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1 :	jhr	16	\documentclass[11pt]{article}
2 :
3 :			\input{defs}
4 :
5 :			\setlength{\textwidth}{6in}
6 :			\setlength{\oddsidemargin}{0.25in}
7 :			\setlength{\evensidemargin}{0.25in}
8 :			\setlength{\parskip}{5pt}
9 :
10 :	jhr	17	\title{Diderot design}
11 :	jhr	16	\author{
12 :			Gordon Kindlmann \\
13 :			University of Chicago \\
14 :			{\small\tt{}glk@cs.uchicago.edu} \\
15 :			\and
16 :			John Reppy \\
17 :			University of Chicago \\
18 :			{\small\tt{}jhr@cs.uchicago.edu} \\
19 :	jhr	17	\and
20 :			Thomas Schultz \\
21 :			University of Chicago \\
22 :			{\small\tt{}t.schultz@uchicago.edu} \\
23 :	jhr	16	}
24 :			\date{\today}
25 :
26 :			\begin{document}
27 :
28 :			\maketitle
29 :			\thispagestyle{empty}
30 :
31 :			\section{Introduction}
32 :	glk	20	This document is a semi-formal design of Diderot.
33 :	jhr	16
34 :			\section{Types}
35 :	jhr	17	The syntax of Diderot types is given in \figref{fig:types}.
36 :			\begin{figure}[t]
37 :			\begin{displaymath}
38 :			\begin{array}{rclr}
39 :	glk	21	\rho & ::= & $\ldots$ & \text{NRRD scalar types} \\[1em]
40 :	jhr	17	\iota & ::= & \TYbool & \text{booleans} \\
41 :			& \mid & \TYint & \text{integers} \\[1em]
42 :	glk	21	% \mu for memory, where images of rawtensors will live (or on disk)
43 :	jhr	43	\mu & ::= & \TYrawten{\Seq{d}}{\rho} & \begin{minipage}[l]{3in}\begin{flushright}
44 :	jhr	44	tensors of order $|\Seq{d}|$ and dimensions $\Seq{d}$,\\with coefficients of type $\rho$
45 :	glk	21	\end{flushright}\end{minipage}\\[1em]
46 :	jhr	43	\theta & ::= & \TYtensor{\Seq{d}} & \begin{minipage}[l]{3in}\begin{flushright}
47 :	jhr	44	tensors of order $|\Seq{d}|$ and dimensions $\Seq{d}$,\\with real coefficients
48 :	glk	21	\end{flushright}\end{minipage}\\[1em]
49 :	jhr	17	\tau & ::= & \iota \\
50 :			& \mid & \theta \\
51 :			& \mid & \TYmatrix{n}{m} & \text{$n\times{}m$ matrix} \\
52 :	glk	21	& \mid & \TYimage{d}{\mu} & \text{$d$-dimension image of $\mu$ values}\\
53 :	jhr	17	& \mid & \TYkern{k} & \text{convolution kernel with $k$ derivatives} \\
54 :	glk	21	& \mid & \TYfield{k}{d}{\theta} & \text{$d$-dimension field of $\theta$ values with $k$ derivatives} \\
55 :	jhr	17	\end{array}%
56 :			\end{displaymath}%
57 :			where $o\in\Nat$ is the tensor order, $d,n,m\in\SET{2,3}$ are dimensions,
58 :			and $k\in\Nat$ is the differentiability of a field.
59 :			\caption{Diderot types}
60 :			\label{fig:types}
61 :			\end{figure}%
62 :			We use some type abbreviations for common cases:
63 :	jhr	16	\begin{eqnarray*}
64 :	jhr	43	\TYreal & = & \TYtensor{\epsilon} \quad\text{for any $d$} \\
65 :			\TYvec{d} & = & \TYtensor{d}
66 :	jhr	16	\end{eqnarray*}%
67 :
68 :			\section{Operations}
69 :
70 :			\subsection{Scalar operations}
71 :	glk	22	\noindent{}\point Arithmetic:
72 :	jhr	16	\begin{displaymath}
73 :	jhr	17	\begin{array}{cl}
74 :			\BinopTy{{\odot}}{\tau}{\tau}{\tau}
75 :	jhr	44	& \text{for $\odot\in\SET{\OP{+},\OP{-},\OP{*},\OP{/}}$ and $\tau\in\SET{\TYint,\TYreal}$} \\
76 :			\UnopTy{\OP{-}}{\tau}{\tau}
77 :	jhr	17	& \text{for $\tau\in\SET{\TYint,\TYreal}$}
78 :			\end{array}%
79 :	jhr	16	\end{displaymath}%
80 :
81 :	glk	22	\noindent{}\point Comparisons:
82 :	jhr	16	\begin{displaymath}
83 :			\BinopTy{{\odot}}{\tau}{\tau}{\TYbool}
84 :	jhr	44	\qquad\text{for $\odot\in\SET{\OP{<},{\leq},\OP{=},{\neq}\OP{>},{\geq}}$ and $\tau\in\SET{\TYint,\TYreal}$}
85 :	jhr	16	\end{displaymath}%
86 :
87 :
88 :			\subsection{Matrix operations}
89 :
90 :			\subsection{Tensor operations}
91 :
92 :	glk	22	\noindent{}\point Addition:
93 :	jhr	16	\begin{displaymath}
94 :	jhr	43	\BinopTy{{\odot}}{\TYtensor{\Seq{d}}}{\TYtensor{\Seq{d}}}{\TYtensor{\Seq{d}}}
95 :	jhr	44	\qquad\text{for $\odot\in\SET{\OP{+},\OP{-}}$}
96 :	glk	22	\end{displaymath}%
97 :
98 :			\noindent{}\point Negation:
99 :			\begin{displaymath}
100 :	jhr	44	\UnopTy{\OP{-}}{\TYtensor{\Seq{d}}}{\TYtensor{\Seq{d}}}
101 :	glk	22	\end{displaymath}%
102 :
103 :			\noindent{}\point Scalar division:
104 :			\begin{displaymath}
105 :	jhr	44	\BinopTy{\OP{/}}{\TYtensor{\Seq{d}}}{\TYreal}{\TYtensor{\Seq{d}}}
106 :	glk	22	\end{displaymath}%
107 :
108 :			\noindent{}\point Scalar multiplication (scalar times order-N):
109 :			\begin{displaymath}
110 :	jhr	16	\begin{array}{c}
111 :	jhr	44	\BinopTy{\OP{*}}{\TYreal}{\TYtensor{\Seq{d}}}{\TYtensor{\Seq{d}}} \\
112 :			\BinopTy{\OP{*}}{\TYtensor{\Seq{d}}}{\TYreal}{\TYtensor{\Seq{d}}}
113 :	jhr	16	\end{array}%
114 :			\end{displaymath}%
115 :
116 :	glk	22	\noindent{}\point Tensor scalar multiplication (contraction of two order-N
117 :			tensors down to a scalar, \eg{} dot product of vectors, double dot
118 :			product of 2nd-order tensors)
119 :	jhr	17	\begin{displaymath}
120 :	glk	22	\begin{array}{c}
121 :	jhr	43	\BinopTy{{?}}{\TYtensor{\Seq{d}}}{\TYtensor{\Seq{d}}}{\TYreal}
122 :	glk	22	\end{array}%
123 :	jhr	17	\end{displaymath}%
124 :	glk	22	Defined by $c = A_{\vec{i}}B_{\vec{i}}$
125 :			~(\eg{} $\mathbf{a}\cdot\mathbf{b} = a_ib_i$ or $\mathbf{A:B} = A_{ij}B_{ij}$)
126 :			NOTE: In this and subsequent index notation expressions, a vector over the index variable
127 :			(\eg{} $\vec{i}$) means that the variable is in fact standing for a sequence of
128 :			contiguous index variables \\
129 :			Possible direct notation syntax: ~~ {\tt .} (period) ~~ {\tt :} (colon) ~~ {\tt dot} ~~ {\tt o}
130 :	jhr	17
131 :	glk	22	\noindent{}\point Tensor product (aka outer product; order output is sum of orders)
132 :	jhr	16	\begin{displaymath}
133 :	glk	22	\begin{array}{c}
134 :	jhr	43	\BinopTy{{?}}{\TYtensor{\Seq{d}}}{\TYtensor{\Seq{d'}}}{\TYtensor{\Seq{d}\Seq{d'}}}
135 :	glk	22	\end{array}%
136 :	jhr	16	\end{displaymath}%
137 :	glk	22	Defined by $C_{\vec{i}\vec{j}} = A_{\vec{i}}B_{\vec{j}}$ \\
138 :			Possible direct notation syntax: ~~ {\tt x} ~~ {\tt (x)} ~~ {\tt out}
139 :	jhr	16
140 :	glk	22	\noindent{}\point Matrix Multiply
141 :	jhr	17	\begin{displaymath}
142 :	glk	22	\begin{array}{c}
143 :	jhr	43	\BinopTy{{?}}{\TYtensor{\Seq{d}}}{\TYtensor{d_1 d_2}}{\TYtensor{\Seq{d}}} \\
144 :			\BinopTy{{?}}{\TYtensor{d_1 d_2}}{\TYtensor{\Seq{d}}}{\TYtensor{\Seq{d}}}
145 :	glk	22	\end{array}%
146 :	jhr	17	\end{displaymath}%
147 :	glk	22	Defined by $C_{i\vec{k}} = A_{ij}B_{j\vec{k}}$ and $C_{\vec{i}k} = B_{\vec{i}j}A_{jk}$
148 :			~(\eg{} $u_i = M_{ij}v_j$ or $T_{ijl} = S_{ijk}M_{kl}$) \\
149 :			Possible direct notation syntax: ~~ ?
150 :	jhr	17
151 :	glk	22	\noindent{}\point Contracting out last or first index of tensor (order $o \geq 1$)
152 :			with vector
153 :			\begin{displaymath}
154 :			\begin{array}{c}
155 :	jhr	43	\BinopTy{{?}}{\TYtensor{d'\Seq{d}}}{\TYvec{d}}{\TYtensor{\Seq{d}}} \\
156 :			\BinopTy{{?}}{\TYvec{d}}{\TYtensor{d'\Seq{d}}}{\TYtensor{\Seq{d}}}
157 :	glk	22	\end{array}%
158 :			\end{displaymath}%
159 :			Defined by $C_{\vec{i}} = A_{\vec{i}j}v_j$ and $C_{\vec{j}} = v_iA_{j\vec{j}}$ \\
160 :			Possible direct notation syntax: ~~ ?
161 :
162 :			\noindent{}\point Arbitrary ``tensor comprehension''. The product can also be
163 :			expressed in general index notation, and which may increase, preserve, or
164 :			decrease the tensor order.
165 :			\begin{displaymath}
166 :			\begin{array}{c}
167 :	jhr	43	\BinopTy{{?}}{\TYtensor{\Seq{d}}}{\TYtensor{p}{d}}{\TYtensor{q}{d}}
168 :	glk	22	\end{array}%
169 :			\end{displaymath}%
170 :			Defined by the conventions of Einstein summation notation. \\
171 :			Possible syntax: {\tt <A.i.j.k,B.j.k.l>} ~~ {\tt <A\_i\_j\_k,B\_j\_k\_l>}
172 :
173 :	jhr	16	\subsection{Field operations}
174 :
175 :	glk	22	\noindent{}\point Creation from an image:
176 :	jhr	16	\begin{displaymath}
177 :	jhr	43	\BinopTy{\OPconvolve}{\TYkern{k}}{\TYimage{d'}{\TYrawten{\Seq{d}}{\rho}}}{\TYfield{k}{d'}{\TYtensor{\Seq{d}}}}
178 :	jhr	16	\end{displaymath}%
179 :
180 :	glk	22	\noindent{}\point Scalar multiplication:
181 :	jhr	17	\begin{displaymath}
182 :			\begin{array}{c}
183 :	jhr	44	\BinopTy{\OP{*}}{\TYreal}{\TYfield{k}{d}{\theta}}{\TYfield{k}{d}{\theta}} \\
184 :			\BinopTy{\OP{*}}{\TYfield{k}{d}{\theta}}{\TYreal}{\TYfield{k}{d}{\theta}}
185 :	jhr	17	\end{array}%
186 :			\end{displaymath}%
187 :
188 :	glk	22	\noindent{}\point Scalar division:
189 :	jhr	17	\begin{displaymath}
190 :	jhr	44	\BinopTy{\OP{/}}{\TYfield{k}{d}{\theta}}{\TYreal}{\TYfield{k}{d}{\theta}}
191 :	jhr	17	\end{displaymath}%
192 :
193 :	glk	22	\noindent{}\point Negation:
194 :	glk	21	\begin{displaymath}
195 :	jhr	44	\UnopTy{\OP{-}}{\TYfield{k}{d}{\theta}}{\TYfield{k}{d}{\theta}}
196 :	glk	21	\end{displaymath}%
197 :
198 :	glk	22	\noindent{}\point Addition:
199 :	jhr	16	\begin{displaymath}
200 :	glk	21	\begin{array}{c}
201 :			\BinopTy{{\odot}}{\TYfield{k}{d}{\theta}}{\theta}{\TYfield{k}{d}{\theta}} \\
202 :			\BinopTy{{\odot}}{\theta}{\TYfield{k}{d}{\theta}}{\TYfield{k}{d}{\theta}} \\
203 :	jhr	17	\BinopTy{{\odot}}{\TYfield{k_1}{d}{\theta}}{\TYfield{k_2}{d}{\theta}}{\TYfield{\min(k_1,k_2)}{d}{\theta}}
204 :	glk	21	\end{array}%
205 :	jhr	44	\qquad\text{for $\odot\in\SET{\OP{+},\OP{-}}$}
206 :	jhr	16	\end{displaymath}%
207 :
208 :	glk	22	\noindent{}\point Differentiation:
209 :	jhr	16	\begin{displaymath}
210 :	jhr	43	\UnopTy{\OPdiff}{\TYfield{k}{d'}{\TYtensor{\Seq{d}}}}{\TYfield{k-1}{d'}{\TYtensor{d'\Seq{d}}}}
211 :	jhr	16	\qquad\text{for $k > 0$}
212 :			\end{displaymath}%
213 :
214 :	glk	22	\noindent{}\point Probing:
215 :	jhr	16	\begin{displaymath}
216 :	jhr	44	\BinopTy{\OP{@}}{\TYfield{k}{d}{\theta}}{\TYvec{d}}{\theta}
217 :	jhr	16	\end{displaymath}%
218 :
219 :			\end{document}

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