 key steps of a communication audit
The 4 key steps of a communication audit
Type: Podcasts
By Caroline Cornell
Zora Artis
Are your communications really adding value? Without conducting regular audits or assessments, you can't be certain. An audit, Zora Artis tells us, provides an objective picture of what is actually happening in the organization, rather than what we think is happening. Most importantly, it helps communicators understand how they are adding value and it can help spot areas for improvement.
"At the end of the day, it's going to help you do a better job as a communicator and it increases your value to your organization, so I can't see why you wouldn't do it," shares Artis, founder of Zora Artis Consulting in Melbourne, Australia, in the latest episode of CW Radio.
But, where does one begin? Artis breaks the audit process down into four key areas, and explains them as follows:
1. Identifying and setting the scope and framework
To start, gain a thorough understanding of the organization and its strategy. More specifically, find out the issues the audit should investigate and understand what you are trying to achieve.
Also in this phase, Artis recommends gaining senior management buy-in, identifying your audiences and agreeing on a methodology.
2. Discovery process
This step is focused on research. Primary research could include interviews, focus groups, surveys, etc. Secondary research will primarily review paid, earned and owned media.
3. Distillation process
The distillation process involves assessing the collected data. At this point, Artis says it's important to identify patterns in your research and draw out key insights.
Finally, identify the communication gaps and opportunities. Review areas like message quality and delivery, the quality of relationships with audiences, and audience communication overall, Artis says.
Listen to the complete interview below to hear case studies illustrating the bottom-line impact a proper communication audit can have on an organization.
You can review the #11ways research mentioned in the podcast here.
Caroline Cornell Caroline Cornell is associate editor for IABC's content department.


47 Nayvadius Cash Famous Sayings, Quotes and Quotation.
I was drawn to love songs, but I was just drawn to great music - no matter if it's hip-hop, pop, R&B or whether it's rock n' roll or country. It could be a Garth Brooks song, and if it's a smash, then I'll love the different wordplay and different melodies. That's what I'm a fan of - great music.
I've made a lot of mistakes in my life. I wouldn't say musically, though.
My three wishes are just be respected out the game. I want respect from everybody. And just make a successful clothing line and company. I'm not going to put a number on it. Just to have a great career.
Where I know I wanna go is being consistent on business, and that's just making another artist, my clothing line - capitalizing off the moment. I wanna be consistently doing that - capitalizing off every move I make.
Rihanna takes risks, and I love a woman who takes risks. It just goes to show you have your own mind and your own way of thinking.
I like all types of women. I accept them as they are when they come into my life ... But I'm not a romantic. I'm just up-front. I like to be a part of something real, not make-believe. I tell women to tell me the truth, to just lay it out. Let me be the judge and decide if I want you around or not. Let me have my choice.
In Atlanta, there's no limitation to where you can take your music. You can be as creative as you want to be.
G-Rock from the Westside, from Allen Temple. He gave my name, The Future; he was like, 'Man, you the future.' Just stuck with the name.
My favorite designer has to be Louis Vuitton.
I need to let people know who I am and instead of just trying to make great records, just be honest and make it more personal and make it more passionate, to make records with emotion and not be afraid to express that.
I wanna make my imprint in the game as far as music - hip-hop, and just music, period. 'Cause I come from hip-hop, that's my background, but I'm not gonna let that limit me from where I can go.
I can't leave home without certain movies. 'The Godfather' is a big one for me. And I've gotta have my beats so I can write new music.
I don't read blogs. I'm living the life they're writing about. So why read about it?
I've always been into fashion since I was a kid. I love fashion. I appreciate it. I just enjoy dressing up and getting all the new sneakers and all the hot exclusive clothes - I did even when I was young.
I take a little bit from everybody and add that to my life to make me, me. I wanna be nobody else but Future. When you look at me, I want you to say, 'Future.' The way I talk, the way I dress, there's nobody in the world but me.
Future's not everybody. The people who are taking my style are like my babies. I've got a tribe of kids that want to be like me ... But I understand why people want to imitate the things I do. They're dope. It comes naturally to me. My fans can expect greatness. If I wasn't me, I would want to be me.
I don't just write hits for myself, or for other artists, or to just be writing it. I write it because I was born to do this. I was given this gift, and I'm making the most of my opportunity.
At times, I think, 'What would I rather be doing than music?' That's what you have to ask yourself, if you feel like you need to be somewhere else ... But there's nothing else I want to do more than music. That's why I stay in the booth.
I like to express myself through the clothes that I wear.
You make music to change the radio, not make music for the radio.
Sometimes it's hard to open up about your personal life, your relationship because you always want the music to be in the forefront. You want the music to be the biggest carrier of everything that you represent.
I gotta have my long trench coats, a nice scarf for the winter time when you're walking around, and some nice fitted jeans to go with the trench coats.


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