[python-dtcwt] 274/497: docs: add description of image registration

Tue Jul 21 18:06:13 UTC 2015

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ghisvail-guest pushed a commit to branch debian/sid
in repository python-dtcwt.

commit 46a42a7f061cc8b80203db93a6fda2924d222dd1
Author: Rich Wareham <rjw57 at cam.ac.uk>
Date:   Thu Jan 30 13:53:21 2014 +0000

    docs: add description of image registration
---
 docs/index.rst             |    1 +
 docs/registration-flow.svg | 3570 ++++++++++++++++++++++++++++++++++++++++++++
 docs/registration.rst      |  269 ++++
 3 files changed, 3840 insertions(+)

diff --git a/docs/index.rst b/docs/index.rst
index 68b57f1..71c5d1e 100644
--- a/docs/index.rst
+++ b/docs/index.rst
@@ -37,6 +37,7 @@ Contents
     gettingstarted
     variant
     backends
+    registration
     examples
     reference
 
diff --git a/docs/registration-flow.svg b/docs/registration-flow.svg
new file mode 100644
index 0000000..001752b
--- /dev/null
+++ b/docs/registration-flow.svg
@@ -0,0 +1,3570 @@
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diff --git a/docs/registration.rst b/docs/registration.rst
new file mode 100644
index 0000000..2e29cd0
--- /dev/null
+++ b/docs/registration.rst
@@ -0,0 +1,269 @@
+Image Registration
+==================
+
+The :py:mod:`dtcwt.registration` module provides an implementation of a
+DTCWT-based image registration algorithm. The output is similar, but not
+identical, to "optical flow". The underlying assumption is that the source
+image is a smooth locally-affine warping of a reference image. This assumption
+is valid in some classes of medical image registration and for video sequences
+with small motion between frames.
+
+Algorithm overview
+------------------
+
+This section provides a brief overview of the algorithm itself. The algorithm
+is a 2D version of the 3D registration algorithm presented in `Efficient
+Registration of Nonrigid 3-D Bodies
+<http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=5936113&tag=1>`_. The
+motion field between two images is a vector field whose elements represent the
+direction and distance of displacement for each pixel in the source image
+required to map it to a corresponding pixel in the reference image. In this
+algorithm the motion is described via the affine transform which can represent
+rotation, translation, shearing and scaling. An advantage of this model is that
+if the motion of two neighbouring pixels are from the same model then they will
+share affine transform parameters. This allows for large regions of the image
+to be considered as a whole and helps mitigate the aperture problem.
+
+The model described below is based on the model in `Phase-based
+multidimensional volume registration
+<http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1176641>`_ with changes
+designed to allow use of the DTCWT as a front end.
+
+Motion constraint
+'''''''''''''''''
+
+The three-element homogeneous displacement vector at location
+:math:`\mathbf{x}` is defined to be
+
+.. math::
+
+    \mathbf{\tilde{v}}(\mathbf{x}) \triangleq \begin{bmatrix}
+        \mathbf{v}(\mathbf{x}) \\ 1
+    \end{bmatrix}
+
+where :math:`\mathbf{v}(\mathbf{x})` is the motion vector at location
+:math:`\mathbf{x} = [ x \, y ]^T`. A motion constraint is a three-element
+vector, :math:`\mathbf{c}(\mathbf{x})` such that
+
+.. math::
+
+    \mathbf{c}^T(\mathbf{x}) \, \mathbf{\tilde{v}}(\mathbf{x}) = 0.
+
+In the two-dimensional DTCWT, the phase of each complex highpass coefficient
+has an approximately linear relationship with the local shift vector. We can
+therefore write
+
+.. math::
+
+    \frac{\partial \theta_d}{\partial t} =
+    \nabla_\mathbf{x} \theta_d \cdot \mathbf{v}(\mathbf{x})
+
+where :math:`\nabla_\mathbf{x} \theta_d \triangleq [(\partial \theta_d/\partial
+x) \, (\partial \theta_d/\partial y)]^T` and represents the phase gradient at
+:math:`\mathbf{x}` for subband :math:`d` in both of the :math:`x` and :math:`y`
+directions.
+
+Numerical estimation of the partial derivatives of :math:`\theta_d` can be
+performed by noting that multiplication of a subband pixels's complex
+coefficient by the conjugate of its neighbour subtracts phase whereas
+multiplication by the neighbour adds phase. We can thus construct equivalents
+of forward-, backward- and central difference algorithms for phase gradients.
+
+Comparing the relations above, it is clear that the motion constraint vector,
+:math:`\mathbf{c}_d(\mathbf{x})`, corresponding to subband :math:`d` at location
+:math:`\mathbf{x}` satisfies the following:
+
+.. math::
+
+    \mathbf{c}_d(\mathbf{x}) = C_d(\mathbf{x}) \begin{bmatrix}
+    \nabla_\mathbf{x} \theta_d  \\ - \frac{\partial \theta_d}{\partial t}
+    \end{bmatrix}
+
+where :math:`C_d(\mathbf{x})` is some weighting factor which we can interpret
+as a measure of the confidence we have of subband :math:`d` specifying the
+motion at :math:`\mathbf{x}`.
+
+This confidence measure can be heuristically designed. The measure used in this
+implementation is:
+
+.. math::
+
+    C_d(\mathbf{x}) = \frac{
+        \left| \sum_{k=1}^4 u_k^* v_k \right|
+    }{
+        \sum_{k=1}^4 (\left|u_k\right|^3 + \left|v_k\right|^3) + \epsilon
+    }.
+
+where :math:`u_k` and :math:`v_k` are the wavelet coefficients in the reference
+and source transformed images, subscripts :math:`k = 1 \dots 4` denote the four
+diagonally neighbouring coefficients and :math:`\epsilon` is some small value
+to avoid division by zero when the wavelet coefficients are small. It is beyond
+the scope of this documentation to describe the design of this metric. Refer to
+the `original paper
+<http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=5936113&tag=1>`_ for more
+details.
+
+Cost function
+'''''''''''''
+
+The model is represented via the six parameters :math:`a_1 \dots a_6` such that
+
+.. math::
+
+    \mathbf{v}(\mathbf{x}) =
+    \begin{bmatrix}
+        1 & 0 & x & 0 & y & 0 \\
+        0 & 1 & 0 & x & 0 & y \\
+    \end{bmatrix}
+    \begin{bmatrix}
+    a_1 \\ \vdots \\ a_6
+    \end{bmatrix}
+    \triangleq
+    \mathbf{K}(\mathbf{x}) \, \mathbf{a}.
+
+We then make the following definitions:
+
+.. math::
+
+    \mathbf{\tilde{K}}(\mathbf{x}) \triangleq \begin{bmatrix}
+        \mathbf{K}(\mathbf{x}) & \mathbf{0} \\
+        \mathbf{0} & 1
+    \end{bmatrix},
+    \quad
+    \mathbf{\tilde{a}} \triangleq \begin{bmatrix}
+        \mathbf{a} \\ 1
+    \end{bmatrix}
+
+and then the homogenous motion vector is given by
+
+.. math::
+
+    \mathbf{\tilde{v}}(\mathbf{x}) =
+    \mathbf{\tilde{K}}(\mathbf{x}) \, \mathbf{\tilde{a}}.
+
+Considering all size subband directions, we have:
+
+.. math::
+
+    \mathbf{c}_d(\mathbf{x}) \, \mathbf{\tilde{K}}(\mathbf{x}) \, \mathbf{\tilde{a}} = 0,
+    \quad \forall \ d \in \left\{ 1, \dots, 6 \right\}.
+
+Each location :math:`\mathbf{x}` has six constraint equations for six unknown
+affine parameters in :math:`\mathbf{\tilde{a}}`. We can solve for
+:math:`\mathbf{\tilde{a}}` by minimising squared error
+:math:`\epsilon(\mathbf{x})`:
+
+.. math::
+    \begin{align}
+        \epsilon(\mathbf{x}) &= \sum_{d=1}^6 \left\|
+            \mathbf{c}_d^T(\mathbf{x}) \mathbf{\tilde{K}}(\mathbf{x}) \mathbf{\tilde{a}}
+        \right\|^2 \\
+        &= \sum_{d=1}^6
+            \mathbf{\tilde{a}}^T \, \mathbf{\tilde{K}}^T(\mathbf{x}) \, \mathbf{c}_d(\mathbf{x})
+            \mathbf{c}^T_d(\mathbf{x}) \, \mathbf{\tilde{K}}(\mathbf{x}) \, \mathbf{\tilde{a}} \\
+        &= \mathbf{\tilde{a}}^T \mathbf{\tilde{Q}}(\mathbf{x}) \mathbf{\tilde{a}}
+    \end{align}
+
+where
+
+.. math::
+
+    \mathbf{\tilde{Q}}(\mathbf{x}) \triangleq \sum_{d=1}^6
+        \mathbf{\tilde{K}}^T(\mathbf{x}) \, \mathbf{c}_d(\mathbf{x})
+        \mathbf{c}^T_d(\mathbf{x}) \, \mathbf{\tilde{K}}(\mathbf{x}).
+
+In practice, in order to handle the registration of dissimilar image features
+and also to handle the aperture problem, it is helpful to combine
+:math:`\mathbf{\tilde{Q}}(\mathbf{x})` matrices across more than one level of
+DTCWT and over a slightly wider area within each level. This results in better
+estimates of the affine parameters and reduces the likelihood of obtaining
+singular matrices. We define locality :math:`\mathbf{\chi}` to represent this
+wider spatial and inter-scale region, such that
+
+.. math::
+
+    \mathbf{\tilde{Q}}_\mathbf{\chi} = \sum_{\mathbf{x} \in \mathbf{\chi}}
+    \mathbf{\tilde{Q}}(\mathbf{x}).
+
+The :math:`\mathbf{\tilde{Q}}_\mathbf{\chi}` matrices are symmetric and so can
+be written in the following form:
+
+.. math::
+
+    \mathbf{\tilde{Q}}_\mathbf{\chi} = \begin{bmatrix}
+        \mathbf{Q}_\mathbf{\chi} & \mathbf{q}_\mathbf{\chi} \\
+        \mathbf{q}^T_\mathbf{\chi} & q_{0,\mathbf{\chi}}
+    \end{bmatrix}
+
+where :math:`\mathbf{q}_\mathbf{\chi}` is a six-element vector and
+:math:`q_{0,\mathbf{\chi}}` is a scalar. Substituting into our squared error
+function gives
+
+.. math::
+
+    \epsilon_\mathbf{\chi} =
+        \mathbf{a}^T \mathbf{Q}_\mathbf{\chi} \mathbf{a} +
+        2 \mathbf{a}^T \mathbf{q}_\mathbf{\chi} + q_{0,\mathbf{\chi}}.
+
+To minimize, we differentiate and set to zero. Hence,
+
+.. math::
+
+    \nabla_\mathbf{a} \epsilon_\mathbf{\chi} =
+    2 \mathbf{Q}_\mathbf{\chi} \mathbf{a} + 2 \mathbf{q}_\mathbf{\chi} = 0
+
+and so the local affine parameter vector satisfies
+
+.. math::
+
+    \mathbf{Q}_\mathbf{\chi} \mathbf{a}_\mathbf{\chi} = - \mathbf{q}_\mathbf{\chi}.
+
+In our implementation, we avoid calculating the inverse of
+:math:`\mathbf{Q}_\mathbf{\chi}` directly and solve this equation by
+eigenvector decomposition.
+
+Iteration
+'''''''''
+
+There are three stres in the full registration algorithm: transform the images
+to the DTCWT domain, perform motion estimation and register the source image.
+We do this via an iterative process where coarse-scale estimates of
+:math:`\mathbf{a}_\mathbf{\chi}` are estimated from coarse-scale levels of the
+transform and progressively refined with finer-scale levels.
+
+The following flow diagram, taken from the paper, illustrates the algorithm.
+
+.. figure:: registration-flow.*
+    :align: center
+
+Using the implementation
+------------------------
+
+The implementation of the image registration algorithm is accessed via the
+:py:mod:`dtcwt.registration` module's functions. The two functions of most
+interest at :py:func:`dtcwt.registration.estimatereg` and
+:py:func:`dtcwt.registration.warp`. The former will estimate
+:math:`\mathbf{a}_\mathbf{\chi}` for each 8x8 block in the image and
+:py:func:`dtcwt.registration.warp` will take these affine parameter vectors and
+warp an image with them.
+
+Registration of a source image to a given reference image can be performed
+using two function calls not counting the DTCWT itself:
+
+.. code::
+
+    import dtcwt.backend.backend_numpy as backend
+    import dtcwt.registration as registration
+
+    # ... load two identically shaped NxM images into reference and source ...
+
+    # Transform reference and source
+    t = backend.Transform2d()
+    ref_t = t.forward(reference, nlevels=6)
+    src_t = t.forward(source, nlevels=6)
+
+    # Estimate registration from source -> reference
+    reg = registration.estimatereg(src_t, ref_t)
+
+    # Warp source image
+    registered_source = registration.warp(source, reg)

-- 
Alioth's /usr/local/bin/git-commit-notice on /srv/git.debian.org/git/debian-science/packages/python-dtcwt.git

