# Elliptic operators on non-compact manifolds have closed range

Luther Rinehart

###### Abstract

We show that a second-order elliptic differential operator $P$, on any
manifold $M$, has closed range in $C^{\infty}(M)$. If $M$ has no compact
components, then $P$ is surjective on $C^{\infty}(M)$. Applications to
Helmholtz decomposition are discussed.

Most literature on elliptic operators deals with bounded domains and compact
manifolds, since in those cases the operators are Fredholm, and the problems
of uniqueness as well as existence are well-understood. In contrast this paper
deals with existence results in unbounded domains. Specifically, we show that
a second-order elliptic operator $P$, on any manifold, compact or not, has
closed range in $C^{\infty}(M)$. Then, for certain manifolds, this combines
with the much easier proof that an elliptic operator has dense range, to give
the result that the operator is surjective on $C^{\infty}(M)$.

Hörmander, in section 10.6 of [1], treats the case of constant coefficient
operators on $\mathbb{R}^{n}$. There they introduced the notion of
$P$-convexity, which for constant-coefficient operators is equivalent to
$C^{\infty}$ surjectivity. This proves, for example, that the Laplacian is
surjective on $\mathbb{R}^{n}$, and indeed any open subset of
$\mathbb{R}^{n}$. The method of $P$-convexity can be extended to more general
subelliptic operators, as explained by Petersen in section 4.7 of [2]. This
approach shows, by functional analytic arguments, that the closed-range
property can be reduced to the property of $P$-convexity, an essentially
topological property of the domain. Petersen’s method is stated for domains in
$\mathbb{R}^{n}$, but is easily adapted to arbitrary manifolds.

##  1 A motivating example

As a motivating example for existence results in unbounded domains, consider
Helmholtz decomposition in elementary vector analysis. It is well known that
any vector field in $\mathbb{R}^{3}$ can be written, non-uniquely, as a sum of
a gradient and a curl. For vector fields decaying sufficiently rapidly at
infinity, the justification of this fact using, say, the fundamental solution
of the Laplacian, is standard textbook material. However, its proof for
general smooth vector fields requires other methods and is less well-known.
Consider the following generalization: Let $M$ be an orientable Riemannian
manifold, with exterior differential $d$, exterior codifferential $\delta$,
and Hodge Laplacian $\Delta=d\delta+\delta d$.

######  Definition 1.

We say $M$ has the _Helmholtz decomposition property_ if, given any
differential form $\omega$, there exist differential forms $\alpha$ and
$\beta$ such that $\omega=d\alpha+\delta\beta$.

Aside from general usefulness in computations, this property is related to the
following facts:

######  Lemma 1.

If $M$ has the Helmholtz decomposition property, and
$\omega=d\alpha+\delta\beta$, then $\alpha$ may be chosen to be coclosed (even
coexact), and $\beta$ may be chosen to be closed (even exact).

###### Proof.

Apply decomposition to $\alpha=d\alpha_{1}+\delta\alpha_{2}$ and replace
$\alpha$ with $\delta\alpha_{2}$. Similarly, apply decomposition to
$\beta=d\beta_{1}+\delta\beta_{2}$, and replace $\beta$ with $d\beta_{1}$. ∎

######  Theorem 1.

$M$ has the Helmholtz decomposition property if and only if $\Delta$ is
surjective on differential forms.

###### Proof.

Clearly, if $\Delta$ is surjective, then given $\omega$, there exists $\phi$
with $\omega=\Delta\phi=d(\delta\phi)+\delta(d\phi)$.

Now suppose $M$ has Helmholtz decomposition. Given $\omega$, we may write

$\omega=d\alpha+\delta\beta$ $\alpha=d\alpha_{1}+\delta\alpha_{2}$
$\beta=d\beta_{1}+\delta\beta_{2}$

By lemma 1, we can choose $d\alpha_{2}=0$ and $\delta\beta_{1}=0$. Then let
$\phi=\alpha_{2}+\beta_{1}$.

$\Delta\phi=(d\delta+\delta
d)(\alpha_{2}+\beta_{1})=d\alpha+\delta\beta=\omega$

∎

######  Theorem 2.

If $M$ has the Helmholz decomposition property, then the _generalized Maxwell
problem_ is solvable. That is, given an exact form $\alpha$ and a coexact form
$\beta$, there exists a differential form $\omega$ satisfying

$d\omega=\alpha\qquad\delta\omega=\beta$ (1)

###### Proof.

$\alpha$ is exact, so write $\alpha=d\alpha_{1}$ where, by Helmholtz
decomposition, we may choose $\delta\alpha_{1}=0$. Likewise, $\beta$ coexact,
so write $\beta=\delta\beta_{1}$ where, by Helmholtz decomposition, we may
choose $d\beta_{1}=0$. Then $\omega=\alpha_{1}+\beta_{1}$ is the required
solution. ∎

##  2 Main results

Let $M$ be a manifold and $E$ a vector bundle on $M$. Let $P\colon
C^{\infty}(E)\rightarrow C^{\infty}(E)$ be a second-order elliptic
differential operator with smooth coefficients. Give $C^{\infty}(E)$ its usual
Frechet space topology. Let $\mathcal{E}^{\prime}(E^{*})=(C^{\infty}(E))^{*}$
denote the space of compactly supported distributional sections. In the
following, the bundle $E$ will be suppressed in the notation. A useful fact we
will use several times is that if $P$ is second-order elliptic and $Pu=0$,
then the support of $u$ must have empty boundary. This first of all gives the
easy result:

######  Theorem 3.

Suppose $M$ is connected and not compact. Then $P$ has dense range.

###### Proof.

The transpose $P^{*}$ acting on $\mathcal{E}^{\prime}$ is also elliptic, so
with $M$ being connected and not compact, the equation $P^{*}u=0$ has no
nontrivial compactly supported solutions. Hence $P^{*}$ is injective, so $P$
has dense range. ∎

The rest of this paper will focus on proving that $P$ has closed range. The
proof that $P$ has closed range has two halves. The first, functional analytic
half, consists of the method laid out in [2] section 4.7. The second half is
topological.

Petersen’s theorem 7.9 first proves a general subelliptic estimate satisfied
by $P$ on compact subsets $K$. This estimate is then used in theorem 7.22 to
prove that the image $P(\mathcal{E}^{\prime}_{K})$ of the space of
distributions supported in $K$, is weak* closed in $\mathcal{E}^{\prime}$. The
key definition for obtaining a global closed-range theorem is the following.

######  Definition 2.

$M$ is $P$_-convex_ if for every compact $K\subseteq M$, there exists a
compact $K^{\prime}\subseteq M$, such that $\forall u\in\mathcal{E}^{\prime},\
\text{supp}Pu\subseteq K\Rightarrow\text{supp}\,u\subseteq K^{\prime}$.

Petersen’s corollary 7.24 shows that if $M$ is $P^{*}$-convex, then
$P^{*}(\mathcal{E}^{\prime})$ is weak*-closed in $\mathcal{E}^{\prime}$. By
the closed-range theorem for Frechet spaces ([2] p. 33), this is then
equivalent to $P(C^{\infty})$ being closed in $C^{\infty}$.

If we want to show that $M$ is $P^{*}$-convex, this amounts to finding, for
each compact set $K$, a larger compact set $K^{\prime}$ such that the
components of $M\setminus K^{\prime}$ are all unbounded, that is, having
noncompact closure. This is because the boundary of the support of $u$ cannot
intersect the region where $P^{*}u=0$, so the support of $u$ cannot extend
into any unbounded component of $M\setminus K^{\prime}$. Thus the key
topological lemma in this regard is the following one. The proof is thanks to
Pierre PC [3] and Aitor Iribar Lopez [4] at mathoverflow.net. It is based on
the proof of theorem 23.5 in [5].

######  Lemma 2.

Let $X$ be locally compact Hausdorff, connected, and locally connected. Given
compact $K\subseteq X$, let $K^{\prime}$ be the union of $K$ with all the
relatively compact components of $X\setminus K$. Then $K^{\prime}$ is compact.

###### Proof.

Index the connected components of $X\setminus K$ as $\\{C_{j}\\}_{j\in J}$,
and set $J_{0}=\\{j\in J\mid\overline{C_{j}}\text{ is compact}\\}$. By
definition, $K^{\prime}=K\cup\bigcup_{j\in J_{0}}C_{j}$ and $X\setminus
K^{\prime}=\bigcup_{j\notin J_{0}}C_{j}$. As $X$ is locally connected, all the
$C_{j}$ are open, so $K^{\prime}$ is closed.

As $X$ is locally compact Hausdorff, let $U$ be a relatively compact open
neighborhood of $K$. As $X$ is connected, $\forall j,\ C_{j}\cap
U\neq\varnothing$.

Also, $\partial U$ is compact, and $\\{C_{j}\\}_{j\in J}$ forms a disjoint
open cover of it. Thus there are only finitely many $j_{1},...,j_{n}\in J_{0}$
such that $C_{j_{i}}\cap\partial U\neq\varnothing$. For all other $j$, since
$C_{j}$ is connected, we mus have $C_{j}\subseteq U$. Then,

$K^{\prime}\subseteq U\cup C_{j_{1}}\cup...\cup C_{j_{n}}$

Since $K^{\prime}$ is closed and the right-hand side is relatively compact, it
follows that $K^{\prime}$ is compact. ∎

######  Theorem 4.

$P$ has closed range.

###### Proof.

First suppose that $M$ is connected. Given a compact $K\subset M$ and
constructing $K^{\prime}$ as in lemma 2, we see that $M$ is $P^{*}$-convex. By
Petersen’s functional analytic argument, it follows that $P$ has closed range.

For non-connected $M$, $C^{\infty}$ decomposes as the topological direct
product of the spaces of smooth sections on the components. Furthermore,
differential operators respect this decomposition, so the range of $P$ is the
direct product of its ranges on the components, hence closed in the product
topology. ∎

######  Corollary.

If $M$ is connected and not compact, or more generally if $M$ has no compact
components, then $P$ is surjective on $C^{\infty}$.

We observe that any failure of surjectivity can be entirely described by the
Fredholm theory on the compact components.

## References

  * [1] L. Hörmander, _The Analysis of Linear Partial Differential Operators II_ , Springer-Verlag, Berlin, Heidelberg, 2005.
  * [2] B. E. Petersen, _Introduction to the Fourier Transform and Pseudo-differential Operators_ , Pitman Advanced Pub. Program, 1983.
  * [3] Pierre PC (https://mathoverflow.net/users/129074/pierre-pc), Is the union of a compact and the relatively compact components of its complementary in a manifold compact?, URL: https://mathoverflow.net/q/410072.
  * [4] Aitor Iribar Lopez (https://mathoverflow.net/users/176397/aitor-iribar-lopez), Is the union of a compact and the relatively compact components of its complementary in a manifold compact?, URL: https://mathoverflow.net/q/410372.
  * [5] O. Forster, _Lectures on Riemann Surfaces_ , Springer-Verlag, New York, 1981.


# Residual Quantity in Percentage of Factory Machines Using Computer Vision
and Mathematical Methods

Seunghyeon Kim1, Jihoon Ryoo2, Dongyeob Lee3, and Youngho Kim4

##  I INTRODUCTION

Figure 1: Image of a vibrating hopper [3]

Computer vision is an advanced computer science technique that has provided
the basis for many new patents and APIs that have informed a multitude of
applications. Computer vision involves all analyses, including machine
learning, performed using pixels of images. Some manufacturing industries have
recently discovered that they need human resources to monitor whether the
machine is working properly and if adequate materials are entering the hopper.
From an economic perspective, using humans to monitor resources manually can
be an inefficient allocation of resources associated with welfare decline.
Furthermore, the hopper is 2m tall, introducing the possibility of falls as
personnel ascends a ladder for monitoring. A fall from the ladder could lead
to an industrial accident and a significant loss of profit for the company.

Using computer vision could eliminate these risks. While it may be possible to
use the loadcell specific to the hopper to monitor the change in mass, such a
solution could be very useful when the hopper does not vibrate. However, many
manufacturing industries have vibrating hoppers,which may not be amenable to
loadcell implementation. Computer vision can provide a solution by using a
specific algorithm that utilizes one camera and a computer to analyze the data
received from the camera. This system can indicate the fullness of the hopper.
It does not require any human resources, minimizing welfare loss and the
possibility of an industrial accident. We provide in this report the precise
algorithm used.

##  II Related Works

###  II-A Loadcell

Figure 2: Simple diagram of a loadcell [6]

First, a load cell that measures the weight of the materials in a hopper
cannot be used because most manufacturing industries use vibrating hoppers,
which can equally distribute the materials inside the hopper and minimize
jamming. The vibrating hopper not only wears out the load cell but could also
exhibit oscillating motion that does not return a precise weight. Therefore,
it is not an economically prudent solution to the problem.

###  II-B Machine Learning Techniques

Figure 3: Simple diagram of how deep learning technique works [2]

Deep learning techniques are very useful in approaching this type of problem
as they offer very precise analyses of the images and are very adept at
computer vision. Nevertheless, they are not recommended as they present time
lags. They do not return the results as quickly as a human eye, requiring more
than 100 sample data points to achieve near-excellent learning. They also
require a high-quality graphics processing unit to offer excellent data
analyses, which is not necessarily profitable.

##  III Challenge and Target Scenario

The challenge was to create an algorithm that could efficiently analyze the
fullness of the hopper: 10%, 25%, 50%, 75%, or 100%. Our first approach was to
use a K means clustering algorithm to simplify the image into two colors
before analysis. However, a significant outlier was identified when I used the
sample data. As a result, a new algorithm was designed that uses OpenCV,
standard deviation, and variance.

###  III-A Theory

The theory of the algorithm is based on the fact that more materials in the
hopper affect the standard deviation of the target line from the image(target
lines will be explained in upcoming subsections).

Figure 4: Sample Data of 10% Full Hopper Figure 5: Modified Sample Data of 10%
Full Hopper Figure 6: Modified Sample Data of 50% Full Hopper

$\sigma=\sqrt{\frac{\sum(r_{i}-r_{avg})^{2}}{N}}$ (1)

Where $\sigma$ = standard deviation, $r_{i}$ = value of each pixel, $r_{avg}$
= average pixel value, and N = size of the pixel array.

$\sigma^{2}=\frac{\sum(r_{i}-r_{avg})^{2}}{N}$ (2)

Where $\sigma^{2}$ = variance and the rest of the variable are the same as
formula number 1.

$A_{1}=\frac{\sigma_{1}+\sigma_{2}}{2}$ (3)

Where ${\sigma_{1}}$ = standard deviation of pixels in L1, ${\sigma_{2}}$ =
standard deviation of pixels in L2, and $A_{1}$ = arithmetic mean of the
standard deviation of L1 and L2.

$A_{2}=\frac{\sigma_{1}^{2}+\sigma_{2}^{2}}{2}$ (4)

Where ${\sigma_{1}^{2}}$ = variance of pixels in L1, ${\sigma_{2}^{2}}$ =
variance of pixels in L2, and $A_{2}$ = arithmetic mean of the variance of L1
and L2.

Figure 4 shows sample data of a 10% filled hopper, and Figure 5 depicts the
two abstract targeted lines used for this algorithm. As seen in Figure 5, the
image is converted into a greyscale image from an RGB image, and there are two
lines: "L1" and "L2." L2 is vertically located at the center of the image so
that L2 can determine the standard deviation at the center and determine the
fullness of the hopper. The line L1 indicates whether the hopper is 75% full
or 100% full (the materials will extend to L1 if it overflows 75%).

The algorithm retrieves the values of the pixels located in the lines. As the
values are retrieved, the code calculates standard deviation using Formula 1.
As seen in Figure 5, the standard deviation of L2 seems more affected (in
terms of standard deviation) by the texture of the materials inside the hopper
than that of L2 in Figure 6. The length of the zigzagged lines inside the
figures depicts the degree of variation within the standard variation created
by the materials inside the hopper. Had the hopper been even more filled, the
zigzag portion would have been larger, increasing the standard deviation
value. However, contrasting the standard deviation values revealed that the
formula involving the standard deviation of L1 and L2 should be determined.

Using the arithmetic mean of among variance, standard deviation, or
$A_{1}^{2}$($A_{1}$ in Formula 3) was proven effective for contrasting with
greater than 85% accuracy (explained in the choice of algorithm section). Each
presented pros and cons. For example, the four-digit magnitude mean of
variance could be used to compare the values between each hopper type. The
squared arithmetic mean of the standard deviation also exhibited four-digit
magnitudes that provided the same benefits as the mean of variance but with
different values. For the arithmetic mean, however, the magnitudes were not
large enough(two-digit) to determine the precise threshold between each
percentage of hoppers which can be problematic, and it is extremely similar to
the squared arithmetic mean. In fact, the larger the magnitude, the greater
the range of errors permitted. Therefore, the choice of algorithm was narrowed
to the mean of variance and the squared value of the arithmetic mean of
standard deviation.

As there are different types and designs of hoppers in industry and smart
factories, the standard deviation of the empty hopper image could be
subtracted from the returned value. The different hopper designs can reflect
different textures. The amount of rust or stain will also differ, which may
affect the pixel value of the image, leading to an inaccurate standard
deviation value. When subtracted, however, the result provides the standard
deviation of the texture of the materials inside the hopper. In that case, the
range in which the standard deviation could indicate 10%, 25%, 50%,75%, or
100% would theoretically be the same as all the other hoppers. Nevertheless,
the precise experimental results could not be produced since there were fewer
images of empty hoppers in modern smart factory industries; thus, this could
be considered an inference.

###  III-B Experimental Results of Algorithms

Figure 7: The Accuracy of the Algorithm

As seen in Figure 7, the accuracy of the algorithms tended to be more than
85%, and the average time taken for the code to analyze the image was less
than a second, substantially less than 1.8s of run time and slightly over 80%
accuracy for existing deep learning techniques in the common smart factory
industry. However, it may be debatable whether 0.003s of run time is worth 5%
accuracy compared to each algorithm mentioned. This question may not be
debatable when only seeing the 5% value, but only 23 data points were given.
1/23 is approximately 4.3%, which is related to the difference in accuracy in
that the accuracy of the algorithm differs only by one test case. As mentioned
regarding the vibrating hopper in the first section, the given test case
images were collected in extreme environments (such as in Figure 6, where the
materials are skewed to one side). Therefore, the mentioned algorithms may
exhibit high accuracy in the real data set. Specific explanations will be
included in the evaluation section.

##  IV Evaluation

The overall algorithm seems useful, and we will now describe the process used
to prove the effectiveness of the algorithm.

###  IV-A Run Time Measurement and Accuracy Measurement

Figure 8: Sample code to determine the run-time

In Figure 7, the algorithm’s run-time and accuracy can be seen. The approach
to measuring the run time was using Python’s "time" library. It is a general
approach to determine the run-time of the code but necessary to determine the
run-time for 23 data points at once. Thus, I used the "time" library to
determine the run-time of each of the 23 test data points and averaged them
(Figure 7). The accuracy also considers all the outliers (an inaccurate output
may be considered to have an algorithm return of 50% instead of 75% when the
actual image is 75%) as wrong outputs, so it only returns

$R/T=accuracy$

where

$R=$ Correct number of outputs

and

$T=$ Total number of outputs.

Figure 8 shows how it was coded to determine the run-time for the code. As a
practical example, FronTech Inc., located in the Republic of Korea, runs 50
machine hoppers in total. Had the algorithm been based on deep learning
techniques, a server computer would require a total of 60 seconds to process
the images from 50 cameras (1.2s for an image to be processed * 50 = 60s). In
a FronTech Inc. industrial operation, a worker can check 9 10 hoppers in one
minute. However, the suggested algorithm requires only 2.3 seconds to process
50 data points from the cameras(0.046 for an image to be processed * 50 =
2.3s). Indeed, the company would require at least 5 GPU-installed computers to
process each camera using deep learning techniques. However, the suggested
algorithm requires only 1 for 50, which is economically and technically
efficient.

###  IV-B Test Data Sets and Definitions

Figure 9: Test Data of 25% Full Hopper Figure 10: Test Data of 25% Full Hopper

Only 23 test data points were provided to verify the algorithm. In addition,
most of the data sets were recorded in extreme cases. For example, Figure 9
shows skewed material in the left top corner, which is biased data and highly
unlikely for a vibrating hopper. Figure 10 shows biased data where the
materials are skewed towards the left bottom. Had the hopper been vibrating,
the materials would likely be nearly equally distributed due to gravitational
law.

Figure 11: Test Data of 10% Full Hopper Figure 12: Test Data of 100% Full
Hopper

Some data, however, were not appropriate. Figure 11 shows a 10% filled hopper,
but the worker’s arm is observed, which would result in an improper output.
Figure 12 also shows data where the picture was taken while the objects were
falling. As a result, 2 of 25 data were inappropriate and had to be removed.

Figure 13: Test Data of 75% Full Hopper

The algorithm could be validated if there were at least 50 appropriate data
points. Currently, however, there is no alternative way to validate the data.
Though the data is biased, using the given data would be the only data
available to validate my algorithm. Figure 13 shows one of the data points
that the algorithm could not correctly identify. FronTech Inc. stated that the
data exhibits a 75% filled hopper, but the algorithm identifies the hopper as
50% full. Visually, people would likely identify the data as a 50%-filled
hopper. They are stacked on top of each other; as the hopper vibrates,
however, they should be distributed equally.

Not only the data set, but how FronTech Inc. defined 10%, 25%, 50%, 75%, and
100% was abstract. There is likely a general tendency for patterns to occur,
but not all the data could be reported in exact percentages. In fact, data
such as that shown in Figure 13 may be problematic because deciding the range
of standard deviation for each percentage may force the user to create a more
incorrect range. Data such as those in Figure 13 could also be considered
outliers.

##  V Future Work & Discussion

The techniques mentioned can be used to monitor hoppers in various smart
factory industries. Certain factors will require manipulation to keep track of
the materials correctly, such as the coordinates of the L1 and L2 line axes.
The algorithm could be modified to identify the coordinates of L1 and L2
automatically from the hopper image, increase the number of lines, and count
the number of specific materials in the hopper.

The suggested algorithm may be viewed as simple. Nonetheless, such algorithms
are not investigated because domestic firms in many countries demand deep
learning techniques and luxury algorithms that could serve to advertise their
companies. These advanced solutions to problems such as the one addressed here
do not improve welfare and are inefficient. In fact, sometimes, a simpler
solution is necessary.

##  VI Conclusion

Overall, it was a lengthy process to create the algorithm and correct the
data. Though there were limitations during algorithm validation, the algorithm
itself still exhibits high accuracy and seems readily deployable in real-world
industrial situations. It was challenging to choose the specific method
between the arithmetic mean of variance and the square of average standard
deviation between L1 and L2. In determining which algorithm to sacrifice, I
decided to include both in the GitHub link[4]. The algorithm may have more
potential in the computer vision field, and it could become an algorithm that
stands as an alternative to deep learning techniques. It requires refinement,
and outliers are present as it relies on the texture of the materials. It is
likely applicable to solutions to many other challenges.

## References

  * [1] Amilkanthawar, Vivek. “Choosing a Deep Learning Framework.” Medium, The HumAIn Blog, 28 Feb. 2019, https://medium.com/in-pursuit-of-artificial-intelligence/choosing-a-deep-learning-framework-5669a85ebc3f.
  * [2] “CNN, Convolutional Neural Network abstract.” Blog main page, 4 Jan. 2018, http://taewan.kim/post/cnn/.
  * [3] “Home.” Magnetic Food Feeding Vibratory Hopper - Buy Magnetic Vibratory Feeder,Food Feeding Conveyor,Food Industry Vibrating Feeder Product on Alibaba.com, https://www.alibaba.com/product-detail/Magnetic-food-feeding-vibratory-hopper_60488056903.html.
  * [4] Kim, Seunghyeon. “Residual Quantity in Percentage of Factory Machines Using Computer Vision and Mathematical Methods.” GitHub, 1 Nov. 2021, https://github.com/seunghk1206/Residual-Quantity-in-Percentage-of-Factory-Machines-Using-Computer-Vision-and-Mathematical-Methods/settings.
  * [5] “Standard Deviation.” Corporate Finance Institute, 31 Jan. 2020, https://corporatefinanceinstitute.com/resources/knowledge/standard-deviation/.
  * [6] WhitePaper. “Load Cell vs. Force Sensor.” Tekscan, 5 June 2020, https://www.tekscan.com/resources/whitepaper/load-cell-vs-force-sensor.