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+Theory of Deep Samples
+######################
+
+
+This document derives the techniques for splitting and combining two non-solid samples of
+equal depth and thickness. These should should be used by deep image "flattening" algorithms to compute the combined colour of two samples. The formulas are defined in the document :doc:`InterpretingDeepPixels`. This document derives those formulas, and is for information only.
+
+
+
+Definitions
+===========
+
++-----------------------+--------------------------------------------------------------------------+
+|       symbol          | descripion                                                               |
++=======================+==========================================================================+
+| :math:`r`             | ratio of length of original sample to a subsample: :math:`0\le r \le 1`  |
++-----------------------+--------------------------------------------------------------------------+
+| :math:`\mathbf{C_i}`  | Colour of sample :math:`i`                                               |
++-----------------------+--------------------------------------------------------------------------+
+| :math:`\mathbf{C'_i}` | Colour of a subsample of :math:`i`                                       |
++-----------------------+--------------------------------------------------------------------------+
+| :math:`\alpha_i`      | alpha value of sample :math:`i`                                          |
++-----------------------+--------------------------------------------------------------------------+
+| :math:`\alpha'_i`     | alpha value of a subsample of :math:`i`                                  |
++-----------------------+--------------------------------------------------------------------------+
+| :math:`T_i`           | transparency/transmission of sample :math:`i`; :math:`T_a=1-\alpha_a`    |
++-----------------------+--------------------------------------------------------------------------+
+| :math:`\mathbf{c_i}`  | colour per unit length (‘instantaneous colour’) of :math:`i`             |
++-----------------------+--------------------------------------------------------------------------+
+| :math:`t_i`           | transparency per unit length (‘intanstaneous transparency’) of :math:`i` |
++-----------------------+--------------------------------------------------------------------------+
+
+Subscripts :math:`_a` and :math:`_b` refer to the input samples;
+subscript :math:`_c` refers to the combined output sample
+
+Sample Model
+============
+
+Many deep compositing operations require subdividing a sample into two
+subsamples, or else clipping a sample at a given point. When this
+happens, the alpha value of the subsample(s) must be recomputed, and the
+colour will also change. To derive an equation for computing the new
+alpha, the amount of light attenuated by each subsample must be
+considered:
+
+
+.. figure:: images/twosamples.png
+   :alt: Light passing through multiple objects from left to right. Each object attenuates a fraction t of the light
+   :width: 200px
+   :align: center
+
+   Light passing through multiple objects from left to right. Each object attenuates a fraction t of the light
+
+In this case, light passes through two objects, from left to right. Each objects attenuates the light. Light passing through any light absorbing object will by attenuated according to the **transparency**  :math:`t` of that object. Thus, if
+:math:`i` is the amount of light entering the object, the light
+:math:`o` leaving the object is given by
+
+
+.. math::
+   o=t\cdot i
+  :label: absorb
+
+Where light passes through two objects, the light is attenuated twice (the output from the first is fed into second)
+
+.. math::
+   o=t_2\cdot(t_1\cdot i)
+   :label: two
+
+where :math:`t_2` and :math:`t_1` are the transmission of the
+individual samples. In general, if the transmission :math:`t` of all
+objects are the same, the light emerging from :math:`n` adjacent samples
+is given by the *Beer-Lambert* law:
+
+.. math::
+   o=i\cdot t^n
+
+.. math::
+   T=\frac{o}{i}=t^n
+   :label: beer
+
+where :math:`T` is the total transmission. If :math:`t` is considered
+the *transmission per unit length* and :math:`n` the length, the
+equation is continuous, applying also for non-integer values of
+:math:`n`\  [1]_.
+
+An OpenEXR deep sample can then be modelled as an object of a specified
+length :math:`n` which absorbs light according to equation Beer's law :eq:`beer`
+
+Alpha vs Transparency
+---------------------
+
+OpenEXR images use alpha :math:`\alpha` instead of transparency.
+:math:`T=1-\alpha`: if :math:`\alpha=1` (so :math:`T=0`) then all light
+is absorbed; if :math:`\alpha=0` (so :math:`T=1`), then all light is
+transmitted and the material is transparent. Subsituting :math:`T` into :eq:`absorb` and :eq:`two` combined alpha of two objects is given by the *screen* compositing operation:
+
+.. math::
+   \alpha_c = 1-(1-\alpha_1)(1-\alpha_2)
+   :label: screen
+
+where :math:`\alpha_c` is the combined alpha value, and
+:math:`\alpha_1` and :math:`\alpha_2` are the alpha values of the two samples.
+
+Throughout this document, :math:`T` is preferred to :math:`\alpha` where
+it gives simpler equations.
+
+Computing :math:`t`
+-------------------
+
+OpenEXR images samples store the :math:`\alpha` of the entire sample.
+This gives us the total transmission of the sample, not the
+transmission per unit length :math:`t`. The total transmission :math:`T` 
+and the sample length :math:`n` can be used to compute the transmission per unit length:
+
+.. math:: t=T^\frac{1}{n}
+  :label: t
+
+Sample Properties
+-----------------
+
+The following are assumptions made by the formulas derived here:
+
+-  Samples have **constant optical density and colour**: if a sample has
+   length :math:`n` and a subsample of length :math:`n'` is extracted
+   from it, the RGBA colour of the subsample will be the same regardless
+   from where in the sample it is extracted. In particular, if a sample
+   is split into :math:`k` subsamples of equal length, each subsample
+   will have the same RGBA colour.
+
+-  Sample attenuation is **non-scattering** and **pixel independent**:
+   light travelling through the sample is either absorbed or
+   transmitted; it is not reflected back down the sample or scattered
+   into neighbouring samples. Scattering causes point lights to appear
+   blurred when passing through fog, and also tends to make fog look
+   more optically dense than it really is, since detail is lost very
+   quickly, even though light is being transmitted. This effect is not
+   modelled with OpenEXR volumetric samples, and must be approximated by
+   including the scattered light within the volume or applied as a
+   post-process. Since the light attenuation profile throughout a pixel
+   will not follow the Beer-Lambert equation :eq:`beer`, extra samples
+   must be used to model the pixel.
+
+-  Sample behaviour is **unit** and **scale independent**: if a sample
+   is divided into :math:`k` subsamples of equal length, then scaling
+   the depth channels :math:`z_\textsf{front}` and
+   :math:`z_\textsf{back}` of the deep image will not change the
+   colour of the samples. When merging two deep images, prescaling the
+   depth of each image by the same amount then merging the images is
+   identical to merging the original images then scaling the depth of
+   the result. This scaling property allows any unit to be used to store
+   depth, and the unit to be changed by scaling the depth channels
+   without modification of the RGBA channels.
+
+-  Sample behaviour is **position independent**: moving a sample in
+   depth will not change the RGBA values obtained by subdividing it. If
+   a sample is :math:`10` units long and a subsample of :math:`2` units
+   extracted from it, the subsample will have the same RGBA values
+   regardless of the position in depth of the sample. Shifting two
+   images by adding a constant :math:`c` to the depth channels of each
+   image then merging them is identical to merging the original images
+   then shifting the result.
+
+-  Sample splitting is **non-destructive**: using the equations derived
+   here, the samples of a pixel can be arbitrarily subdivided.
+   Recombining the subsamples with the *over* compositing operation will
+   yield the RGBA colour of the original sample (assuming sufficient
+   numerical precision).
+
+Sample Splitting
+================
+
+Alpha of a Subsample
+--------------------
+
+When a subsample of length :math:`n'` is extracted from an original
+sample of length :math:`n`, the alpha value :math:`\alpha'` for
+subsample must be computed.
+
+The total transmission :math:`T'=t^{n'}`. Substituting for :math:`t`
+computed for the entire sample length using :eq:`t` gives:
+
+.. math::
+
+   \begin{aligned}
+   T'&=\left(T^\frac{1}{n}\right)^{n'} \nonumber\\
+   T'&=T^\frac{n'}{n}\end{aligned}
+
+Here, :math:`n` refers to the *length* of the original sample, but
+samples are specified in OpenEXR with a front and back depth. Thus, if a
+sample :math:`T` with front and back depths :math:`z_\textsf{front}`
+and :math:`z_\textsf{back}` is split at point :math:`z` in space,
+the transmission :math:`T'` of the front subsample is given by
+
+.. math::
+
+   \begin{aligned}
+   T'&=T^\frac{z-z_\textrm {front}}{z_\textrm {back} - z_\textrm {front}}\\
+   \alpha'&=1-\left(1-\alpha\right)^\frac{z-z_\textrm{front}}{z_\textrm {back} -
+   z_\textrm{front}}\end{aligned}
+
+Since
+:math:`({z-z_\textrm{front}})/({z_\textsf{back} - z_\textsf{front}})`
+is the *ratio of the original sample length to the extracted sample
+length*, we can substitute this ratio for :math:`r` for the remainder.
+This is equivalent to considering subdivision of a sample of unit
+length.
+
+Colour of a Subsample
+---------------------
+
+A sample with RGB information is modelled as a cylinder. The alpha
+channel(s) of the sample are represented by some non-scattering,
+absorbent material within it, which attenuates the light passing through
+the sample. The colour channel(s) of the sample is represented by
+infinitely many light sources, which emit coloured light (but don’t
+absorb anything), with identical intensities :math:`\mathbf{c}`, evenly
+spaced through the sample:
+
+.. figure:: images/drawing.png
+   :alt: Model for a sample with colour and alpha.
+   :align: center
+   :width: 300px
+
+   Model for a sample with colour and alpha. The grey box is light
+   absorbent material with transparency :math:`t`; the red dots are
+   light sources each of which has colour :math:`\mathbf{c}`.
+
+A light source at distance :math:`x` into the sample will be attenuated
+by the amount of absorber between it and the front of the sample.
+According to Beer-Lambert, the total transmission of that part of the
+sample :math:`t` is given by
+
+.. math::
+
+   \begin{aligned}
+   t'&=T^x\\
+   \alpha'&=1-(1-\alpha)^x\end{aligned}
+
+The light which reaches the front of the sample from the individual
+light source at distance :math:`x` is :math:`\mathbf{c} T^{x}`. If there
+were a finite number :math:`N` sources, the total light reaching the
+front of the sample would be:
+
+.. math:: \mathbf{C} = \sum_{k=0}^{N}\mathbf{c}{T}^{k/N}
+
+where :math:`k/N` is the position of light :math:`k`, equal to
+:math:`x`. As :math:`N` tends to infinity, this becomes
+
+.. math::
+   \begin{aligned}
+   \mathbf{C} &= \int_0^1\mathbf{c}{T}^x {\mathop{}\!\textrm{d}}x\\
+   &=\left.\frac{\mathbf {c}{T}^x}{\log(T)}\right|_{x=0}^{x=1} \nonumber\\
+   &=\frac{\mathbf{c}(T-1)}{\log(T)}\end{aligned}
+   :label: solve
+
+
+The value :math:`\mathbf{C}` is the sample’s RGB value, as stored in the
+OpenEXR image. Rearranging gives us the colour of each
+light (or perhaps the ‘instantaneous colour’ of the sample)
+
+.. math:: \mathbf{c} = \frac{\mathbf{C}\log(T)}{T-1}
+  :label: little
+
+If the entire sample is reduced to a subsample length :math:`r`, only a
+subsection of lights are included, and we must compute the new RGB
+colour :math:`\mathbf{C'}` for the subsample. Hence:
+
+.. math::
+
+   \begin{aligned}
+   \mathbf{C'} &= \int_0^{r}\mathbf{c}{T}^x {\mathop{}\!\textrm{d}}x \\
+   &=\frac{\mathbf{c}(T^{r}-1)}{\log(T)}
+   \end{aligned}
+
+Substituting for :math:`\mathbf{c}` from :eq:`little` gives
+
+.. math::
+
+   \begin{aligned}
+   \mathbf{C'}&=\frac{\mathbf{C}\log(T)}{T-1}\frac{(T^{r}-1)}{\log(T)}   \\
+   &=\mathbf{C}\frac{(T^{r}-1)}{T-1}\end{aligned}
+
+Noting that :math:`T-1=-\alpha` and :math:`T^{r}-1=T'-1=-\alpha'` and
+multiplying top and bottom by :math:`-1` gives:
+
+.. math:: \mathbf{C'}=\mathbf{C}\frac{\alpha'}{\alpha}
+  :label: splitcolour
+
+This is exactly equivalent to **unpremultiplying** the colour by the
+original alpha value, computing the new alpha value, and
+**premultiplying** by the new value.
+
+Splitting Transparent Samples
+-----------------------------
+
+When :math:`\alpha=0` (or alternatively, :math:`T=1`), unpremultiplying in :eq:`splitcolour` would require a division by zero.
+A special case is derived for :math:`\alpha=0`. The sample model
+of the colour provided by many light sources still applies, but there
+is now no attenuation. Therefore, each light contributes equally to the
+observed colour :math:`\mathbf{C}`. The ‘instantaneous’ colour equals
+the final colour, (assuming sample length is normalised to 1)
+
+.. math::
+   \begin{aligned}
+   \mathbf{C}&=\int_0^1 \mathbf{c} {\mathop{}\!\textrm{d}}x\\
+   \mathbf{C}&=\mathbf{c}\end{aligned}
+   :label: transparent
+
+A subsection :math:`\mathbf{C'}` of length :math:`r` is given by
+
+.. math::
+
+   \begin{aligned}
+   \mathbf{C'}&=\int_0^r \mathbf{c} {\mathop{}\!\textrm{d}}x \nonumber \\
+   \mathbf{C'}&=r\mathbf{C}\end{aligned}
+
+That is, if :math:`\alpha=0`, then scaling a sample’s length by
+:math:`r` scales the colour by the same amount.
+
+Sample Merging
+==============
+
+Now, consider combining two samples :math:`a` and :math:`b` together.
+This operation is required when “tidying” a deep sample list, which is
+essential before flattening a deep image into a regular one. We assume
+that :math:`z_{\textsf{front}_a}=z_{\textsf{front}_b}` and
+:math:`z_{\textsf{back}_a}=z_{\textsf{back}_b}` so the lengths
+of each sample are the same, and they fully overlap in space. If this is
+not the case, the samples should be subdivided and those subsamples
+merged separately as described in the document “Interpreting OpenEXR
+Deep Samples”
+
+The same sample model using discrete light sources is used. However, at
+each location, there are now two light sources, :math:`\mathbf{c_a}` and
+:math:`\mathbf{c_b}`, and the light is attenuated by both :math:`t_a`
+and :math:`t_b`. We can treat this as being attenuated first by
+:math:`t_b`, then by :math:`t_a`, so the total light reaching the end of
+the sample is:
+
+.. math:: \mathbf{c_c}=(\mathbf{c_a}+\mathbf{c_b})(T_{a}T_b)^{x}
+
+By substitution into :eq:`solve` and then substituting for
+:math:`\mathbf{c_a}` and :math:`\mathbf{c_b}` from :eq:`little` gives
+
+.. math::
+
+   \begin{aligned}
+   \mathbf{C_c}&=\frac{(\mathbf{c_a}+\mathbf{c_b})(T_aT_b-1)}{\log(T_aT_b)}\\
+   \mathbf{C_c}&=\frac{\left(\frac{\mathbf{C_a}\log(T_a)}{T_a-1} +
+   \frac{\mathbf{C_b}\log(T_b)}{T_b-1}\right)\left(T_aT_b-1\right)}{\log(T_aT_b)}\\
+   \mathbf{C_c}&=\frac{\left(\frac{\mathbf{C_a}\log(1-\alpha_a)}{-\alpha_a} +
+   \frac{\mathbf{C_b}\log(1-\alpha_b)}{-\alpha_b}\right)\left((1-\alpha_a)(1-\alpha_b)-1\right)}{
+   \log\left((1-\alpha_a)(1-\alpha_b)\right)}\\\end{aligned}
+
+Multiplying both parts of the top line by -1 gives
+
+.. math::
+   \mathbf{C_c}=\frac{\left(\frac{\mathbf{C_a}\log(1-\alpha_a)}{\alpha_a} +
+   \frac{\mathbf{C_b}\log(1-\alpha_b)}{\alpha_b}\right)\left(1-(1-\alpha_a)(1-\alpha_b)\right)}{
+   \log\left((1-\alpha_a)(1-\alpha_b)\right)}
+   :label: long
+
+Note that the combined transmission is given by :math:`T_c=T_{a}T_{b}`,
+implying that the combined alpha follows the screen equation
+:eq:`solve`. Substituting into both top and bottom of :eq:`long`
+gives the formula for final colour:
+
+.. math::
+   \mathbf{C_c}=\frac{\left(\frac{\mathbf{C_a}\log(1-\alpha_a)}{\alpha_a} +
+   \frac{\mathbf{C_b}\log(1-\alpha_b)}{\alpha_b}\right)\alpha_c}{\log(1-\alpha_c)}
+   :label: combine
+
+This is the **premultiplied** combined colour. For the unpremultiplied
+colour, the :math:`\alpha_c` term can be omitted. This gives the
+transmission weighted average of the unpremultiplied input colours.
+
+Merging when One Sample is Transparent
+--------------------------------------
+
+Now suppose one of the samples (say, :math:`b`) has no alpha, so
+:math:`\alpha_b=0` and :math:`T_b=1`. From :eq:`transparent`
+:math:`\mathbf{c_b}=\mathbf{C_b}`. The combined colour
+:math:`\mathbf{C_c}` is attenuated according to :math:`T_a`, the
+transmission of :math:`a` alone, and the combined colour is given by:
+
+.. math:: \mathbf{c_c} = (\mathbf{c_a}+\mathbf{C_b})(T_a)^{x}
+
+and substituting into :eq:`solve` gives:
+
+.. math::
+
+   \begin{aligned}
+   \mathbf{C_c}&= \frac{(\mathbf{c_a}+\mathbf{C_b})(T_a-1)}{\log(T_a)}\nonumber\\
+   &=\frac{\left(\frac{\mathbf{C_a}\log(1-\alpha_a)}{-\alpha_a} +
+   \mathbf{C_b}\right)\left((1-\alpha_a)-1\right)}{\log(1-\alpha_a)} \nonumber\\
+   \mathbf{C_c}&=\frac{\left(\frac{\mathbf{C_a}\log(1-\alpha_a)}{\alpha_a} -
+   \mathbf{C_b}\right)\alpha_a}{\log(1-\alpha_a)}\end{aligned}
+
+Merging Two Transparent Samples
+-------------------------------
+
+Where both samples are transparent, the colours simply add together:
+
+.. math:: \mathbf{C_c}=\mathbf{C_a}+\mathbf{C_b}
+
+Merging Solid Samples
+---------------------
+
+For numerical stability, we must give a sensible value for
+:math:`\mathbf{C_c}` when :math:`\alpha_b=1`. Equation :eq:`combine`
+is undefined, since :math:`\log(1-\alpha_b)=\log(0)=-\infty`. We follow
+the initial model of light sources in an absorbing material, but now the
+absorbing material absorbs all light. Thus, we must be observing only
+its closest light source (that at :math:`x=0`) in the solid sample: the
+sample will absorb light from its other sources. Thus, we can treat
+:math:`b` as an infinitely thin ‘discrete’ sample at the front of the
+sample, which will block all light behind it; we can simply composite
+:math:`b` over :math:`a`. Therefore, if :math:`\alpha_b=1` and
+:math:`\alpha_a<1`, the total observed colour will be
+:math:`\mathbf{C_b}`. Transposing :math:`a` and :math:`b` gives
+:math:`\mathbf{C_c}=\mathbf{C_a}` if :math:`\alpha_a=1`
+
+Following the definitions above, when :math:`\alpha_a=1` and
+:math:`\alpha_b=1`, then :math:`\mathbf{C_c}=\mathbf{C_a}+\mathbf{C_b}`.
+However, for stability, it makes sense to define
+
+.. math:: \mathbf{C_c}=\frac{\mathbf{C_a}+\mathbf{C_b}}{2}
+
+This approach gives more stable results when there is random sampling
+error in the depth channel. Assume that the front depth of sample
+:math:`A` differs by some small amount from that of :math:`B`, so that
+:math:`z_a=\delta+z_b`. Only the front sample will be visible: if
+:math:`\delta<0`, then :math:`\mathbf{C_c}=\mathbf{C_a}`; if
+:math:`\delta>0`, then :math:`\mathbf{C_c}=\mathbf{C_b}`. Setting
+:math:`\mathbf{C_c}=\mathbf{C_a}+\mathbf{C_b}` when :math:`\delta=0`
+gives an image which is (potentially) twice as bright. If :math:`\delta`
+is random noise due to sampling, we will get a random pattern of bright
+pixels whenever :math:`\delta=0`. Using the mean colour seems a safer
+option. However, this is *not* associative: when combining three solid
+samples :math:`d`,\ :math:`e` and :math:`f` does not give the same
+result as combining first :math:`d` and :math:`e`, then combining the
+result with :math:`f`.
+
+Summary
+=======
+
+To **clip a sample** covering range :math:`z_\textsf{front}` —
+:math:`z_\textsf{back}` into a subsample
+:math:`z_\textsf{front}` — :math:`z`, or **extract a subsample** of
+length :math:`l'` from a sample of length :math:`l`:
+
+.. math::
+
+   \begin{aligned}
+   r&=\frac{z-z_\textsf{front}}{z_\textsf{back}-z_\textsf{front}} = \frac{l'}{l}\\
+   \alpha' &= 1-(1-\alpha)^r\\
+   \mathbf{C'} &=
+   \begin{cases}
+   r\mathbf{C},& \text{ if } \alpha=0\\
+   \mathbf{C}\frac{\alpha'}{\alpha},& \text{ otherwise}
+   \end{cases}\end{aligned}
+
+where :math:`\alpha` and :math:`\alpha'` are the alpha values of the
+original sample and the subsample respectively, and :math:`\mathbf{C}`
+and :math:`\mathbf{C'}` are the colours of the original sample and the
+subsample respectively
+
+To **merge two samples** :math:`a` and :math:`b` into a combined sample
+:math:`c`:
+
+.. math::
+
+   \begin{aligned}
+   \alpha_c &= 1-(1-\alpha_a)(1-\alpha_b)\\
+   \mathbf{C_c} &=
+   \begin{cases}
+   \mathbf{C_a} + \mathbf{C_b}, & \text{if } \alpha_a=0 \text { and } \alpha_b=0\\
+   \mathbf{C_a},& \text{if } \alpha_a=1 \text{ and } \alpha_b <1 \\
+   \mathbf{C_b},& \text{if } \alpha_b=1 \text{ and } \alpha_a <1 \\
+   (\mathbf{C_a} + \mathbf{C_b})/2, & \text{if } \alpha_a=1 \text { and } \alpha=1\\
+   \alpha_a\left(\frac{\mathbf{C_a}\log(1-\alpha_a)}{\alpha_a} - \mathbf{C_b}\right)/\log(1-\alpha_a),&
+   \text {if } 0<\alpha_a<1 \text { and } \alpha_b=0 \\
+   \alpha_b\left(\frac{\mathbf{C_b}\log(1-\alpha_b)}{\alpha_b} - \mathbf{C_a}\right)/\log(1-\alpha_b),&
+   \text {if } 0<\alpha_b<1 \text { and } \alpha_a=0 \\
+   \alpha_c\left(\frac{\mathbf{C_a}\log(1-\alpha_a)}{\alpha_a} +
+   \frac{\mathbf{C_b}\log(1-\alpha_b)}{\alpha_b}\right)/\log(1-\alpha_c),& \text{otherwise}
+   \end{cases}\end{aligned}
+
+.. [1]
+   Various forms of the Beer-Lambert law exist, using different
+   constants. For example, :math:`o=i\cdot10^{-\epsilon l c}`, where
+   :math:`\epsilon` and :math:`c` are the molar absorptivity and
+   concentration respectively. This is equivalent to :eq:`beer` if
+   :math:`t=10^{-\epsilon c}` and :math:`l=n`
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diff --git a/docs/index.rst b/docs/index.rst
index 485e788246..859129bf4c 100644
--- a/docs/index.rst
+++ b/docs/index.rst
@@ -59,6 +59,7 @@ Technical Documents
    OpenEXRFileLayout
    MultiViewOpenEXR
    InterpretingDeepPixels
+   TheoryDeepPixels
    StandardOptionalAttributes
               
 * :ref:`genindex`