# Computational Technologies for Brain Morphometry

Zicong Zhou University of Texas at Arlington, Mathematics Department,
Arlington, Texas 76019-0408<EMAIL_ADDRESS>Ben Hildebrandt
University of Texas at Arlington, Mathematics Department, Arlington, Texas
76019-0408<EMAIL_ADDRESS>Xi Chen University of Texas at
Arlington, Mathematics Department, Arlington, Texas 76019-0408,
<EMAIL_ADDRESS>Guojun Liao University of Texas at Arlington, Mathematics
Department, Arlington, Texas 76019-0408<EMAIL_ADDRESS>
(October 10, 2018)

###### Abstract

In this paper, we described a set of computational technologies for image
analysis with applications in Brain Morphometry. The proposed technologies are
based on a new Variational Principle which constructs a transformation with
prescribed Jacobian determinant (which models local size changes) and
prescribed curl-vector (which models local rotations). The goal of this
research is to convince the image research community that Jacobian determinant
as well as curl-vector should be used in all steps of image analysis.
Specifically, we develop an optimal control method for non-rigid registration;
a new concept and construction of average transformation; and a general robust
method for construction of unbiased template from a set of images.
Computational examples are presented to show the effects of curl-vector and
the effectiveness of optimal control methods for non-rigid registration and
our method for construction of unbiased template.

Keywords: Brain Morphometry, non-rigid registration, tensor based morphometry,
Jacobian determinant, curl-vector, diffeomorphism

##  1 Introduction

In this paper, we address a key mathematical issue in brain morphometry which
investigates variabilities in the shape and size of brain structures from
image data. The issue is how to characterize diffeomorphisms. The prevailing
paradigm in this field is to focus on Jacobian determinant $J(\boldsymbol{T})$
of the transformation $\boldsymbol{T}$, which models local size changes. We
show that $J(\boldsymbol{T})$ alone cannot completely determine a
transformation. Instead, we must use both $J(\boldsymbol{T})$ and the curl-
vector of $\boldsymbol{T}$, $\text{curl}(\boldsymbol{T})$, which models local
rotaions, in all steps of brain morphometry studies. Specifically, we propose
a new set of computational tools for morphometry studies.

  1. (1)

For the non-rigid registration step, we develop an optimal control method
which minimizes a dis-similarity measure ($DS$) under the constraints of
partial differential equations such as $\text{div}(\boldsymbol{T})=f$,
$\text{curl}(\boldsymbol{T})=g$ with respect to control functions $f$ and $g$.

  2. (2)

For the construction of an unbiased template, we propose an innovative concept
of averaging a set of diffeomorphisms based on averaging the Jacobian
determinants and curl-vectors of the diffeomorphisms.

  3. (3)

For group differences, we propose to use Jacobian determinants and curl-
vectors as features in TBM (tensor based morphometry) studies.

In this paper, we will focus on (1) and (2). We will demonstrate the
effectiveness of our optimal control registration method mentioned in (1) and
use it to perform the needed registration for constructing an unbiased
template from a set of images in (2). In a future work, we will implement (3)
in TBM (tensor based morphometry) studies of real data sets.

##  2 Effect of curl-Vectors, Variational Principle, Average of
Transformations

The computational technologies described in this paper are based on a newly
formulated Variational Principle [11]. We begin with an example to show the
effects of the curl-vector.

###  2.1 Effect of curl-Vectors

Started with a ground truth image (GT), based on portrait of Mona Lisa with
128*128 pixels shown in Figure 1(a). Then we generate two deformations
$\boldsymbol{D}_{\boldsymbol{1}}$ and $\boldsymbol{D}_{\boldsymbol{2}}$ from
the 128*128 uniform grid Figure 1(d) with spacing 1 (shown in 32*32) by
opposite rotations such as that their Jacobian determinants are very close to
1 (in fact, between 0.996 and 1.003), and the average of their curl is 0 (the
two rotations are cancelled). Figure 1(b) and 1(c) are the re-sampled images
of Figure 1(a) on these two deformations Figure 1(e) and 1(f), respectively.

#### Resampling $I_{0}$ by $\boldsymbol{D}_{1}$ and $\boldsymbol{D}_{2}$,
which $J(\boldsymbol{D}_{2})\approxeq 1\approxeq J(\boldsymbol{D}_{1})$ but
$\text{curl}(\boldsymbol{D}_{1})\neq\text{curl}(\boldsymbol{D}_{2})$

Figure 1:

(a) $I_{0}$ — Ground Truth

(b) $I_{1}$

(c) $I_{2}$

(d) $\boldsymbol{Id}$ — uniform grid

(e) $\boldsymbol{D}_{1}$

(f) $\boldsymbol{D}_{2}$

###  2.2 Variational Principle

In this section, we describe the principles in 2D [11], where
$\text{curl}(\boldsymbol{T})$ is a scalar function. A diffeomorphism
$\boldsymbol{T}_{1}$ on a square domain $\mathbb{D}$ can be deformed to
$\boldsymbol{T}_{2}$ with prescribed Jacobian determinant
$J(\boldsymbol{T}_{2}(\boldsymbol{x}))=f_{0}(\boldsymbol{x})$ and
$\text{curl}(\boldsymbol{T}_{2}(\boldsymbol{x}))=g_{0}(\boldsymbol{x})$ in the
interior of $\mathbb{D}$, and $\boldsymbol{T}(\boldsymbol{x})=\boldsymbol{x}$
on the boundary of $\mathbb{D}$ ($f_{0}$ is properly normalized for
solvability). $\boldsymbol{T}_{2}$ is constructed by computational
minimization of the functional:

$\iint_{\mathbb{D}}{[(J(\boldsymbol{T}(\boldsymbol{x}))-f_{0}(\boldsymbol{x}))^{2}+(\text{curl}(\boldsymbol{T}(\boldsymbol{x})-g_{0}(\boldsymbol{x}))^{2}]}d\boldsymbol{x}$
(2.1)

subject to the constraints $\mathrm{\Delta}\boldsymbol{T}=\boldsymbol{F}$. The
variational problem is computed by a gradient decent method with respect to
the control function $\boldsymbol{F}=(F_{1},F_{2})$. We demonstrate the
principle in next two examples below.

#### Example 2: Numerical Performances of Variational Principal

Deform $\boldsymbol{D}_{1}$ to $\boldsymbol{D}_{2}$ by the Variational
Principle. Taking $f_{0}=J(\boldsymbol{D}_{2})$ and
$g_{0}=\text{curl}(\boldsymbol{D}_{2})$ in the Variational Principle, in 25
iteration steps, $\boldsymbol{D}_{1}$ is deformed to a transformation —
$\hat{\boldsymbol{D}}_{12}$, as shown in Figure 2(f) that is almost identical
to $\boldsymbol{D}_{2}$ by prescribing its Jacobian as $J(\boldsymbol{D}_{2})$
and its curl as $\text{curl}(\boldsymbol{D}_{2})$. Total elapsed time for 25
iteration steps is 0.955899 seconds using a laptop.

Figure 2: $\boldsymbol{D}_{1}$ to $\boldsymbol{D}_{2}$

(a) $\boldsymbol{D}_{1}$

(b) $\boldsymbol{D}_{2}$

(c) step 0

(d) step 2

(e) step 10

(f) $\hat{\boldsymbol{D}}_{12}$ superimposed on $\boldsymbol{D}_{2}$(black)

We next add noise to $\boldsymbol{D}_{1}$ and deform the distorted
$\boldsymbol{D}_{1}$, denoted as $\boldsymbol{Dn}_{1}$, to
$\boldsymbol{D}_{2}$ as before. The calculated transformation
$\hat{\boldsymbol{Dn}}_{12}$ (in red lines) by The Variational Principle is
superimposed on $\boldsymbol{D}_{2}$ as it shows in Figure 3(f). The red lines
are almost identical to the black lines. This indicates that the results of
the Variational Principle are very accurate.

Figure 3: $\boldsymbol{Dn}_{1}$ to $\boldsymbol{D}_{2}$

(a) $\boldsymbol{Dn}_{1}$

(b) $\boldsymbol{D}_{2}$

(c) step 0

(d) step 550

(e) step 7200

(f) $\hat{\boldsymbol{Dn}}_{12}$ superimposes on $\boldsymbol{D}_{2}$(black)

###  2.3 Average of Transformations

Based on the Variational Principle, we propose to average a set of
transformations by averaging their Jacobian determinants and curl vectors.
Given a set of transformations $\boldsymbol{T}_{i}$ where $i=1,2,\dots,N$, we
construct their average in the following steps:

  1. (1)

Let $f_{0}=\frac{1}{N}\sum_{i}^{N}{J(\boldsymbol{T}_{i})}$ and
$g_{0}=\frac{1}{N}\ \sum_{i=1}^{N}{\text{curl}(\boldsymbol{T}_{i=1})}$ as in
[12]. (weighted averages can also be used)

  2. (2)

Use $f_{0}$ and $g_{0}$ in the Variational Principle to calculate a
transformation $\boldsymbol{T}$ such that $J(\boldsymbol{T})=f_{0}$ and
$\text{curl}(\boldsymbol{T})=g_{0}$.

  3. (3)

We define this $\boldsymbol{T}$ as the average of $\boldsymbol{T}_{i}$, where
$i=1,\dots,N$.

Our definition of average deformation has clear geometrical meaning: The local
size change ratios modeled by $J(\boldsymbol{T}_{i})$ and the local rotations
modeled by $\text{curl}(\boldsymbol{T}_{i})$ are averaged to determine the
average deformation. These two geometrical features also are biologically
meaningful: the Jacobian determinant models the tissue size changes; the curl
vector models the local shape change. Moreover, we directly work with
transformations and thus the method applies to different registration methods.

##  3 Construction of Unbiased Template

We propose a new method for construction of unbiased template from the members
of an image set. The general method is described in Example 3 and a
competitive computation friendly approach in Example 4. First, we obtain six
brain images by re-sampling a ground truth brain image on six intentionally
designed transformations $\boldsymbol{D}_{i}$, where $i=1,\dots,6$.

Figure 4: $I_{0}$ ground truth (GT) image

We constructed six transformations $\boldsymbol{D}_{i}$ where $i=1,\dots,6$,
which are shown below:

Figure 5: Transformations $\boldsymbol{D}_{1-6}$

(a) $\boldsymbol{D}_{1}$

(b) $\boldsymbol{D}_{2}$

(c) $\boldsymbol{D}_{3}$

(d) $\boldsymbol{D}_{4}$

(e) $\boldsymbol{D}_{5}$

(f) $\boldsymbol{D}_{6}$

These transformations are constructed in such a way that their Jacobian
determinants have average equal to 1, and their curls have average 0. In fact,
we have

$\displaystyle 0.999986099590103$
$\displaystyle\leq\frac{1}{6}\sum_{i=1}^{6}{J(\boldsymbol{D}_{i})}\leq
1.000016719949570$ $\displaystyle-6.4029*(10^{-6})$
$\displaystyle\leq\frac{1}{6}\sum_{i=1}^{6}{\text{curl}(\boldsymbol{D}_{i})}\leq\
6.4029*(10^{-6})$

Six modeled brain images $I_{i}$, where $i=1,\dots,6$ are generated by re-
sampling GT on each of the six transformations $\boldsymbol{D}_{i}$, where
$i=1,\dots,6$. These images are shown Figure 6.

Figure 6: Image $I_{1-6}$

(a) $I_{1}$

(b) $I_{2}$

(c) $I_{3}$

(d) $I_{4}$

(e) $I_{5}$

(f) $I_{6}$

Now when we take $f_{0}=\frac{1}{6}\sum_{i}^{6}{J(\boldsymbol{D}_{i})}\approx
1$ and $g_{0}=\frac{1}{6}\ \sum_{i}^{6}{\text{curl}(\boldsymbol{D}_{i})}\
\approx 0$, by the Variational Principle, the average of these images is
expected to be a good approximation to GT. We will verify this in Example 3.
The Sum of Squared Differences $(SSD)$ between $I_{i}$ and GT —
$SSD(I_{i},I_{0})$ (they can be seen as the initial difference from $I_{i}$ to
$I_{0}$) are shown in the following table:

$i$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$
---|---|---|---|---|---|---
$SSD(I_{i},I_{0})=(10^{6})*$ | 1.8923 | 1.8906 | 1.8986 | 1.9610 | 1.8309 | 1.8353

#### Example 3: General Approach

In this example, we will walk through our method for the construction of
unbiased template in a general sense. Step 1: take one image $I_{i}$ out of
the six images as the initial template, then register $I_{i}$ to all six
images $I_{j}$ for $j=1,2,\dots,6$ (arrowed dash-lines) to find six
registration transformations — $\boldsymbol{\phi}_{ij}$ for $j=1,2,\dots,6$;
Step 2: find the average transformation
$\boldsymbol{avg}_{i}=avg(\boldsymbol{\phi}_{ij=1,\dots,6})$ of the six
registration deformations by the Variational Principle (arrowed solid-line),
as are demonstrated in the following diagram.

$I_{1}$ $I_{2}$ $I_{3}$ $I_{4}$ $I_{5}$ $I_{6}$
$\boldsymbol{\phi}_{i1}$$\boldsymbol{\phi}_{i2}$$\boldsymbol{\phi}_{i3}$$\boldsymbol{\phi}_{i4}$$\boldsymbol{\phi}_{i5}$$\boldsymbol{\phi}_{i6}$$\boldsymbol{avg}_{i}$
$I_{i}$ (initial template)

Step 3, re-sample $I_{i}$ on the average transformation
$\boldsymbol{avg}_{i}$, indicated as the next diagram and shown in Figure 7,
to get the biased temporary templates
${\hat{T}emplate}_{i}=I_{i}(\boldsymbol{avg}_{i})$ for each $i=1,2,\dots,6$
which are shown in Figure 8.

$I_{i}$ (initial template) $\boldsymbol{avg}_{i}$ ${\hat{T}emplate}_{i}$

Figure 7: Average transformation —
$\boldsymbol{avg}_{i}=avg(\boldsymbol{\phi}_{ij=1,\dots,6})$

(a) $\boldsymbol{avg}_{1}$

(b) $\boldsymbol{avg}_{2}$

(c) $\boldsymbol{avg}_{3}$

(d) $\boldsymbol{avg}_{4}$

(e) $\boldsymbol{avg}_{5}$

(f) $\boldsymbol{avg}_{6}$

Figure 8: Biased Temporary Templates —
$\hat{T}emplate_{i}=I_{i}(\boldsymbol{avg}_{i})$

(a) $\hat{T}emplate_{1}$

(b) $\hat{T}emplate_{2}$

(c) $\hat{T}emplate_{3}$

(d) $\hat{T}emplate_{4}$

(e) $\hat{T}emplate_{5}$

(f) $\hat{T}emplate_{6}$

The SSD between $\hat{I}_{i}$ with the GT — $SSD({\hat{T}emplate}_{i},I_{0})$
and $Error_{i}=\frac{SSD({\hat{T}emplate}_{i},I_{0})}{SSD(I_{i},I_{0})}$ are
shown in the following table. $Error_{i}$ measures the error reduction in
temporary template ${\hat{T}emplate}_{i}$ compared to the image $I_{i}$.

$i$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$
---|---|---|---|---|---|---
$SSD(\hat{T}emplate_{i},I_{0})=(10^{6})*$ | 2.0504 | 3.4073 | 0.4963 | 0.4905 | 0.4835 | 0.5116
$Error_{i}$ | 0.1084 | 0.1802 | 0.0261 | 0.0250 | 0.0264 | 0.0279

As it can be seen, ${\hat{T}emplate}_{2}$ has the highest error $=0.1802$. At
this stage, ${\hat{T}emplate}_{4}$ is the best with an error 0.0250 (this
means $97.5\%$ of the initial $SSD$ between image $I_{4}$ and GT is reduced by
our optimal control registration method). The large difference between
$Error_{2}=0.1802$ and $Error_{4}=0.0250$ indicates the existence of bias
towards the image that is used as an initial template. In order to quantify
the bias, we calculate the sample mean and standard deviation of the six
$SSD({\hat{T}emplate}_{i},I_{0})$’s: Sample Mean $=1.2400*(10^{6})$, Sample
Standard Deviation $=1.2306*(10^{6})$. So, $SSD(\hat{T}emplate_{6},I_{0})$ is
the closest to the Sample Mean. But the large standard deviation of this
sample disqualifies any one of these templates as unbiased.

This leads to Step-4: repeat Step-1 to Step-3 on biased temporary templates —
$\hat{T}emplate_{i}$ to reduce their bias. So we get a new group of average
transformation $\boldsymbol{Avg}_{i}$ as shown in Figure 9 and the re-sampled
images are the Unbiased Templates —
$Template_{i}=\hat{T}emplate_{i}(\boldsymbol{Avg}_{i})$ as shown in Figure 10.

Figure 9: Average transformation — $\boldsymbol{Avg}_{i}$

(a) $\boldsymbol{Avg}_{1}$

(b) $\boldsymbol{Avg}_{2}$

(c) $\boldsymbol{Avg}_{3}$

(d) $\boldsymbol{Avg}_{4}$

(e) $\boldsymbol{Avg}_{5}$

(f) $\boldsymbol{Avg}_{6}$

Figure 10: Unbiased Templates —
$Template_{i}=\hat{T}emplate_{i}(\boldsymbol{Avg}_{i})$

(a) $Template_{1}$

(b) $Template_{2}$

(c) $Template_{3}$

(d) $Template_{4}$

(e) $Template_{5}$

(f) $Template_{6}$

The SSD between $Template_{i}$ with the GT — $SSD({Template}_{i},I_{0})$ and
$Error_{i}=\frac{SSD({Template}_{i},I_{0})}{SSD(I_{i},I_{0})}$ are shown in
the following table which measures the error reduction in temporary template
${Template}_{i}$ compared to the image $I_{i}$.

$i$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$
---|---|---|---|---|---|---
$SSD(Template_{i},I_{0})=(10^{5})*$ | 6.7786 | 6.6041 | 5.5583 | 5.6649 | 5.7469 | 5.8088
$Error_{i}$ | 0.0358 | 0.0349 | 0.0292 | 0.0289 | 0.0314 | 0.0317

The statistics are significantly improved: New Sample Mean $=6.0269*(10^{5})$;
New Sample Standard Deviation $=5.2436*(10^{4})$. $Template_{6}$ is closest to
the New Sample Mean. The New Sample Standard Deviation $=5.2436*(10^{4})$ is
now only $4.3566\%$ of the previous Sample Standard Deviation
$=1.2306*(10^{6})$. This means, the effectiveness of repeating Step-1 to
Step-3 on the biased temporary templates has greatly reduced the bias of
biased tmporary template $\\{\hat{T}emplate_{i}\\}$. Hence, we can take any of
the new templates $Template_{i}$ as an unbiased template. To check that, we
may register each of $Template_{i}$ to $I_{0}$ to get six register
transformations and all of them are expected to be close to the identity map
$\boldsymbol{Id}$ as the results are shown in Figure 11.

Figure 11: Average transformation — $\boldsymbol{\hat{I}d}_{i}$

(a) $\boldsymbol{\hat{I}d}_{1}$

(b) $\boldsymbol{\hat{I}d}_{2}$

(c) $\boldsymbol{\hat{I}d}_{3}$

(d) $\boldsymbol{\hat{I}d}_{4}$

(e) $\boldsymbol{\hat{I}d}_{5}$

(f) $\boldsymbol{\hat{I}d}_{6}$

And the behaviors of their Jacobian determinant and curl are shown in the
following table.

$i$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$
---|---|---|---|---|---|---
max $J(\boldsymbol{\hat{I}d}_{i})$ | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000
min $J(\boldsymbol{\hat{I}d}_{i})$ | 0.9999 | 0.9999 | 0.9999 | 0.9999 | 0.9999 | 0.9999
max $\text{curl}(\boldsymbol{\hat{I}d}_{i})=10^{-14}*$ | 0.3553 | 0.3553 | 0.1776 | 0.1776 | 0.3553 | 0.3553
min $\text{curl}(\boldsymbol{\hat{I}d}_{i})=10^{-14}*$ | -0.3553 | -0.3553 | -0.1776 | -0.1776 | -0.3553 | -0.3553

#### Example 4: Computation Friendly Approach

In Example 3, the crucial part is Step-3 for our general approach of
constructing an unbiased template, which requires to find all
$\hat{T}emplate_{i}$ for all $i=1,\dots,6$ and it is not a cheap task to do in
terms of computational costs. In this example, we construct a correction
transformation to reduce the bias of any initial template $I_{i}$, by using an
approximation to composition of small deformations, which “allows” us to do a
similar job without finding all $\hat{T}emplate_{i}$. This can be done under
the assumption that ground-truth Jacobian determinant and curl-vector are
known.

Let’s take one of the $I_{i}$ as the initial template, then we perform Step-1
and Step-2 as they were in Example 3, to get an biased temporary template
$\hat{T}emplate_{i}$. To see the process is independent of choosing initial
template, we will show the results of all six image $I_{i}$ as the initial
template individually.

#### Correction transformation $\boldsymbol{H}_{i}$’s

We have derived a mathematical formula to generate correction transformations
$\boldsymbol{H}_{i}$’s for each of $\hat{T}emplate_{i}$ where $i=1,\dots,6$.
Let $\\{I_{i}\\}$ be the given images and $\\{{\hat{T}emplate}_{i}\\}$ be the
temporary templates by above procedures. As it can be seen in Example 3, using
$I_{1}$ and $I_{2}$ to form the temporary templates would give strongly biased
results. Therefore, in order to construct unbiased template, a correction
transformation $\boldsymbol{H}_{i}$ needs to be introduced which depends on
the initial template $I_{i}$.

To construct $\boldsymbol{H}_{i}$(arrowed solid-line on the diagram) for a
fixed $i$: Firstly, applying our image registration method to find
registration transformations
$\hat{\boldsymbol{\phi}}_{ij}:{\hat{T}emplate}_{i}\longrightarrow I_{j}$,
$\forall$ $j=1,2...,N$ (arrowed dash-line on the diagram). Let consider the
following diagram:

$I_{1}$ $I_{2}$ $I_{3}$ $I_{4}$ $I_{5}$ $I_{6}$
$\boldsymbol{\hat{\phi}}_{i1}$$\boldsymbol{\hat{\phi}}_{i2}$$\boldsymbol{\hat{\phi}}_{i3}$$\boldsymbol{\hat{\phi}}_{i4}$$\boldsymbol{\hat{\phi}}_{i5}$$\boldsymbol{\hat{\phi}}_{i6}$
$\hat{T}emplate_{i}$ $\boldsymbol{H}_{i}$ $Template_{i}$

The construction of biased temporary templates $\hat{T}emplate_{i}$ are
actually expected to be the unbiased template $Template_{i}$ in Step-1 and
Step-2 of Example 3. Due to computational limitations, they did not reduce the
bias enough, so we got only a biased temporary template $\hat{T}emplate_{i}$.
However, the original bias of the initial template $I_{i}$ has been reduced to
a level that makes the temporary templates $\hat{T}emplate_{i}$ closer to
$I_{0}$ than the initial template $I_{i}$. We would like to ask, does it exist
a small deformation $\boldsymbol{H}_{i}(\boldsymbol{x})$ that deforms the
unbiased template $Template_{i}$ to $\hat{T}emplate_{i}$? If it exists, how to
find it?

Secondly, according to our definition of unbiased template, if
$\boldsymbol{H}_{i}$ exists, then it must satisfy the following equations:

$\left\\{\begin{aligned}
\frac{\sum_{j=1}^{N}J(\hat{\boldsymbol{\phi}}_{ij}\circ\boldsymbol{H}_{i}(\boldsymbol{x}))}{N}&=1\\\
\sum_{j=1}^{N}\text{curl}(\hat{\boldsymbol{\phi}}_{ij}\circ\boldsymbol{H}_{i}(\boldsymbol{x}))&=0\end{aligned}\right.$
(3.1)

This means, if there exists an unbiased template which should be almost
identical to GT — $I_{0}$, then the each compositions of
$\hat{\boldsymbol{\phi}}_{ij}\circ\boldsymbol{H}_{i}$ should behaves very
close to the inverse transformation of $\boldsymbol{D}_{i}$, where
$\boldsymbol{D}_{i}$ were shown in Figure 5. Therefore the average of
$J(\hat{\boldsymbol{\phi}}_{ij}\circ\boldsymbol{H}_{i}(\boldsymbol{x}))$
should be $1$ and the average
$\text{curl}(\hat{\boldsymbol{\phi}}_{ij}\circ\boldsymbol{H}_{i}(\boldsymbol{x}))$
should be $0$ for each $i$.

By property
$J(\boldsymbol{T}_{2}\circ\boldsymbol{T}_{1})=J(\boldsymbol{T}_{1})J(\boldsymbol{T}_{2})$,
we know

$\frac{\sum_{j=1}^{N}J(\hat{\boldsymbol{\phi}}_{ij}\circ\boldsymbol{H}_{i}(\boldsymbol{x}))}{N}=J(\boldsymbol{H}_{i}(\boldsymbol{x}))\frac{\sum_{j=1}^{N}J(\hat{\boldsymbol{\phi}}_{ij}(\boldsymbol{x}))}{N}=1\hskip
28.45274pt\Longrightarrow\hskip
28.45274ptJ(\boldsymbol{H}_{i}(\boldsymbol{x}))=\frac{N}{\sum_{j=1}^{N}J(\hat{\boldsymbol{\phi}}_{ij}(\boldsymbol{x}))}$
(3.2)

$\boldsymbol{H}_{i}(\boldsymbol{x})$ is small deformation and it can be
represented as
$\boldsymbol{H}_{i}(\boldsymbol{x})=\boldsymbol{x}+\boldsymbol{u}_{i}(\boldsymbol{x})$,
therefore, we have

$\text{curl}(\boldsymbol{H}_{i}(\boldsymbol{x}))=\text{curl}(\boldsymbol{u}_{i}(\boldsymbol{x}))$
(3.3)

And composition of two small deformations, let
$\boldsymbol{T}_{1}(\boldsymbol{x})=\boldsymbol{x}+\boldsymbol{u}_{1}(\boldsymbol{x})$
and
$\boldsymbol{T}_{2}(\boldsymbol{x})=\boldsymbol{x}+\boldsymbol{u}_{2}(\boldsymbol{x})$
where $\boldsymbol{u}_{i}(\boldsymbol{x})$ are displacement fields, can be
approximated [5] by

$\boldsymbol{T}_{2}\circ\boldsymbol{T}_{1}(\boldsymbol{x})\approx\boldsymbol{x}+\boldsymbol{u}_{1}(\boldsymbol{x})+\boldsymbol{u}_{2}(\boldsymbol{x})=\boldsymbol{x}+\boldsymbol{u}_{2}(\boldsymbol{x})+\boldsymbol{u}_{1}(\boldsymbol{x})=\boldsymbol{T}_{2}(\boldsymbol{x})+\boldsymbol{u}_{1}(\boldsymbol{x})$

this leads to

$\boldsymbol{H}_{i}\circ\hat{\boldsymbol{\phi}}_{ij}(\boldsymbol{x})\approx\hat{\boldsymbol{\phi}}_{ij}(\boldsymbol{x})+\boldsymbol{u}_{i}(\boldsymbol{x})$

Therefore, by (3.3), we have

$\text{curl}(\boldsymbol{H}_{i}\circ\hat{\boldsymbol{\phi}}_{ij}(\boldsymbol{x}))\approx\text{curl}(\hat{\boldsymbol{\phi}}_{ij})+\text{curl}(\boldsymbol{u}_{i}(\boldsymbol{x}))=\text{curl}(\hat{\boldsymbol{\phi}}_{ij})+\text{curl}(\boldsymbol{H}_{i}(\boldsymbol{x}))$
(3.4)

Now, by (3.2) and (3.4), the approximations of Jacobian determinant and curl
of $\boldsymbol{H}_{i}$ in (3.1), denoted as
$J(\hat{\boldsymbol{H}}_{i}(\boldsymbol{x}))$ and
$\text{curl}(\hat{\boldsymbol{H}}_{i}(\boldsymbol{x}))$, respectively, which
are in the form of

$\left\\{\begin{aligned}
J(\hat{\boldsymbol{H}}_{i}(\boldsymbol{x}))&=\frac{N}{\sum_{j=1}^{N}J(\hat{\boldsymbol{\phi}}_{ij}(\boldsymbol{x}))}\\\
\text{curl}(\hat{\boldsymbol{H}}_{i}(\boldsymbol{x}))&=\frac{-\sum_{j=1}^{N}\text{curl}(\hat{\boldsymbol{\phi}}_{ij}(\boldsymbol{x}))}{N}\end{aligned}\right.$
(3.5)

Finally, we find $\hat{\boldsymbol{H}}_{i}$ by our Variational Principle. The
approximated transformations $\hat{\boldsymbol{H}}_{i}$’s are shown in Figure
12.

Figure 12: Correction Transformation, $\hat{\boldsymbol{H}}_{1-6}$

(a) $\hat{\boldsymbol{H}}_{1}$

(b) $\hat{\boldsymbol{H}}_{2}$

(c) $\hat{\boldsymbol{H}}_{3}$

(d) $\hat{\boldsymbol{H}}_{4}$

(e) $\hat{\boldsymbol{H}}_{5}$

(f) $\hat{\boldsymbol{H}}_{6}$

After re-sampling each $\hat{T}emplate_{i}$ on transformation
$\hat{\boldsymbol{H}}_{i}$’s, we obtain a set of six new templates
$Template_{i}$ as shown in Figure 13, which are much more uniform now.

Figure 13: Unbiased Templates — $Template_{i}$

(a) $Template_{1}$

(b) $Template_{2}$

(c) $Template_{3}$

(d) $Template_{4}$

(e) $Template_{5}$

(f) $Template_{6}$

Similarly as the results from Example 3, the highest error is $0.0372$, downed
from $0.1802$. The $SSD$ between new templates $Template_{i}$’s and $I_{0}$
are reduced by an order of $10$ as $SSD(Template_{i},I_{0})$ and
$Error_{i}=\frac{SSD(Template_{i},I_{0})}{SSD(I_{i},I_{0})}$ shown in the
table. It measures the relative error in the new template $Template_{i}$
compared to the image $I_{i}$.

$i$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$
---|---|---|---|---|---|---
$SSD(Template_{i},I_{0})=(10^{5})*$ | 6.6972 | 7.0447 | 5.0143 | 4.9911 | 5.0290 | 5.2408
$Error_{i}$ | 0.0354 | 0.0372 | 0.0264 | 0.0254 | 0.0274 | 0.0286

The statistics are also significantly improved just like in the results in
Example 3: New Sample Mean $=5.6695*(10^{5})$; New Sample Standard Deviation
$=9.4137*(10^{4})$. $Template_{6}$ is closest to the New Sample Mean. The New
Sample Standard Deviation $=9.4137*(10^{4})$ is now only $7.65\%$ of the
previous Sample Standard Deviation $=1.2306*(10^{6})$. This means, with the
help of approximated correction transformations $\hat{\boldsymbol{H}}_{i}$, we
have greatly reduced the bias. And $\hat{\boldsymbol{H}}_{i}$ has been a good
approximation to $\boldsymbol{H}_{i}$. To check that, we registered
$Template_{i}$ to $I_{0}$ for each $i$, denoted as $\boldsymbol{\hat{I}d}_{i}$
which are shown in next Figure 14. They are expected to be close to the
identity map $\boldsymbol{Id}$. Therefore, we can take any of the new
templates $Template_{i}$ as an unbiased template.

Figure 14: $\boldsymbol{\hat{I}d}_{i}$

(a) $\boldsymbol{\hat{I}d}_{1}$

(b) $\boldsymbol{\hat{I}d}_{2}$

(c) $\boldsymbol{\hat{I}d}_{3}$

(d) $\boldsymbol{\hat{I}d}_{4}$

(e) $\boldsymbol{\hat{I}d}_{5}$

(f) $\boldsymbol{\hat{I}d}_{6}$

$i$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$
---|---|---|---|---|---|---
max $J(\boldsymbol{\hat{I}d}_{i})$ | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000
min $J(\boldsymbol{\hat{I}d}_{i})$ | 0.9999 | 0.9999 | 0.9999 | 0.9999 | 0.9999 | 0.9999
max $\text{curl}(\boldsymbol{\hat{I}d}_{i})=10^{-14}*$ | 0.3553 | 0.3553 | 0.1776 | 0.1776 | 0.1776 | 0.1776
min $\text{curl}(\boldsymbol{\hat{I}d}_{i})=10^{-14}*$ | -0.3553 | -0.3553 | -0.1776 | -0.1776 | -0.1776 | -0.1776

##  4 Optimal Control Method for Non-rigid Image Registration

In this section, we describe the image registration method that we used to
calculate the examples. Image registration is the process of establishing
pixel (voxel) correspondence between two or more images so that certain
similarity measure is maximized, or a dis-similarity measure ($DS$) is
minimized. The images can be taken from the same individual at different posts
and different times; or from different individuals. The images can also be in
different modalities.

Over last decades, many sophisticated methods have been developed. Most of
these methods regularize the ill-posed registration problem by various penalty
terms that are either based on physical models or geometric considerations.
The cost to be minimized is the sum of $D$S and the regularizing term Reg:
$C=DS+\alpha Reg$, where $\alpha$ is a parameter controlling the trade-off
between the data term $DS$ and the regularizing term $Reg$. There are two
issues with this framework:

  1. (1)

the addition of the second term distorted the problem; For instance, the
regularizing term changed the optimal transportation flow to be non-gradient
flow.

  2. (2)

The determination of the value of $\alpha$ is more like an art than science.
If it is too large, the registration transformation becomes too smooth; while
if it is too small the computation becomes unstable.

Some methods introduce regularization implicitly. For instance, the splines-
based method minimizes $DS$ on a coarse grid (say 8 by 8 pixels are
represented in a grid cell) whose nodes are the control points for splines.
After the new locations of the control points are determined, locations of the
remaining pixels are interpolated by the splines formulas.

In [4, 6] we proposed an optimal control approach that has no explicit penalty
terms (and hence no parameters). Instead, we minimize $DS$ subject to the
constraints $\mathcal{L}[\boldsymbol{u}]=\boldsymbol{F}$ with respect to the
control function $\boldsymbol{F}$. The registration transformation T is
iteratively determined: at the $k+1$ step,
$\boldsymbol{T}^{k+1}(\boldsymbol{x})=\boldsymbol{T}^{k}(\boldsymbol{x}+\boldsymbol{u}^{k+1})$.
$\mathcal{L}$ is a partial differential operator providing regularity. For
instance, in 2D, we can take $\text{div}(\boldsymbol{u})=F_{1}$, and
$\text{curl}(\boldsymbol{u})=F_{2}$. Or we can simply take
$\mathrm{\Delta}\boldsymbol{u}=\boldsymbol{F}$. In the version described in
[4], there is a mechanism that keeps the Jacobian determinant positive, which
in turn assures that $\boldsymbol{T}$ is a diffeomorphism (invertible and
smooth).

The optimal control approach is inspired by the success of fluid control
algorithms. The variational problem is solved by the gradient decent method.
As indicated in the survey paper [9], one could add a penalty term to $DS$ for
specific application with a small parameter value $\alpha$ in order to use
prior knowledge. We now present a teapot example.

#### Example 5: Registration of 3D Teapot Images by Optimal Control Method

We now describe the volume image $I_{0}$ and the twisted image $I_{t}$ in
Step-1. In Step-2, we register $I_{t}$ to $I_{0}$, and out put the
registration transformation and the deformed image $I_{t}$. The results show
that the twisted image $I_{t}$ is deformed back to the $I_{0}$ with high
accuracy.

Step-1: A teapot is rotated about its vertical axis passing through the tip of
its lid. Snapshots are taken from a fixed camera at $5^{\circ}$ intervals. A
total of 72 photos are taken as the teapot completes $360^{\circ}$ rotation
and return to the initial position. Each of these photos is re-sampled as a
$72\times 72$ grayscale image $I_{0i}$. These 72 images $I_{0i}$ for
$i=1,\dots,72$ are used to form a three dimensional volume image of size
$72\times 72\times 72$. We refer to this volume image as the Rotating Teapot
Image, denoted as $I_{0}$.

Figure 15: Teapot—$I_{0}$

(a) $0^{\circ}$

(b) $45^{\circ}$

(c) $90^{\circ}$

(d) $135^{\circ}$

(e) $180^{\circ}$

(f) $225^{\circ}$

(g) $270^{\circ}$

(h) $315^{\circ}$

(i) $360^{\circ}$

As the teapot rotates, we deform each slice $I_{0i}$ (of $I_{0}$) to $I_{ti}$
by a rotation $\boldsymbol{T}_{i}$ as show in Figure 16 that is cut-off near
the boundary for $i=1,\dots,72$.

Figure 16: $\boldsymbol{T}_{i}$’s

(a) $\boldsymbol{T}_{1}$

(b) $\boldsymbol{T}_{9}$

(c) $\boldsymbol{T}_{18}$

(d) $\boldsymbol{T}_{27}$

(e) $\boldsymbol{T}_{36}$

(f) $\boldsymbol{T}_{45}$

(g) $\boldsymbol{T}_{54}$

(h) $\boldsymbol{T}_{63}$

(i) $\boldsymbol{T}_{72}$

These 72 twisted $72\times 72$ images form a $72\times 72\times 72$ volume
image referred to as the Twisted Teapot, denoted as $I_{t}$ as shown in Figure
17.

Figure 17: Twisted Teapot—$I_{t}=\boldsymbol{T}(I_{0})$

(a) $0^{\circ}$

(b) $45^{\circ}$

(c) $90^{\circ}$

(d) $135^{\circ}$

(e) $180^{\circ}$

(f) $225^{\circ}$

(g) $270^{\circ}$

(h) $315^{\circ}$

(i) $360^{\circ}$

The deformations $\boldsymbol{T}_{i}$ for $i=1,\dots,72$ and the Twisted
Teapot image $I_{t}$ can be seen from Twisted Teapot(CLICK HERE)

Step-2: The registration transformation $\boldsymbol{\phi}$ from $I_{0}$ to
$I_{t}$ is calculated by minimizing

$SSD(\boldsymbol{\phi}(\boldsymbol{x}))=\iiint(I_{t}(\boldsymbol{\phi}(\boldsymbol{x}))-I_{0}(\boldsymbol{x}))^{2}d\boldsymbol{x}$
(4.1)

The resulting registration deformation $\boldsymbol{\phi}$, restricted to each
$i$th slice, is expected to be very close to $\boldsymbol{T}_{i}$ as shown in
Figure 18.

Figure 18: $\boldsymbol{\phi}_{i}$’s

(a) $\boldsymbol{\phi}_{1}$

(b) $\boldsymbol{\phi}_{9}$

(c) $\boldsymbol{\phi}_{18}$

(d) $\boldsymbol{\phi}_{27}$

(e) $\boldsymbol{\phi}_{36}$

(f) $\boldsymbol{\phi}_{45}$

(g) $\boldsymbol{\phi}_{54}$

(h) $\boldsymbol{\phi}_{63}$

(i) $\boldsymbol{\phi}_{72}$

and the deformed images of the twisted teapot — $I_{t}(\boldsymbol{\phi})$ is
expected to be close to $I_{0}$ as it shows on next Figure 19. The total
elapsed time for Step-2 is 927.601032 seconds.

Figure 19: Reversed Twisted Teapot—$I_{t}(\boldsymbol{\phi})$

(a) $0^{\circ}$

(b) $45^{\circ}$

(c) $90^{\circ}$

(d) $135^{\circ}$

(e) $180^{\circ}$

(f) $225^{\circ}$

(g) $270^{\circ}$

(h) $315^{\circ}$

(i) $360^{\circ}$

The registration deformation $\boldsymbol{\phi}$ from Step-2 and the
corresponding reversed twisted teapot image $I_{t}=\boldsymbol{T}(I_{0})$ are
shown from: Reversed Twisted Teapot(CLICK HERE). From these visualization
files, we conclude that our optimal control method correctly recovered the
ground truth deformations $\boldsymbol{T}_{i}$ for $i=1,\dots,72$ and the
grand truth images $I_{0i}$ for $i=1,\dots,72$, as expected.

##  5 Conclusion

In this paper, a set of new computational techniques is described. They are
based on both the Jacobian determinant and the curl-vector. Specifically, The
Variational Principle is used to calculate a transformation with prescribed
Jacobian determinant and curl-vector. A new method of averaging deformations
based on averaging the Jacobian determinants and the curl-vectors is used to
construct an unbiased template from the members of a set of images. We will
further refine the techniques and generalize them to three dimensional real
world images.

## References

  * [1] Ashburner, J.: VBM Tutorial. http://www.fil.ion.ucl.ac.uk/ john/misc/VBMclass10.pdf, Mar 15, 2010.
  * [2] Ashburner, J and Friston, K. J.: COMMENTS AND CONTROVERSIES, Why Voxel-Based Morphometry Should Be Used. NeuroImage, Vol. 14, pp. 1238-1243, 2001. doi:10.1006/nimg.2001.0961
  * [3] Bookstein, F. L.: Voxel-Based Morphometry Should Not Be Used with Imperfectly Registered Images. NeuroImage, Vol. 14, Issue 6, pp. 1454-1462, 2001.
  * [4] Hsiao, H-Y., Hsieh, C-Y., Chen, X., Gong, Y., Luo, X. and Liao, G.: New Development of Nonrigid Registration. ANZIAM Journal, Vol. 55, pp. 289-297, 2014. doi:10.1017/S1446181114000091
  * [5] Hoshi, S., Davis, B., Jomier, M. and Gerig, G.:: Unbiased Diffeomorphic Atlas Construction for Computational Anatomy NeuroImage, Vol. 23, pp151-160, 2004. doi:10.1016/j.neuroimage.2004.07.068
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  * [8] Novak, G. and Einstein, S. G.: Structural Magnetic Resonance Imaging as a Biomarker for the Diagnosis, Progression, and Treatment of Alzheimer Disease. Janssen Research and Development, 1125 Trenton-Harbourton Road, Titusville, NJ 08560, USA. Translational, Tools for CNS Drug Discovery, Development and Treatment, 2013.
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# A standard form of incompressible surfaces in 3-dimensional handlebodies

Wei Lin, Fengchun Lei

###### Abstract

Applying Morse theory, we give a standard form for a class of surfaces which
includes all the properly embedded incompressible surfaces in 3-dimensional
handlebodies. We also give a necessary and sufficient condition to determine
the incompressibility of such surfaces placed in our standard form. Our
algorithm is practical. Several examples are given to test the algorithm.

Keywords: incompressible surface, handlebody, Morse theory, general position

Mathematics Subject Classification 2020: 57K30, 57N75, 57R65

##  1 Introduction

The question of constructions for incompressible surface in genus two
handlebody $H_{2}$ was first studied by Jaco [7], who showed there exists a
properly embedded, nonseparating, incompressible surface with arbitrarily
large genus in $H_{2}$. Jaco then asked whether there exists a separating
incompressible surface with arbitarily high genus in $H_{2}$. This question
was answered in the affirmative independently by Eudave-Muñoz [1], Howards
[3], and Qiu [8]. Nogueria and Segerman [6] showed that for every compact
surface with boundary, orientable or not, there is an incompressible embedding
of the surface into $H_{2}$.

Several standard positions of incompressible surfaces in the handlebodies were
given by different authors. Charitos, Papadoperakis, Tsapogas [4] gave a
canonical gluing method where the pieces are disks, so that every properly
embedded incompressible surface in a handlebody can be constructed with. They
also gave a simple sufficient condition which asserts that the result of the
gluing process is incompressible. Let $F$ be a compact surface and let $I$ be
the unit interval. Green [2] gave a standard position for all 2-sided
incompressible surfaces in the 3-manifold $F\times I$. It also supplies a
simple sufficient condition for when 2-sided surfaces in this form are
incompressible. And, since $F\times I$ is a handlebody when $F$ has boundary,
the result applies to incompressible surfaces in handlebodies. Lei, Liu, Li,
Vesnin [5] gave a necessary and sufficient condition for a surface sum of two
handlebodies along a connected surface to be a handlebody.

We work on differentiable manifolds, including the handlebodies and the
surfaces embedded in it. To construct the interested surfaces, we first
associate the handlebody $V$ whose genus is larger than or equal to two, and
the surface $S$ properly embedded in it with a height function $h$. We divide
$V$ through a maximal essential disk system $\mathcal{D}$, and study $S$ as
the union of the subsurfaces in each division. We say a properly embedded
surface $S$ in $V$ is _pseudo-essential_ , if for each horizontal disk $D$
that intersects $S$ transversely, the intersection $S\cap D$ consists of
arc(s) alone (for details, see $\S$2). We prove a result that completely
determines whether a given psuedo-essential surface in standard form is
incompressible or not.

We will show that every pseudo-essential surface can be constructed through a
simple canonical gluing of disk components. We can also show that, the class
of pseudo-essential surfaces includes the class of incompressible surfaces.
Each disk component as a gluing piece of the surface is isotoped to a
particular form, so that some disks will be considered interesting in the
sense that they contain critical points of $h$. We will formulate the above
mentioned in $\S$3.

Furthermore, when a pseudo-essential surface is placed in our standard form,
we claim there exists an algorithm to tell if it is incompressible or not. And
we give a complexity measurement $\mathcal{L}$ in $\S$4, so that $S$ is placed
in _standard position_ when $\mathcal{L}$ is minimized. $S$ under such a
placement can be considered well-behaved, in the sense that, if $S$ is
incompressible, it possesses certain properties listed in Proposition 4.3. We
determine whether $S$ is incompressible or not by simply examining the
combination of disks called _polygons_ in each component of
$\overline{\\{V\setminus\mathcal{D}\\}-S}$, that can be altered as described
in Definition 5.1. We let $S$ retract to a graph $\mathcal{G}_{S}$, whose
vertices mark the critical points of $S$. Then we have the following statement
as our main result:

######  Theorem 1.1.

Let $V$ be a handlebody whose genus is larger than or equal to two,
$\mathcal{D}$ be a maximal essential disk system of $V$. Suppose $S\subset V$
is a connected pseudo-essential surface, that is placed in standard position
with respect to $\mathcal{D}$. Then there is an algorithm of finite many steps
to determine whether $S$ is incompressible or not. $S$ is compressible if and
only if $\mathcal{G}_{S}$ is trivial, or there exists a set of polygons
satisfy the following linear equation set:

$\begin{cases}\frac{1}{2}(n_{2}+(-1)\cdot n_{6}+...+(2-m)\cdot n_{2m})=1\\\
n_{2}+(2)\cdot n_{4}+...+(m)\cdot
n_{2m}=|\mathcal{E}_{\mathcal{D}}|=|\mathcal{E}_{S}|\\\
n_{2}\leq\mathcal{N}_{c}\cdot|\mathcal{P}_{2}|\end{cases}$ (1)

where $m$, $\mathcal{N}_{c}$, $|\mathcal{E}_{\mathcal{D}}|$,
$|\mathcal{E}_{S}|$ and $|\mathcal{P}_{2}|$ are positive integers determined
by $S$, and $n_{2m}$ is the amount of polygons with $2m$ edges.

See $\S$5 for the algorithm mentioned in the above theorem. We notice that the
above equation set has at most finitely many positive integer solutions.

If $S$ is properly embedded in a thickened surface $F\times[0,1]$, we have the
following corollary that coincides with a probably well-known result, stated
as Proposition 3.1 in [9]:

######  Corollary 1.2.

Let $V$ be a handlebody, $\mathcal{D}$ be a maximal essential disk system of
$V$. Suppose $S$ is a pseudo-essential surface that can be isotoped so that
$S\subset\partial V\times[0,1]\subset V$. When $S$ is placed in standard
position with respect to $\mathcal{D}$, it is incompressible if and only if
$\mathcal{G}_{S}$ is nontrivial, and $S$ is $\partial$-parallel to $\partial
V$ .

In $\S$2 we give some basic notations applied in this paper. In $\S$3 we give
a construction of pseudo-essential surfaces. In $\S$4 we give a _standard
position_ for pseudo-essential surfaces. In $\S$5 we give a _normal position_
to regulate the compressing disks. We claim that any compressing disks, if
there exists any, can be put or divided into a union of disk(s) in such a
position. In $\S$6 we demonstrate the application of our algorithm through
some examples.

##  2 Preliminary Background

Let $M$ be a 3-dimensional orientable differentiable manifold with or without
boundary. Let $S\subset M$ be a properly embedded surface, i.e., the interior
Int$(S)$ and the boundary $\partial S$ of $S$ satisfy the inclusions
Int$(S)\subset$ Int$(M)$ and $\partial S\subset\partial M$. A _compressing
disk_ for $S\subset M$ is an embedded disk $D\subset M$ such that $\partial
D\subset S$, and Int$(D)\subset M\setminus S$ and $\partial D$ is an essential
loop in $S$, i.e. the map $\partial D\to S$ induces an injection
$\pi_{1}(\partial D)\to\pi_{1}(S)$. A properly embedded surface $S\subset M$
is _incompressible_ if there are no compressible disks for $S$ and no
component of $S$ is a sphere that bounds a ball. If $S\subset M$ is connected
and 2-sided (i.e., if $S$ is orientable) then $S$ is incompressible if and
only if the induced map $\pi_{1}(S)\to\pi_{1}(M)$ is injective and $S$ is not
a sphere that bounds a ball. Incompressible surface is an important subject of
research in the theory of 3-manifolds.

Throughout the paper, we assume all the manifolds involved are differentiable.
Let $V$ be a 3-dimensional handlebody, generally assumed to have genus larger
than or equal to two. We place $V$ vertically, choose a Morse function
$h:\mathbb{R}^{3}\to\mathbb{R}$, $\forall u=(x,y,z)\in\mathbb{R}^{3}$,
$h(u)=z$, and a _maximal essential disk system_
$\mathcal{D}=\\{D_{1},\cdots,D_{3g-3}\\}$, such that each of $D_{1}$, $D_{2}$,
$\\{D_{3},D_{4}\\}$, $\cdots,\\{D_{3g-6},D_{3g-5}\\}$, $D_{3g-4}$, and
$D_{3g-3}$ is a preimage of a regular value of $h$ at different heights, shown
as in Figure 1 below. A disk $D$ in $V$ is called a _horizontal disk_ if it is
a component of the preimage of a regular value of $h$. We give the definition
for the _pseudo-essential surfaces_ in handlebodies as the class of subjects
we study in this paper.

Figure 1:

######  Definition 2.1.

A properly embedded surface $S$ in $V$ is _pseudo-essential_ (with respect to
a maximal essential disk system $\mathcal{D}$ and the height function $h$), if
for each horizontal disk $D$ that intersects $S$ transversely, the
intersection $S\cap D$ consists of arc(s) alone.

Let $C$ be the spine (a 3-valent graph) that $V$ retracts to, so that a disk
in $\mathcal{D}$ retracts to a point in the interior of an edge in $C$, as in
Figure 2(a). For a surface $S$ properly embedded in $V$, we isotope $S$ in
$V$, while keeping $\partial S\subset\partial V$, so that $|S\cap C|$ reaches
the minimum, and we fix the points of $S\cap C$ prior to any isotopies of $S$
for the rest of this paper. We use the notation
$h_{M_{S}}=h|_{M_{S}}:M_{S}\to\mathbb{R}$ to denote the Morse function on the
submanifold $M_{S}$. Following from Morse theory, we can require the critical
heights of $h_{\partial V}$, $h_{S}$, and $h_{\partial S}$ are all regular and
distinct. We may also require these values to be distinct from the discrete
values $h_{\mathcal{D}}$. And we say that $S$ is in a _general position_ with
$\mathcal{D}$ and $C$, if $S$ meets all the conditions above.

$\mathcal{D}$ cuts $V$ into $2g-2$ 3-balls, we consider the closure of them,
$\\{P_{k},1\leq k\leq 2g-2\\}$, each is called a pant-shaped ball, or simply,
a ps-ball, see (b) and (c) in Figure 2. A ps-ball $P_{i}$ has 3 horizontal
disks in $\partial P_{k}$ from copies of $\mathcal{D}$, see Figure 2(c).

Figure 2:

Let $s_{1},s_{2},\cdots,s_{K}$ be all the critical heights of $h_{S}$ and
$h_{\partial S}$ with $s_{1}<s_{2}<\cdots<s_{K}$. We can take an enough small
$\varepsilon_{i}>0$, so that $s_{i}$ is the only critical height of all the
critical heights of $h_{S}$ and $h_{\partial S}$ in
$I_{i}=[s_{i}-\varepsilon_{i},s_{i}+\varepsilon_{i}]$. Let $W_{i}$ denote the
$I$-product structures, so that each $W_{i}=h^{-1}(I_{i})=E_{i}\times I_{i}$,
and that $E_{i}\times\\{z_{i}\\},z_{i}\in I_{i}$ is a horizontal disk.
Additionally, we can adjust each $W_{i}$, so that each component of the
complements of $\\{\cup W_{i}\\}$ (the union of all such $W_{i}$’s) in $V$ is
one of the following two cases:

(I) A closed ball that has three disk intersections with the boundaries of
some $W_{i}$’s. and such a component is called a _short ps-ball_ , denoted as
$X_{j}$, $X_{j}\in\overline{V\setminus\\{\cup W_{i}\\}}$, $1\leq j\leq 2g$;

(II) A closed ball which contains the highest or lowest critical height, that
has only one disk intersection with the boundary of one of the $W_{i}$’s.

We may require (II) to have empty intersections with $S$, and it is considered
as the trivial case. It follows from Morse theory that $S$ can be chosen as a
representative of its isotopy class to satisfy the above conditions.

##  3 The Construction of pseudo-essential Surfaces

We now give the definition for the gluing disk components mentioned in the
introduction.

######  Definition 3.1.

Let $S_{2n}^{i}$ denote a disk in a product $W_{i}$ whose boundary $\partial
S_{2n}^{i}$ consists of $n$ edges in $E_{i}\times\partial I_{i}$, and $n$
edges in $\partial E_{i}\times I_{i}$, where $n=1,2,3$, or $4,6,...,2k$, and k
is a positive integer. Similarly, we use the notation $S_{2n}^{j}$ to denote a
disk in a short ps-ball $X_{j}$, where $n=1,2,3$. Such a disk is called a
_2n-disk_ if it subjects to the following restrictions:

(1) For each horizontal disk that intersects $S^{i}_{2n}$ or $S_{2n}^{j}$
transversely, the intersection consists of properly embedded arcs.

(2) Each $2$-disk contains a critical point of $h_{\partial S}$ on
$S\cap\partial V$. See Figure 3(a).

(3) Each $4$-disk in $W_{i}$ contains no critical point, and each $4$-disk in
$X_{j}$ may contain a critical point of $h_{\partial S}$ in each arc on
$S\cap\partial V$. See Figure 3(b).

(4) Each $6$-disk contains a critical point of $h_{\partial S}$ on
$S\cap\partial V$. See Figure 3(c).

(5) Each $2N$-disk (where $N$ is an even integer, $N\geq 4$) is a non-
degenerate saddle that contains a critical point $c_{i}$ of $h_{S}$ in its
interior. $c_{i}$ is contained in a horizontal disk $E_{i}\times\\{s_{i}\\}$,
so that $E_{i}\times\\{s_{i}\\}\cap S_{2N}$ is the only non-transverse
intersection of $E_{i}\times\\{z_{i}\\}\cap S_{2N}$ for all $z_{i}\in I_{i}$.
Moreover, $E_{i}\times\\{s_{i}\\}\cap S_{2N}$ is a union of $N$ arcs
intersecting each other(s) at $c_{i}$. See Figure 3(d) for an example of a
$12$-disk.

Figure 3:

We say a $2n$-disk is _trivial_ if it contains no critical points, otherwise
we say it is _nontrivial_. We now show that given a maximal disk system
$\mathcal{D}$ and a height function $h$, any properly embedded incompressible
surface $S\subset V$ is pseudo-essential.

######  Proposition 3.1.

A pseudo-essential surface can be isotoped so that each of its disk component
is as described in Definition 3.1. Let $S$ be an incompressible surface
properly embedded in $V$, then $S$ can be isotoped so it is pseudo-essential.

###### Proof.

As in the above section, we place the handlebody $V$ vertically. Let $C$ be
the spine (a 3-valent graph) that $V$ retracts to. We pick a maximal essential
disk system $\mathcal{D}$, so that a disk in $\mathcal{D}$ retracts to a point
in the interior of an edge in $C$. Then we isotope $S$ so that $|S\cap C|$
reaches minimum and $S$ is in a general position with $\mathcal{D}$ and $C$.
We consider a connected component $S^{i}_{2n}\subset S\cap W_{i}$, or
$S^{j}_{2n}\subset S\cap X_{j}$. Since $S$ is incompressible in $V$,
$S^{i}_{2n}$ (or $S^{j}_{2n}$) must be an incompressible surface properly
embedded in the 3-ball $W_{i}$ or $X_{j}$, which implies $S^{i}_{2n}$ (or
$S^{j}_{2n}$) is a disk. We assume $S^{i}_{2n}$ (or $S^{j}_{2n}$) has $2n$
edges, and all isotopies mentioned below are perform in either $W_{i}$ or
$X_{j}$.

When $n=1$, obviously we can isotope $S^{i}_{2n}$ or $S^{j}_{2n}$ so that it
only contains a critical point of $h_{\partial S}$ on $S\cap\partial V$, and
it is a $2$-disk.

When $n=2$, in $W_{i}$ we can isotope the arcs of $S^{i}_{2n}\cap\partial V$,
so that it is “vertical”, i.e., it intersects each horizontal disk
$E_{i}\times\\{z_{i}\\}$ in single point, and it consists of no critical
points of $h_{\partial S}$. In $X_{j}$ we can either perform the same isotopy,
or when the edges of $S^{j}_{2n}$ intersect two disks in $\mathcal{D}$, we
isotope $\partial S^{j}_{2n}$ so that $S^{j}_{2n}\cap\partial V$ contains only
one critical point of $h_{\partial S}$ in each edge of its boundary in
$\partial V$. In either case, the interior of $S^{i}_{2n}$ or $S^{j}_{2n}$
consists of no critical point of $h_{S}$, and we can isotope $S^{i}_{2n}$ or
$S^{j}_{2n}$ so that it is a $4$-disk.

When $n=3$, we consider the edges of $\partial S^{i}_{2n}$ (or $S^{j}_{2n}$).
We can isotope $S^{i}_{2n}$ such that each of two of its edge in $\partial
S^{i}_{2n}\cap\partial E_{i}\times I_{i}$, $A_{1},A_{2}\subset
S^{i}_{2n}\cap\\{\partial E_{i}\times I_{i}\\}$ intersects each horizontal
disk $E_{i}\times\\{z_{i}\\}$ in single point, and the edge $A_{3}$ is the
only edge that contains the critical point of $h_{\partial S}$. $S^{i}_{2n}$
(or $S^{j}_{2n}$) is not regarded as a non-degenerate saddle, and the interior
of it can be altered so it contains no critical point of $h_{S}$. Therefore,
we can isotope $S^{i}_{2n}$ such that it is a $6$-disk.

When $n=N$, $N\geq 4$, and if we can isotope $S^{i}_{2N}$ such that each edge
of $\partial S^{i}_{2N}$, $A_{t}\subset S^{i}_{2N}\cap\\{\partial E_{i}\times
I_{i}\\}$ intersects each horizontal disk $E_{i}\times\\{z_{i}\\}$ in single
point. Then $\partial S^{i}_{2N}$ does not contain a critical point. We can
isotope $S^{i}_{2N}$ so that it is a non-degenerate saddle, and the only non-
transverse intersection between any of the horizontal disks with $S\cap W_{i}$
is in $E_{i}\times\\{s_{i}\\}$, so that $E_{i}\times\\{s_{i}\\}\cap S_{2N}$ is
a union of $N$ arcs intersecting each other(s) at the critical point of
$h_{S}$. See Figure 3(d). Therefore we have shown that $S^{i}_{2N}$ is
isotoped to a $2N$-disk. Notice we can always adjust $S$ so that a $2N$-disk,
$N\geq 4$ is away from any $X_{j}$.

When $n=N$, $N\geq 4$, and there exists $m$ edges in $\partial V$ that
intersects a horizontal disk of $E_{i}\times\partial I_{i}$ twice. As in the
above arguments, we can deform $\partial S^{i}_{2N}$ so that each of these
edges contains a critical point of $h_{\partial S}$. Then we divide $I_{i}$
into the union of $m+1$ subintervals, $I_{i}=I_{i}^{1}\cup
I_{i}^{2}\cup...\cup I_{i}^{m+1}$, and regard $W_{i}$ as a union of sub-
products, $W_{i}=E_{i}\times I^{1}_{i}\cup E_{i}\times I^{2}_{i}\cup...\cup
E_{i}\times I_{i}^{m+1}$. We can isotope $S^{i}_{2n}$ so that each critical
point of $h_{\partial S}$ is the only critical point of $h_{\partial S}$ in
one of the sub-products. By previous arguments, we can have each component of
$S^{i}_{2n}\cap E_{i}\times I^{s}_{i}$, $1\leq s\leq m+1$ that contains a
critical point of $h_{\partial S}$ isotoped to $6$-disk. And we can isotope
the other component of $S^{i}_{2n}\cap E_{i}\times I^{s}_{i}$ in each sub-
product to a $4$-disk, with an exception that one of them is possibly a
$2n^{\prime}$-disk, $n^{\prime}=n-2m\geq 8$. By the previous argument, we can
deform this $2n^{\prime}$-disk in the sub-product so that it is a non-
degenerate saddle. See Figure 4 for an example: We separate $W_{i}$ into a
union of two products, and we isotope the disk so that each product contains
only one critical point of $h_{S}$ or $h_{\partial S}$. On the right, the
upper product contains a $4$-disk and a $6$-disk, and the lower product
contains a $8$-disk. Now that each product contains only one critical point of
$h_{\partial S}$ or $h_{S}$, we can change the subscripts of them, so they are
in accord with the other $W_{i}$’s. As such, each sub-product contains only
disk(s) satisfying the conditions in Definition 3.1.

Figure 4:

Notice that, we have also proved $S$ can be isotoped so it is pseudo-essential
with respect to $\mathcal{D}$ and $h$. Since for each horizontal disk that
intersects a $2n$-disk transversely, the intersection consists of arcs alone.
∎

##  4 Placing Surfaces in the Standard Position

Consider a pseudo-essential surface $S$. Let $\eta(C)$ be a regular open
neighborhood of $C$, then
$\overline{V-\eta(C)\setminus\mathcal{D}}=\cup_{k=1}^{2g-2}PT_{k}$ is a union
of “thickened pair of pants”. Moreover, $PT_{k}=p_{k}\times[0,1]$, and
$\partial PT_{k}$ is the union of $p_{k}\times\\{0\\}$, $p_{k}\times\\{1\\}$.
Here, $p_{k}$ is a “pair of pants”, and $p_{k}\times\\{0\\}$ is regarded as
the outer one. Let $A^{x}_{k}\subset\mathcal{D}\setminus\eta(C),1\leq x\leq 3$
denote the annulus on $\partial PT_{k}$ that intersects $\mathcal{D}$. The
conditions of general position require the number $|S\cap C|=|S\cap\eta(C)|$
to reach minimum, and each of the three annuli $A^{x}_{k}$ on the boundary of
$PT_{k}$ contains transverse intersections of $A^{x}_{k}\cap S$, which are
properly embedded arcs whose boundaries are on the outer circle of $\partial
A^{x}_{k}$.

We say two arcs of $S\cap A^{x}_{k}$ has an _inner-outer relation_ if they co-
bound a sub-disk in $A^{x}_{k}$ with $\partial A^{x}_{k}$. The arc that is
closest to the outer boundary of the annulus is called an _outermost arc_ ,
and the arc that is closest to the inner boundary of the annulus is called an
_innermost arc_. The arcs of $S\cap A^{x}_{k}$ in each annulus can be regarded
as a family of (not necessarily disjoint) sets $\cup\mathcal{A}^{x}_{km}$,
$m\geq 1$, so that each $\mathcal{A}^{x}_{km}$ either consists of only one
arc, or any two arcs in $\mathcal{A}^{x}_{km}$ admits an inner-outer relation.
Moreover, each $\mathcal{A}^{x}_{km}$ is maximal in the sense that all arcs
that has an inner-outer relation with the outermost arc, including the
outermost arc, are all contained in it. For example, in Figure 5(a), the
annulus consists of two such families, each contains the red arc. And the arcs
that are the same color has an inner-outer relation with each other. In
particular, the red arc is an innermost arc that has an inner-outer relation
with all other arcs.

Figure 5:

As such, we call each $\mathcal{A}^{x}_{km}$ a _maximal arc set_ , and we can
define a counting function $f$ whose domain is such families of arcs:
$\\{\mathcal{A}^{x}_{km},1\leq x\leq 3,1\leq k\leq 2g-2,m\geq 1\\}$, so that
$f(\mathcal{A}^{x}_{km})=|\mathcal{A}^{x}_{km}|$. Let $L$ be the maximal value
of the counting function $f$. For each $2n$-disk $S^{i}_{2n}$ that has
nonempty intersections with an annulus $A$ parallel to some $A^{x}_{k}$, we
can consider the arcs of $S^{i}_{2n}\cap A$. If $S$ intersects $A$
transversely, then the definitions above stil applies on $A$ and
$S^{i}_{2n}\cap A$. If $S^{i}_{2n}\cap A$ consist of two arcs that admit an
inner-outer relation, then we call $S^{i}_{2n}$ a _movable saddle_. We denote
the set of all movable saddles as $\mathcal{D}_{ms}$. It is obvious we can
isotope $S$ so that all movable saddles are contained in $W_{i}\cap PT_{k}$
for some $i$’s and $k$’s, i.e., $\mathcal{D}_{ms}\cap\eta(C)=\emptyset$. We
can also assume without loss of generality, that $S$ is still in standard
position after such isotopies.

######  Definition 4.1.

We say a pseudo-essential surface $S$ is in _standard position_ (with respect
to $\mathcal{D}$ and $h$) if $S$ satisfies all the conditions in Definition
3.1, and the $(L+1)$-tuple

$\mathcal{L}=(|S\cap\mathcal{D}|,|f^{-1}(L)|,|f^{-1}(L-1)|,...,|f^{-1}(2)|,|\mathcal{D}_{ms}|)$

is minimized in the lexicographic order, where each $|f^{-1}(l)|,2\leq l\leq
L$, is the amount of maximal arc sets $\mathcal{A}^{x}_{km},1\leq x\leq
3,1\leq k\leq 2g-2,m\geq 1$, so that $f(\mathcal{A}^{x}_{km})=l$.

We study how certain isotopies will affect $\mathcal{L}$. Let $S_{2n}\subset
PT_{k}$ be a movable saddle, we say $S_{2n}$ is an _outermost movable saddle_
with respect to an annulus $A^{x}_{k}$, if there does not exists any arcs of
intersection on the sub-disk corresponding to the $\frac{n}{2}$ arcs of
$S_{2n}\cap A^{x}_{k}$. For example, in Figure 5(b), each of the red dotted
arcs is on the sub-disk corresponding to the 3 arcs of some 12-disk
$S_{12}\cap A^{x}_{k}$.

######  Lemma 4.1.

Given an outermost movable saddle $S_{2n}\subset PT_{k}$ with respect to an
annulus $A^{x}_{k}$, there is an isotopy which induces $(\frac{n}{2}-1)$
$\partial$-compressions on the subsurface $S\cap PT_{K}$ and decreases
$\mathcal{L}$.

###### Proof.

Given an outermost movable saddle $S_{2n}\subset PT_{k}$ with respect to an
annulus $A^{x}_{k}$, we can fix the boundary of $S_{2n}$ on the “pair of
pants”, i.e., $p_{k}\times\\{0\\}$. Then we isotope the interior of $S_{2n}$,
so that the critical point contained in it is moved outside of $PT_{k}$, and
$A^{x}_{k}$ intersects with the $(\frac{n}{2}-1)$ sub-surfaces in $PT_{k}$ in
$(\frac{n}{2}-1)$ arcs after such an isotopy. This isotopy will simultaneously
induce $(\frac{n}{2}-1)$ $\partial$-compressions on the compressed surface
$S_{2n}$ with the boundary $\partial PT_{k}$. It is easy to check that each of
the induced $\partial$-compressions will reduce $\mathcal{L}$, even if $S$ has
other intersections with the corresponding annulus. We notice that the number
$|S\cap\mathcal{D}|$ does not change after such $\partial$-compressions. See
the isotopy shown in Figure 5(c) for an example, where $n=6$. ∎

We induce a graph related to $2n$-disks. Let each $4$-disk retract to an arc
that connects two horizontal disks; Let each $6$-disk retract to 3 arcs that
intersect at a $3$-valent vertex as the critical point of $h_{\partial S}$ in
$\partial S$. And let each $2n$-disk, $n=4,6$, … ,$2k$, retract to $n$ arcs
that intersect a $n$-valent vertex as the critical point of $h_{S}$ in its
interior. Then we have $S$ retracted to a graph $\mathcal{G}_{S}$, with each
vertex marking a critical point of $h_{S}$ or $h_{\partial S}$. For
convenience, we assume an edge $e$ of $\mathcal{G}_{S}$ may contains vertices.

######  Lemma 4.2.

Let $S$ be an incompressible surface in standard position, $P_{k}$ be a ps-
ball, then there exists no such pair of edges, $e_{1}$,
$e_{2}\subset\mathcal{G}_{S}\cap P_{k}$, that they intersect at a vertex $v$,
and that they intersect some annulus $A_{k}^{x}\subset\partial P_{k}$ in two
arcs, $\alpha_{1}$ and $\alpha_{2}$, which have an inner-outer relation.

Figure 6:

###### Proof.

We first claim that neither $e_{1}$ nor $e_{2}$ consists of a 3-valent vertex
$v_{3}$ that marks a critical point of a $6$-disk $S_{6}$, which intersects
$A_{k}^{x}$ in two arcs. Because, if either $e_{1}$ or $e_{2}$ consists such a
vertex, then obviously we can isotope $S_{6}$ while moving $v_{3}$ out of
$P_{k}$ to reduce $|S\cap\mathcal{D}|$, and therefore we can reduce
$\mathcal{L}$. We notice that the condition $S$ being incompressible is needed
(but not necessary), since otherwise there could be a loop of
$\mathcal{G}_{S}$ that consists of $e_{1}$ and $e_{2}$, and
$|S\cap\mathcal{D}|$ would not be necessarily reduced.

If both $e_{1}$ and $e_{2}$ consist of a single vertex $v$, then we have an
outermost movable saddle, whose critical point is $v$. Therefore, by Lemma
4.1, we can reduce $\mathcal{L}$. Now we can assume without of loss of
generality that, other than $v$, $e_{1}$ contains only one more vertex
$v^{\prime}$, and $e_{2}$ does not contain any vertices. Then since
$\mathcal{G}_{S}$ consists no loops, we can reduce the number of vertices by
one through the isotopy shown in Figure 6. And then we have an outermost
movable saddle, so we can reduce $\mathcal{L}$. We notice Figure 6 only
illustrates the situation where both $2n$-disk are away from $\eta(C)$. If one
of them intersects $\eta(C)$, we can also have a similar isotopy to reduce
$\mathcal{L}$. The situation of either $e_{1}$ or $e_{2}$ consists of more
than two vertices is similar, since we can reduce the vertex number through
the above-mentioned isotopies. ∎

######  Proposition 4.3.

Let $S$ be an incompressible surface properly embedded in standard position
with respect to $\mathcal{D}$. Then $S$ has the following properties:

1\. The subgraphs $\mathcal{G}_{S}\setminus\mathcal{D}$ are acyclic;

2\. $S$ contains no movable saddles;

3\. A $2N$-disk $S_{2N}$, $N\geq 5$, must intersect $C$;

4\. Any $2$-disk is contained in some short ps-ball $X_{j}$;

5\. A 3-valent vertex in a component of $\mathcal{G}_{S}\setminus\mathcal{D}$
meets edges that intersect with all three annuli on a ps-ball $P_{k}$.

###### Proof.

Property 1 is obvious, since otherwise it would contradict the condition $S$
is incompressible.

Assume $S$ is in standard position. Let $P_{k}$ be a ps-ball, $S_{2n}$ be an
movable saddle which consists of a vertex $v$ of $\mathcal{G}_{S}$ in its
interior. Because $\mathcal{G}_{S}\cap P_{k}$ consists of no loops, by Lemma
4.2, any two edges $e_{1}$ and $e_{2}$ which intersect at a vertex $v$ must
also intersect each annulus $A_{k}^{x}\subset\partial P_{k}$ in two arcs that
do not has an inner-outer relation. Therefore we can isotope $S\cap P_{k}$ to
reduce the critical point number, and reduce $|\mathcal{D}_{ms}|$, see Figure
7(a). We have shown that property 2 is true.

Figure 7:

To see property 3, we assume a $2n$-disk $S_{2n}$, $n=4,6$, … , $2k$, consists
of a critical point in its interior, and $S_{2n}$ does not intersect $C$. Then
$S_{2n}$ must be a movable saddle, unless it is a $8$-disk shown as in Figure
7(b).

To see property 4, we assume by contradiction that, if there exists a $2$-disk
in some $W_{i}$ which does not intersect $\mathcal{D}$, we can always isotope
$S$ so the $2$-disk is removed form $W_{i}$. Then, if the $2$-disk were in a
product $W_{i}$ that intersects with $\mathcal{D}$, $|S\cap\mathcal{D}|$ would
be reduced by the isotopy that removes the $2$-disk from $W_{i}$. Therefore
the $2$-disk must be embedded in some short ps-ball $X_{j}$.

To see property 5, if two edges of $\mathcal{G}_{S}\setminus\mathcal{D}$ that
meet at a 3-valent vertex $v_{3}$ in $P_{k}$ both intersect with an annulus
$A_{k}^{x}$, then apparently we can reduce $|S\cap\mathcal{D}|$ through an
isotopy of $S$ while moving $v_{3}$ out of $P_{k}$. ∎

As a result, it is reasonable to give the following definition:

######  Definition 4.2.

Let $S$ be a pseudo-essential surface properly embedded in $V$. If $S$ is in
standard position, and there exists a component of
$\mathcal{G}_{S}\setminus\mathcal{D}$ that consists of a cyclic graph, then we
say $\mathcal{G}_{S}$ is _trivial_. Otherwise, we say $\mathcal{G}_{S}$ is
_nontrivial_. Additionally, if a saddle contained in $S$ is not a movable
saddle, we call it an _essential saddle_.

##  5 Compressing disks in Normal Position

Due to Theorem 3.1, any incompressible surface $S$ properly embedded in $V$
can be put in standard position, and according to Proposition 4.3, when $S$ is
in standard position, it has no movable saddles. But the contrary is not
necessarily true. We want to study the necessary and sufficient conditions for
a compressing disk to exist when $S$ is in standard position, so that we can
determine incompressibility of $S$.

We say a properly embedded arc $\beta$ in $S$ is a _horizontal arc_ , if for a
horizontal disk $D$, $\beta\subset S\cap D$; We say $\beta$ is an _essential
arc_ provided $\beta$ cannot be deformed (fixing $\partial b$) in $S$, into
$\partial S$; Otherwise we say it is an _inessential arc_.

######  Definition 5.1.

A sub-disk $p_{D_{c}}$ in $D_{c}$ with
$p_{D_{c}}\cap\\{S_{k}\cup\mathcal{D}\\}=\partial p_{D_{c}}$ is called a
_polygon_ if it satisfies the following conditions:

1\. The boundary $\partial p_{D_{c}}$ of each polygon $p_{D_{c}}$ in $D_{c}$
intersects each essential horizontal arc $\beta$ at most once, and $D_{c}$
does not transversely intersect any inessential horizontal arcs in one point;

2\. For each nontrivial $2n$-disk $S_{2n}$ that intersects $p_{D_{c}}$, where
$n=2,3,4,6$, … ,$2k$, $D_{c}\cap S_{2n}$ is a properly embedded arc in
$\partial D_{c}$ which consists of a single critical point of $h|_{\partial
D_{c}}$ that is in a small neighborhood of the critical point in $S_{2n}$;

3\. A bi-gon $p_{2}$ has to intersect an essential saddle;

4\. A polygon $p_{2m}$ is a non-degenerate saddle disk if and only if there
exists 4 edges (arranged in order, if we orient $\partial p_{2m}$) of
$p_{2m}\cap\mathcal{D}$ that intersect with disks of different heights in
$\mathcal{D}$ alternatingly.

######  Definition 5.2.

Let $S\subset V$ be a compressible surface in standard position that contains
no movable saddles. We use $N_{m}$ to denote the total number of $2m$-gons,
for all $m\geq 4$. If $S$ has no movable saddles, we say a disk $D_{c}$
properly embedded in the complement of $S$ is in _normal position_ , if
$D_{c}$ intersects each $P_{k}$ in polygons, and the $2$-tuple

$\mathcal{L}^{\prime}=(|D_{c}\cap\mathcal{D}|,N_{m})$

is minimized in the lexicographic order.

######  Remark 5.1.

The second tuple $N_{m}$ in the above definition is minimized to reduce
similar situations in Figure 9(d). In which, it shows such a reduction through
an isotopy that is the “inverse” of a $\partial$-compression induced by the
compression along the green disk.

######  Proposition 5.1.

Let $S$ be a surface properly embedded in standard position with respect to
$\mathcal{D}$, $D_{c}$ be a compressing disk of $S$. Assume $\mathcal{G}_{S}$
is nontrivial, then $D_{c}$ can replaced by a union of disk(s)
$\\{D^{(l)}_{c}\\}$ , so that each $D^{(l)}_{c}$ is a compressing disk of $S$
in normal position.

Figure 8:  Figure 9:

###### Proof.

Suppose a compressing disk $D_{c}$ for $S$ is placed so that
$|D_{c}\cap\mathcal{D}|$ is minimized. $D_{c}\cap\mathcal{D}$ consists of
arc(s), since $D_{c}$ is a disk any loops can be isotoped-off by a normal
innermost-loop argument.

Let $\beta$ denote the essential horizontal arc intersects with $\partial
p_{D_{c}}$. We assume there exists an edge $\alpha\subset\partial p_{D_{c}}$
intersecting $\beta$ more than once, then we can deform $\partial p_{D_{c}}$
on $S$, as shown in Figure 88(a), so that $\alpha$ intersects each horizontal
arc no more than once, otherwise $|D_{c}\cap\mathcal{D}|$ can be reduced.
Therefore each edge of $\partial p_{D_{c}}$ must intersect $\beta$ for at most
once. We assume there exists two edges of a polygon $p_{D_{c}}$ intersecting
$\beta$. Then for the horizontal disk $D$ containing $\beta$, we have $D\cap
p_{D_{c}}=\gamma$ is an arc whose boundary points are both contained in
$\beta$. Without loss of generality, we assume the disk $D_{\beta\gamma}$
bounded by $\beta$ and $\gamma$ does not intersect $D_{c}$ in its interior,
then we can perform a surgery along $D_{\beta\gamma}$, shown in Figure 88(b).
The resultant of this surgery performed on $D_{c}$ is a disjoint union of two
disks, that we denote as $D^{\prime}_{c}$ and $D^{\prime\prime}_{c}$. At least
one of $D^{\prime}_{c}$ and $D^{\prime\prime}_{c}$ is still a compressing
disk, otherwise the original disk $D_{c}$ would be trivial. We ignore the
trivial disk(s) and consider the resultant compressing disk(s) after the
surgery, and we assume without loss of generality that $D^{\prime}_{c}$ is a
compressing disk. Then we can deform $D^{\prime}_{c}$ to strictly reduce the
intersection number $|S\cap\mathcal{D}|$, until it is minimized.
$D^{\prime}_{c}$ still has nonempty intersections with $\mathcal{D}$,
otherwise it would contradict the assumption that $\mathcal{G}_{S}$ is
nontrivial. Due to the compactness of $S$ and the disks, we can repeat the
above procedure for finite many times on each of the resultant compressing
disks and ignore the trivial disks, and then the resultant disks are a union
of compressing disk(s) $\\{D^{(l^{\prime})}_{c}\\}$ each satisfying condition
1.

Consider each of the polygons $D^{(l^{\prime})}_{c}\cap P_{k}$. We apply the
notation $p_{D_{c}}$ to represent such a polygon for convenience. The
construction of $S$ in standard position, and condition 1 implies we can
deform the boundary of $p_{D_{c}}$, so that there is only one critical point
$c$ in an edge of $p_{D_{c}}\cap S$, so that $h(c)$ is either the local
maximal or the local minimum. We can also alter $\partial p_{D_{c}}$ so that
$c$ is in a small neighborhood of the critical point in $S_{2n}$, to satisfy
condition 2. For the rest of this proof we assume condition 1 and 2 are always
satisfied by any compressing disks mentioned below.

According to property 5 in Proposition 4.3, a $6$-disk has to intersect three
disks in $\mathcal{D}$. And a $4$-disk that contains a critical point has to
intersect two disks of $\mathcal{D}$. Therefore, the sub-arc of a bi-gon which
contains a local extreme point cannot intersect with a $4$-disk or a $6$-disk.
Thus, we have shown that if condition 1 and 2 are guaranteed, then condition 3
has to be satisfied by each compressing disk $D^{(l^{\prime})}_{c}$. See
Figure 88(c) for an example of the intersection of two bi-gons with a
$12$-disk in some product $W_{i}$.

It is easy to see the “if” part, since similar to the proof of Theorem 3.1, we
can isotope $p_{2m}$ in this situation so that it is a non-degenerate saddle
disk. See Figure 9(c) for two examples of a octagon, and see the lower picture
in Figure 9(d) for an example of a $16$-gon, in which there exists $6$ edges
intersect with disks of different heights in $\mathcal{D}$ alternatingly. To
see the “only if” part, we suppose there are no such edges which satisfy the
assumptions in condition 4. Then if $m\leq 3$, we can place $p_{2m}$ so that
it is not a non-degenerate saddle. See Figure 9(a), (b). If $m\geq 4$, we can
always isotope $p_{2m}$ so that it consists of critical point(s) of
$h_{\partial D_{c}}$ on its edges in $p_{2m}\cap S$, but it does not consist
of any critical points of $h_{D_{c}}$ in its interior. See the upper picture
in Figure 9(d) for such an example. ∎

The above proposition allows us to divide a compressing disk, so that it can
be replaced by a union of compressing disks in normal position. Consider
$D_{c}$ a disk embedded in the complement of $S$, so that
$|D_{c}\cap\mathcal{D}|$ is minimized, and $D_{c}$ intersects each ps-ball
$P_{k}$ in polygons. Each polygon cosists of two type of boundary edges
(arcs): The set consisting all arcs of $D_{c}\cap\mathcal{D}$ will be denoted
as $\mathcal{E}_{\mathcal{D}}$. And, the set consisting all arcs of $D_{c}\cap
S-\mathcal{D}$ will be denoted as $\mathcal{E}_{S}$. Moreover, we use $n_{2}$,
$n_{4}$, … ,$n_{2m}$, where $2m$ is an even integer, to denote the number of
bi-gons, rectangles, …, and $2m$-gons in all $D_{c}\cap P_{k},1\leq k\leq
2g-2.$ We give an Euler characteristic calculation for the polygons numbers in
a compressing disk.

Let $\mathcal{P}_{2m}$ denote the set of distinct $2m$-gons, so that in each
$\mathcal{P}_{I},I=2,4,...,2m$, any two of the polygons are not parallel to
each other; Let $\mathcal{P}^{J}_{I}$ denote the subset of $\mathcal{P}_{I}$
including all the $I$-gons that share common edge(s) with some $J$-gon(s). If
we can choose elements from $\mathcal{P}^{J}_{I},I,J=2,4,...,2m,$ (possibly
identical and parallel to each other) to glue a connected embedded surface
$S_{D}$, so that $\partial S_{D}$ only consists of edges in
$\mathcal{E}_{\mathcal{D}}$, and the number of polygons and their edges in
$S_{D}$ satisfy the a linear equation set, we simply say some elements in
$\mathcal{P}^{J}_{I},I,J=2,4,...,2m$, satisfy the equation set.

######  Lemma 5.2.

Let $S$ be a surface properly embedded in standard position with respect to
$\mathcal{D}$. Suppose $S$ has no movable saddles, and $\mathcal{G}_{S}$ in
nontrivial. Then $S$ is compressible if and only if some elements in
$\mathcal{P}^{J}_{I},I,J=2,4,...,2m,$ satisfy the following equation set:

$\begin{cases}\frac{1}{2}(n_{2}+(-1)\cdot n_{6}+...+(2-m)\cdot n_{2m})=1\\\
n_{2}+(2)\cdot n_{4}+...+(m)\cdot
n_{2m}=|\mathcal{E}_{\mathcal{D}}|=|\mathcal{E}_{S}|\\\ \end{cases}$ (2)

###### Proof.

If some elements in $\mathcal{P}^{J}_{I},I,J=2,4,...,2m,$ satisfy equation set
(2), we can find a disk $D_{c}$ in the complement of $S$, since in a
handlebody the only connected, embedded surface with euler characteristic
equal to one must be a disk. We claim $D_{c}$ is a compressing disk of $S$.
Because if we assume $D_{c}$ is a trivial disk, then we can isotope $D_{c}$
with $\partial D_{c}$ fixed on $S$ to a disk $D_{S}\subset S$ bounded by
$\partial D_{c}$. But this cannot be true, because near the local extreme
point of such a disk, there has to exist an edge of a polygon that intersects
$\mathcal{D}$ twice, which would contradict condition 1 of Proposition 5.1. We
notice that the compressing disk $D_{c}$ glued by such elements in
$\mathcal{P}^{J}_{I},I,J=2,4,...,2m,$ satisfying equation set (2) is not
necessarily in normal position.

By Proposition 5.1, if $S$ is compressible, then there exists a compressing
disk $D_{c}$ in normal position. $D_{c}$ will be divided by arcs of
$D_{c}\cap\mathcal{D}$ into polygons. We use $\mathcal{V}_{D_{c}}$,
$\mathcal{E}_{D_{c}}$, and $\mathcal{F}_{D_{c}}$ to denote the vertex number,
edge number, and face number, respectively, in a compressing disk $D_{c}$.
Then since the pull-back graph on $D_{c}$ of $D_{c}\cap\mathcal{D}$ is
3-valent, we have
$\chi(D_{c})=\mathcal{V}_{D_{c}}-\mathcal{E}_{D_{c}}+\mathcal{F}_{D_{c}}=\frac{1}{2}(2n_{2}+4n_{4}+6n_{6}+...+2m\cdot
n_{2m})-\frac{3}{2}(n_{2}+2n_{4}+3n_{6}+...+m\cdot
n_{2m})+(n_{2}+n_{4}+n_{6}+...+n_{2m})=\frac{1}{2}(n_{2}+(-1)\cdot
n_{6}+...+(2-m)\cdot n_{2m})=1$. Half the edge number in the pullback graph of
$D_{c}$ is equal to $\frac{1}{2}(2n_{2}+4n_{4}+6n_{6}+...+2m\cdot
n_{2m})=|\mathcal{E}_{\mathcal{D}}|=|\mathcal{E}_{S}|$. This is because each
polygon has half of its edges in the interior of $D_{c}$, which will intersect
$\mathcal{D}$ in $|\mathcal{E}_{\mathcal{D}}|$ arcs; And each polygon has the
other half of its edges in $\partial D_{c}$, which will intersect $S$ in
$|\mathcal{E}_{S}|$ arcs. ∎

Let $C_{2x}$ denote the number of $2x$-disks that contains a critical point.
We now prove a technical lemma before we state the algorithm mentioned in
Theorem 1.1.

######  Lemma 5.3.

Let $S$ be a surface properly embedded in standard position with respect to
$\mathcal{D}$, so that $S$ consists of no movable saddles. Suppose $D_{c}$ is
a compressing disk in standard position, then the amount of bi-gons contained
in $D_{c}$ which are parallel and considered the same element in
$\mathcal{P}_{2}$ is bounded by
$\mathcal{N}_{C}=\frac{1}{2}\sum_{x=3}^{N}{x\cdot C_{2x}}$, where $2N$ is the
largest possible number for the edges of a $2n$-disk.

###### Proof.

We consider two identical elements in $\mathcal{P}_{2}$, i.e., two bi-gons,
$p_{2}$ and $p^{\prime}_{2}$ that are parallel to each other. Assume $p_{2}$
and $p^{\prime}_{2}$ are both sub-disks of a compressing disk $D_{c}$ in
normal position. Then in $D_{c}$ there must exist a polygon $p_{2m}$ that is
path-connected to both $p_{2}$ and $p^{\prime}_{2}$. Moreover, $p_{2m}$ must
consists of a critical point, because $p_{2}$ and $p^{\prime}_{2}$ both
consist of a local maximum or local minimum point of $D_{c}$. But this implies
there must exist at least one $2N$-disk $D_{2N}$ which consists of a critical
point, such that $D_{2N}$ also intersects with two polygons $p_{a}$ and
$p_{b}$. And each of $p_{a}$, $p_{b}$ is path-connected to either $p_{2}$ or
$p^{\prime}_{2}$ in $D_{c}$. Because otherwise, $p_{2m}$ would be a polygon
that consists of a critical point, and two edges of $p_{2m}$ both intersect
with one $2n$-disk $D_{2N^{\prime}}$. This leads to a contradiction due to
property (1) of Proposition 5.1. Suppose there are $x_{0}$ parallel polygons,
it would require at least one pair of $2n$-disks with $2x_{1},2x_{2}$ edges,
where $x_{1},x_{2}\geq 4$, $2x_{0}\leq x_{1}\leq x_{2}\leq N$ (or $x_{1}=3$,
$2=x_{0}<x_{1}\leq x_{2}\leq N$), so that it would not lead to the similar
contradiction mentioned above. Therefore the possible times that an element in
$\mathcal{P}_{2}$ being selected is bounded by
$\frac{1}{2}\sum_{x=3}^{N}{x\cdot C_{2x}}$. ∎

### The Algorithm

For any pseudo-essential surface $S$, we have the start from step $I$ of the
algorithm:

$I$ According to Proposition 3.1, $S$ can be isotoped so that each of its disk
component is as described in Definition 3.1. Then we proceed to $II$.

$II$. We reducing the complexity
$(|S\cap\mathcal{D}|,|f^{-1}(L)|,|f^{-1}(L-1)|,...,|f^{-1}(2)|)$ in
lexicographical order through the isotopies described in $\S$4. Then we
minimize the complexity $\mathcal{L}$ by reducing movable saddles, so that $S$
is in standard position. According to property 1 of Proposition 4.3, if we
find a loop of $\mathcal{G}_{S}$ in some $P_{k}$, we conclude that $S$ is
compressible, and this algorithm finishes; Otherwise, according to the proof
of Lemma 4.2 and the proof of property 2 in Proposition 4.3, $S$ can be
isotoped so it contains no movable saddle, and we proceed to $III$.

$III$. By Lemma 5.2 we know that we can determine the compressibility or
incompressibility of $S$ by confirming whether there exists a compressing disk
in normal position, through examining whether the elements in
$\mathcal{P}^{J}_{I},I,J=2,4,...,2m,$ that satisfy equation set (2). According
to property 3 of Proposition 5.1, a bi-gon can only intersects with a
$2n$-disk, where $n\geq 4$. Otherwise we can reduce $|S\cap\mathcal{D}|$.

$III.(a)$ Consider $\mathcal{P}_{2}$, if it is empty, then apparently we can
conclude that there does not exist any compressing disks of $S$. We conclude
that $S$ is incompressible, and this algorithm finishes; Otherwise we proceed
to $III.(b)$

$III.(b)$ If $\mathcal{P}_{2}$ is nonempty, we suppose $S$ is compressible.
There must be at least a pair of bi-gons in the selection. Each element in
$\mathcal{P}_{2}$ can be selected for at most $\mathcal{N}_{c}$ times due to
Lemma 5.3. If two elements in $\mathcal{P}_{2}$, say $b_{1}$ and $b_{2}$,
intersect each other, then $b_{1}$ and $b_{2}$ cannot be both selected,
because the surface glued would not be an embedded surface. Therefore we can
examine all the possible selections of bi-gons, and check the polygons
connected to them, (and the polygons connected to those mentioned above …), to
see of there exists a selection of polygons which satisfies equation set (2).
If the equation set has a solution, then we conclude that $S$ is compressible,
otherwise $S$ is incompressible.

We notice that the algorithm is finite and it always concludes, since $n_{2}$
is bounded by $|\mathcal{P}_{2}|\times\mathcal{N}_{c}$, and the equation set
has at most finitely many positive integer solutions.

###### Proof of Corollary 1.2.

Since $S$ is an incompressible, properly embedded surface in $V$, we can put
$S$ in standard position by Proposition 3.1. Moreover, since $S$ can be
isotoped so that $S\subset\partial V\times[0,1]$, we have $|S\cap C|=0$.
Assume $S$ is in standard position, and not $\partial$-parallel. Then there
must exists a movable saddle in $S$, which contradicts property 2 of
Proposition 4.3. Therefore, we have proved the first claim. The second claim
is obvious from property 1 of Proposition 4.3. ∎

######  Remark 5.2.

A compression body $M_{C}$ is a connected 3-manifold obtained by adding some
1-handles to $F\times I$ with all the gluing disks lying in $F\times 1$, where
$F$ is an orientable closed surface (possibly non-connected), and filling in
3-balls to the resulting 3-manifold along all the 2-sphere components in
$F\times 0$, if any. The surface $F\times 0$ with all 2-sphere components
removed is denoted by $\partial_{-}M_{C}$, and $\partial
M_{C}\setminus\partial_{-}M_{C}$ is denoted by $\partial_{+}M_{C}$. It is
clear that $M_{C}$ is a handlebody if $\partial_{-}M_{C}=\emptyset$. As in the
handlebody case, we may put the part $F\times I$ of $M_{C}$ in a position such
that at a regular height, the intersection of the height plane and the part
$F\times I$ in $M_{C}$ is either a disk, or an annulus, or two annuli, and by
a similar argument, we may generalize Theorem 1.1 to the case of compression
bodies. But the statement of a similar theory on compression bodies is
tediously long and repetitive, we omit the details of this case.

##  6 Some Examples

We demonstrate the application of our geometric algorithm mentioned in Theorem
1.1 through some examples.

Figure 10: Placing $Q_{4}$ in standard position.

######  Example 6.1.

In [8], Qiu has constructed a series of properly embedded incompressible
surfaces $Q_{2n}$, $n\geq 1$, of arbitrarily high genus in a genus two
handlebody $H_{2}$. He has proved each $Q_{2n}$ is incompressible by checking
that $\pi(Q_{2n})$ is $\pi_{1}$-injective in $\pi_{1}(H_{2})$. We give a proof
using our geometric algorithm to show that each $Q_{2n}$ is incompressible for
an arbitrary $n$.

###### Proof.

Let $D_{0}$ be a separating disk of $H_{2}$, and let $(D_{1},D_{2})$ be a set
of meridian disks of $H_{2}$. Then we take
$\mathcal{D}=\\{D_{0}\\}\cup\\{D_{1}\\}\cup\\{D_{2}\\}$, so that
$\overline{H_{2}-\mathcal{D}}=P_{1}\cup P_{2}$. It is easy to see that
$|Q_{2n}\cap C|$ is at least 1. When $n=1$, $Q_{2n}$ is already in standard
position, since any isotopy of $Q_{2n}$ would not decrease the complexity
$\mathcal{L}$. Also when $n=1$, it is easy to check that $Q_{2n}$ is
incompressible.

We are going to picture the surfaces in the style of “Heegaard diagrams”, in
which the red arcs represent unions of $4$-disks, and each closed loop which
does not include the parts in the boundaries of $\mathcal{D}$, bounds a disk
in $P_{1}$ or $P_{2}$. When $n=2$, the graph of $Q_{2n}$ is shown in Figure
10(a), and the alignments of $u_{i},v_{i},1\leq i\leq 4,$ are as given in [8]
. We can isotope $Q_{4}$ so that it is in standard position as shown in Figure
10(b). In $P_{1}$ of Figure 10(b) there exists only one bi-gon $p_{2}$ in
$\mathcal{P}_{2}$. Therefore, by Lemma 5.2 $Q_{2n}$ is incompressible when
$n=2$. For $N\geq 2$, we can show by “induction” that: if $Q_{2N}$ is put in
standard position, then we can put $Q_{2N+2}$ in standard position in a
similar pattern. Compare $Q_{4}$ with the case of $Q_{6}$ in standard position
shown as in Figure 11 will help us see this. Therefore, we can see for an
arbitrary $n$, $Q_{2n}$ always admits only one bi-gon $p_{2}$ in $P_{1}$ and
$P_{2}$. Therefore, by Lemma 5.2 we have proved $Q_{2n}$ is incompressible for
an arbitrary $n$. ∎

Figure 11: $Q_{6}$ in standard position. Figure 12:

######  Example 6.2.

In [8], if we replace the arc of $u_{1}v_{1}$ by the arc shown in Figure
12(a), we can still show that $Q_{2n}$ is incompressible for an arbitrary $n$.
If we replace the arc of $u_{1}v_{1}$ by the arc shown in Figure 12(b), we can
show that for $n\geq 2$, $Q_{2n}$ is compressible. The proof is similar to
that of the preceding example.

######  Example 6.3.

In [7], Jaco constructed a class of non-separating incompressible surfaces
$J_{n}$, $n\geq 1$, of arbitrarily high genus properly embedded in $H_{2}$. We
apply the formal construction of $J_{n}$ shown in [4]. We take
$\mathcal{D}=\\{D_{0},D_{i}|i=1,2\\}$, and place $J_{n}$ in standard position.
Then the submanifolds are symmetrical with respect to $D_{0}$ after we cut
$H_{2}$ and $J_{n}$ with $\mathcal{D}$. $(H_{2},J_{n})$ can be considered as
two ps-ball with subsurfaces of $J_{n}$ embedded in it, shown as in Figure 13,
glued together. There exists no bi-gon in each ps-ball for an arbitrary $n$.
Therefore, by Lemma 5.2 we have proved an arbitrary $n$, $J_{n}$ is
incompressible.

Figure 13:

Acknowledgement The authors would like to express their deep gratitude to the
referees for many helpful suggestions.

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