Polygonal Meshes of Highly Noisy Images based on a New Symmetric

Thinning Algorithm with Theoretical Guarantees

Mohammed Arshad Siddiqui

1 a

and Vitaliy Kurlin

2 b

PDPM Indian Institute of Information Technology Design and Manufacturing, Jabalpur, India

University of Liverpool, U.K.

Keywords:

Edge Detection, Thinning, Skeletonization, Polygonal Mesh.

Abstract:

Microscopic images of vortex ﬁelds are important for understanding phase transitions in superconductors.

These optical images include noise with high and variable intensity, hence are manually processed to extract

numerical data from underlying meshes. The current thinning and skeletonization algorithms struggle to ﬁnd

connected meshes in these noisy images and often output edge pixels with numerous gaps and superﬂuous

branching point. We have developed a new symmetric thinning algorithms to extract from such highly noisy

images 1-pixel wide skeletons with theoretical guarantees. The resulting skeleton is converted into a polygonal

mesh that has only polygonal edges at sub-pixel resolution. The experiments on over 100 real and 6250

synthetic images establish the state-of-the-art in extracting optimal meshes from highly noisy images.

1 INTRODUCTION

Microscopic image analysis of topological defects

(called quantum vortices) in symmetry-breaking

phase transitions is now a hot topic in cosmology

and high-temperature superﬂuids and superconduc-

tors (Lin et al., 2014). Figure 1 shows a piezo-

response force microscopic image of a vortex ﬁeld.

The underlying vortex ﬁeld is a complicated mesh of

non-convex polygons that is hard to recognize even by

human experts because of the highly variable noise.

To understand phase transitions, physicists ur-

gently need to count various statistics of vortex ﬁelds

such as the number and positions of vertices, the num-

ber and sizes of polygonal domains. Since past algo-

rithms produced incomplete meshes with discontinu-

ities or hanging endpoints, see section 6, human ex-

perts had to manually count features of vortex images.

This paper automates the collection of statistical data

that will enable a discovery of high-temperature su-

perconductors as motivated in (Lin et al., 2014).

Deﬁnition 1. A vortex mesh on an image Ω is a skele-

ton or an embedded graph splitting Ω into domains

with polygonal edges that consist of straight line seg-

ments and meet at vertices of degree at least 3.

https://orcid.org/0000-0002-9747-3132

https://orcid.org/0000-0001-5328-5351

Figure 1: Our new mesh on top of a typical microscopic im-

age, see high resolution images in supplementary materials.

Optimal Mesh Problem. For any highly noisy vor-

tex image as in Figure 1, ﬁnd a connected polygonal

mesh that minimizes the average intensity (to get a

darkest skeleton) and has a minimum number of ver-

tices.

Siddiqui, M. and Kurlin, V.

Polygonal Meshes of Highly Noisy Images based on a New Symmetric Thinning Algorithm with Theoretical Guarantees.

DOI: 10.5220/0009340301370146

In Proceedings of the 15th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2020) - Volume 4: VISAPP, pages

137-146

ISBN: 978-989-758-402-2; ISSN: 2184-4321

137

Figure 2: Past edge detection on the input in Figure 1. 1st: many disjoint components based on a small number of Canny

edge pixels (Canny, 1986). 2nd: many superﬂuous components based on a large number of Canny edge pixels. 3rd: fuzzy

logic skeletonization (Kamani et al., 2018). 4th: (Asghari and Jalali, 2015). 5th: watershed by (Soille and Vincent, 1990).

The primary minimization is for the average in-

tensity, because the underlying vortex mesh is rep-

resented by darker pixels. The secondary minimiza-

tion is for the number of degree 2 vertices to simplify

polygonal lines between higher degree vertices.

Here are the challenges in solving the above problem

for the highly noisy microscopic images in practice.

• Polygonal lines have a large and variable thickness

and can meet at higher degree vertices at any angle.

• The intensity largely varies and leads to many noisy

edges, compare with a ground truth in Figure 18.

• Microscopic images often contain isolated noisy

blobs that should be ignored and also larger blobs that

should be treated as vertices or parts of lines.

Here are the novelty and contributions of the paper. to

the state-of-the-art skeletonization of noisy images.

• New Deﬁnition 5 formalizes a 1-pixel wide skeleton

guaranteed in the output by Theorem 6.

• The new erosion algorithm in section 4 preserves the

homotopy type (stronger than a graph connectivity)

for any binary image, see Deﬁnition 3, Theorem 8.

• Table 1 in section 6 shows that our thinning out-

performs all past algorithms on the minimum average

intensity on over 100 real images and also produces

simpler meshes without erroneous degree 1 vertices.

• Table 2 has standard accuracy measures averaged

over 6250 ground truths and shows that the new sym-

metric thinning in section 5 is robust under noise.

Here is the pipeline for the optimal mesh extraction.

Stage 1: the Otsu thresholding (Otsu, 1979) is

adapted for microscopic images with variable noise

in section 3 to get a connected binary image.

Stage 2: any binary image is symmetrically thinned

to a 1-pixel wide skeleton without degree 1 vertices,

see proofs of Theorems 6 and 8 in section 4.

Stage 3: a 1-pixel wide skeleton from Stage 2 is con-

verted to a mesh whose polygonal lines are further

straightened for a statistics analysis in section 6.

2 A REVIEW OF PAST WORK

Edge Detection Algorithms. The Canny edge de-

tector (Canny, 1986) depends on 3 thresholds and can

output many disjoint components, see Figure 2.

Morse-Smale skeleton (Delgado-Friedrichs et al.,

2014) consists of gradient ﬂow curves joining local

extrema and saddle points of the grayscale intensity.

Figure 3 shows a Morse-Smale skeleton with many

superﬂuous black vertices joined by white curves.

Figure 3: The Morse-Smale skeleton (Delgado-Friedrichs

et al., 2014) with black dots for local minima and saddles.

The recent survey on skeletonization (Saha et al.,

2016) is a comprehensive review of past approaches

and current challenges. Typical requirements for a

skeleton representing a binary (black-and-white) im-

age is (1) centrality, (2) invariance under rotations and

other deformations, (3) connectedness or preservation

of a topological type (Kurlin, 2015a), (Kurlin, 2015b),

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138

(Kalisnik et al., 2019), (Smith and Kurlin, 2019).

Another aim from Bernard in (Bernard and Man-

zanera, 1999) is to guarantee that a skeleton is 1-pixel

wide, because a big block of black pixels such as

(Saha et al., 2016, Figure 1) or Figure 11 is an obsta-

cle for many thinning algorithms. The new thinning

algorithm in sections 4-5 solves problem by introduc-

ing a 1-pixel wide skeleton in Deﬁnition 5.

OpenCV offers the thinning by (Zhang and Suen,

1984) and (Guo and Hall, 1989). Both methods use

a 3 × 3 neighborhood to decide which pixels to re-

move. The ﬁrst method (Zhang and Suen, 1984) con-

sists of two sub-iterations where the pixels are re-

moved from north-east and south-west corners, then

from the south-east and north-west corners. The sim-

ilar second method guarantees (Guo and Hall, 1989,

Proposition A.3) the global connectivity of clusters,

not the homotopy type in Deﬁnition 3, so a circular

cluster can become a non-closed polygonal line.

Fuzzy Logic: Kamani (Kamani et al., 2018) proposed

a skeletonization method based on fuzzy logic, which

can take any grayscale image, see Figure 2(3).

CNNs: convolutional neural networks (Wang et al.,

2018), (Shen et al., 2017), (Panichev and Voloshyna,

2019) were particularly successful in cases when a lot

of ground truth data is available, see Fig. 4.

Figure 4: The output by (Asghari and Jalali, 2015) on the

image from Figure 1. One could zoom in and ﬁnd many

disjoint components instead of one big connected skeleton.

3 THE NEW ADAPTIVE

THRESHOLDING

This section discusses Stage 1 of the proposed solu-

tion to the Optimal Mesh Extraction Problem from

section 1. Since a mesh is still detectable by hu-

man eye in grayscale, the mesh problem was stated in

terms of minimizing the average grayscale intensity.

Stage 1 will build a connected binary image.

Deﬁnition 2. For a given width w and height h, a

binary image on the continuous domain Ω = [0,w] ×

[0,h] ⊂ R

is a set of black unit square pixels whose

complement in Ω consists of the white unit square pix-

els in the background. An image I is connected if the

union of its black pixels forms a connected sub-graph

in the 8-connected grid of Ω.

We have tried many thresholdings to get a con-

nected image that can be further thinned. Our micro-

scopic images have uneven illumination with darker

regions towards the image boundary. This gradual

change in intensity was not uniform in given images.

The global and local thresholding methods (Bernsen,

1986), (Niblack, 1985), (Phansalkar et al., 2011),

(Sauvola and Pietik

ainen, 2000), (Soille, 2013) often

fail to preserve continuity of edges in the noisy im-

age from Figure 1, see outputs of above binarizations

in Figure 6. The clustering-based thresholding (Otsu,

1979) couldn’t ﬁnd an optimal threshold. Figure 5

shows the Otsu thresholding with many components.

To resolve uneven illumination, we ﬁnd optimal

parameters for Otsu binarization in sub-images of size

M × M within 2584× 1936 images. Otsu binarization

chooses an optimal threshold using the histogram of

all intensities. Since some sub-images are dark, we

shift any intensity i lower than a certain bound b for

all sub-images by i 7→ (i + b)/2.

The parameters M and b are dynamically selected

for each image as follows. The original image is di-

vided into M × M sub-images whose average inten-

sities are calculated. The value of M increases from

50 to 400 in increments of 25. We choose M so that

the standard deviation of the average intensity in sub-

images is minimal. Then b is set as the average in-

tensity of the optimal sub-image. This shift implicitly

increases the optimal Otsu threshold and yields a con-

nected binary image in Figure 5 (middle).

A resulting image can have small holes as in the

middle of Figure 5, green box. Physicists advised us

that expected domains have at least 50 pixels, so we

have ﬁlled all small holes as in the right picture of

Figure 5, instead of the persistence-based hole detec-

tion (Kurlin, 2014b), (Kurlin, 2014a), (Kurlin, 2016).

The output of Stage 1 in Figure 5 (right) has few small

black components, which are removed in sections 4-5.

Polygonal Meshes of Highly Noisy Images based on a New Symmetric Thinning Algorithm with Theoretical Guarantees

139

Figure 5: Left: original Otsu thresholding (Otsu, 1979) breaks connectivity in Figure 1. Middle: the new adaptive threshold-

ing outputs a better connected image, compare the bottom parts. Right: output of Stage 1 after ﬁlling small holes.

Figure 6: Outputs of several local thresholding methods on the input image in Figure 1. 1st: (Bernsen, 1986), 2nd: (Niblack,

1985), 3rd: (Phansalkar et al., 2011), 4th: (Soille, 2013) and 5th: (Sauvola and Pietik

ainen, 2000).

4 THE NEW EROSION WITH

GUARANTEES

The output of Stage 1 is a binary image that covers

polygonal lines of a required vortex mesh. The next

stage is to thin (or erode) the binary image to preserve

its homotopy type without making any holes.

Deﬁnition 3. A subimage of a binary image I con-

sisting of black pixels on a continuous domain Ω is a

subset J ⊂ I of black pixels considered on the same

domain Ω of I. A homotopy (or a deformation re-

traction) f : I → J is a continuous family of functions

: Ω → Ω, t ∈ [0,1], such that f

= id

Ω

, f

= f and

all f

keep the points of J ﬁxed.

Figure 7: A homotopy

of an image I to its

subimage J is a contin-

uous deformation.

Figure 8: Examples of Erosion 2

when we remove a pixel p with

two changes of colors in neigh-

bors from Deﬁnition 4.

A homotopy can be visualized as a sequence of

pixel removals that preserve the topological shape

of a binary image. A removal of a pixel with only

one neighbor (in the 3 × 3 neighborhood) is a homo-

topy by Deﬁnition 3. A removal of a pixel whose

two neighbors become disconnected (as formalized in

Deﬁnition 4) cannot be realized by a continuous fam-

ily of deformations, so isn’t a homotopy.

Deﬁnition 4. (neighbors) Any non-boundary black

(foreground) pixel p has 8 neighbors in the 3 × 3-

neighborhood. Any white (background) pixel has only

4 neighbors. When we walk around the pixel p (say,

clockwisely starting from any neighbor), we count the

number of color changes in the 8 neighbors of p from

black to white and from white to black.

Section 5 describes the new algorithm that sym-

metrically removes pixels starting from an external

layer and outputs a better skeleton with a lower av-

erage intensity. We now discuss a simpler thinning

when pixels are removed one by one being processed

from top to bottom and from left to right.

Erosion 1. Remove a black pixel that has at most 1

neighbor, e.g. any isolated pixel is removed as an out-

lier. We remove pixels with 1 neighbor as in Figure 7

since a vortex mesh has no degree 1 vertices.

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Erosion 2. Remove a black pixel p if colors change

twice around p, see Deﬁnition 4 and Figure 8.

Erosion 3. Remove a black pixel p if p has 4

changes of colors, but all its neighbors are pairwise

adjacent to each other in the 8-connected grid, see

Figure 9.

Figure 9: Erosion 3 removes

a pixel p with four changes

of colors. in neighbors.

Figure 10: Erosion 4 re-

moves a pixel p when all

black neighbors are con-

nected.

Erosion 4. We remove a black pixel p if p wasn’t

removed by any of Erosions 1-3, p has at least two

white neighbors and all its black neighbors form a

connected sub-graph (in the 8-connected grid), see

Figure 10. This erodes busy junctions as in Fig-

ure 11 (right), which were obstacles for homotopy-

preserving thinning (Saha et al., 2016). Erosion 3 was

separated from Erosion 4 to speed up the algorithm.

The erosion algorithm scans a binary image row by

row, processing black pixels in each row from left to

right. Black pixels are stored in a map-structure for

quick access by a pixel position. For any black pixel,

we try Erosions 1,2,3,4 in this order until all remain-

ing pixels cannot be removed by Erosions 1-4.

New Erosion 4 resolves busy junctions to get a 1-

pixel wide skeleton (see Theorem 6) and keeps the ho-

motopy type of a binary image (modulo 1-pixel holes)

due to Theorem 8. Theorems 6 and 8 are two ‘edges

of the thinning sword’ justifying wider applications of

the thinning based on Erosions 1-4.

Deﬁnition 5. A binary image I consisting of black

pixels is called a 1-pixel wide skeleton if any 2 × 2

block contains at most 3 black pixels of I except the

case in Figure 11 (left), where each of 4 black pixels

has a diagonal neighbor.

Figure 11: Left: the 2 × 2 block cannot be removed because

of 4 diagonal neighbors. Right: Erosion 4 removes 4 pixels

p in the previously unresolved junction (Saha et al., 2016).

The diagonal neighbors in Figure 11(left) might

be kept by Erosions 1-4 if each of 4 black corner lines

contains at least one black pixel. Figure 11(right)

shows the power of new Erosion 4, which can pen-

etrate big black blocks by creating only 1-pixel holes

that can be later ﬁlled if needed. Our microscopic im-

ages had no superﬂuous 1-pixel holes from Erosion 4.

Theorem 6. For any binary image I with ﬁxed bound-

ary black (background) pixels, the algorithm based on

Erosions 1-4 outputs a 1-pixel wide skeleton.

Proof of Theorem 6. If I isn’t a 1-pixel wide skeleton,

we can ﬁnd a 2 × 2 block B of black pixels adjacent

(by a side) to at least one white pixel q. The side

neighbor p ∈ B of q cannot be removed by Erosion 4

only if its side neighbor s is white and corner neighbor

r is black as in the left picture of Figure 11 so that r

becomes disconnected from other neighbors of p if

q is eroded. The other 3 pixels in B cannot be eroded

only if they have diagonal neighbors in Figure 11.

Deﬁnition 7. A 1-pixel hole in a binary image I of

black pixels is a white pixel sharing 4 sides with black

pixels, see the central white pixel in the last picture of

Figure 10. A connected component C of an image I is

trivial if C has no holes larger than 1-pixel holes.

Theorem 8. For any image I without trivial compo-

nents and 1-pixel holes, the erosion after ﬁlling 1-

pixel holes outputs a subimage homotopic to I.

Proof of Theorem 8. If a central black pixel p shares a

side with a white pixel, the removal of p is realized by

a homotopy in the sense of Deﬁnition 3, see Figure 7.

If p is eroded and the 4 neighbors sharing sides with

p are black, Erosion 4 was applied and p becomes a

1-pixel hole by Deﬁnition 7. Erosions 1-4 preserve

the local connectivity, i.e. the neighbors of p remain

connected in the 8-connected grid.

Each of the 4 side neighbors q of p in Figure 10

cannot be later eroded, otherwise the two corner black

pixels (sharing sides with p and corners with q) be-

come disconnected in the 3 × 3 neighborhood of q.

Hence p will keep all its 4 side neighbors and will re-

main a 1-pixel hole that will not affect the homotopy

type (after ﬁlling 1-pixel holes in J).

Figure 12: Ablation study: past thinning vs symmetrical.

Left: too straight lines after erosion in section 4. Right:

more accurate polygonal lines after symmetric thinning.

Polygonal Meshes of Highly Noisy Images based on a New Symmetric Thinning Algorithm with Theoretical Guarantees

141

5 SYMMETRIC THINNING AND

A MESH

Figure 13: Errors of past methods in subareas of Figure 1.

1st: extra edges (Zhang and Suen, 1984), 2nd: extra de-

gree 1 vertex (Guo and Hall, 1989), 3rd: lost connec-

tion (Arganda-Carreras et al., 2010), 4th: many superﬂuous

edges (Delgado-Friedrichs et al., 2014). Right: our mesh.

We improve the erosion algorithm from section 4 by

processing pixels not row by row, but in a data-driven

recursive way starting from the external layer.

Deﬁnition 9. For any binary image I ⊂ Ω consisting

of black pixels on a white background Ω − I, the ex-

ternal layer of I is the subset L ⊂ I of all black pixels

adjacent to a white pixel in the 4-connected grid.

Figure 14: The red external layer from Def. 9 is removed.

A continuous deformation of the image domain Ω

is any bijection Ω → Ω such that any adjacent pixels

(in the 8-connected grid of Ω) map to adjacent pixels.

Corollary 10. For any binary image I ⊂ Ω, the sym-

metric thinning algorithm outputs a skeleton S that is

invariant under any continuous deformation of Ω and

satisﬁes the conclusions of Theorems 6 and 8. Any

pixel of S has only 2 or 3 neighboring pixels in S.

The symmetric skeletons obtained from all our

microscopic images had no black 2× 2 blocks, which

were allowed in a 1-pixel wide skeleton from Deﬁni-

tion 5. To guarantee that every skeleton has vertices

of only degrees 2 or 3, the ﬁnal step of Stage 2 after

symmetric thinning is to resolve few cases of degree 4

vertices in Figure 15, which preserve connectivity in

5 × 5 or 6 × 6 neighborhood.

Proof of Corollary 10. The only difference between

the erosion algorithm in section 4 and the new sym-

metric thinning in Figure 14 is the order in which pix-

els are processed. Since Deﬁnition 9 uses adjacency

of pixels in the 4-connected grid, the output remains

the same under any continuous deformation of Ω. The

conclusion about 2 or 3 neighbors follows from re-

solving all degree 4 cases in Figure 15.

The symmetric skeleton is better centered along

darkest pixels, compare the ﬁrst two pictures in Fig-

ure 13. The average intensity (over more than 100

images) of the symmetrically thinned skeleton has

dropped from 142.95 to 122.77 in comparison with

section 4. The conversion of a pixel-based skele-

ton into a polygonal mesh uses the 8-connected

grid. To minimize the number of vertices accord-

ing to the problem in section 1, ﬁnal Stage 3 ap-

plies the Douglas-Peucker straightening (Douglas and

Peucker, 1973) of polygonal edges.

6 COMPARISONS WITH PAST

METHODS

We chose algorithms (Zhang and Suen, 1984), (Guo

and Hall, 1989), (Arganda-Carreras et al., 2010), be-

cause they are offered by OpenCV and ImageJ, while

(Delgado-Friedrichs et al., 2014), (Kamani et al.,

2018) are most recent since 2015. The methods

(Zhang and Suen, 1984), (Guo and Hall, 1989) are

implemented in OpenCV and require a binary image

as input, which can be produced by the original Otsu

thresholding (also by OpenCV). The ablation study in

Table 1 shows these thinning algorithms are substan-

tially improved by our adaptive thresholding in sec-

tion 3.

Figure 16 shows 6 example outputs on the im-

age from Figure 1. The authors of (Arganda-Carreras

et al., 2010) kindly distributed their software, which

produced the 3rd skeleton in Figure 16. The 4th

skeleton in Figure 16 was obtained by the code from

the authors of the Morse-Smale skeleton (Delgado-

Friedrichs et al., 2014). The authors of (Kamani et al.,

2018) had provided their ﬂux-based skeletonization

code whose output is the 5th picture of Figure 16.

The ﬁnal picture of Figure 16 shows our mesh af-

ter Stage 3. In Figure 16 the past 5 methods either

lost connectivity (ImageJ) or kept too many small

loops (Delgado-Friedrichs et al., 2014) or superﬂuous

hanging edges (Zhang and Suen, 1984), (Arganda-

Carreras et al., 2010). The connectivity guarantees in

(Guo and Hall, 1989, Proposition A.3) are too weak

for microscopic images that need correct homotopy

types for 1-pixel wide skeletons in Theorems 6, 8.

Table 1 shows the averages over more than 100

images. The Optimal Mesh Problem stated in sec-

tion 1 required to minimize the average grayscale in-

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Figure 15: Resolving 2 × 2 black blocks and black pixels with 4 neighbors while preserving the local connectivity.

Figure 16: Output skeletons from past algorithms on the input image in Figure 1 from left to right: (Zhang and Suen,

1984), (Guo and Hall, 1989), (Arganda-Carreras et al., 2010), (Delgado-Friedrichs et al., 2014), (Kamani et al., 2018), ours.

tensity over the mesh, because polygonal edges are

expected to pass through darkest pixels. In addition

to the average intensity, Table 1 includes the number

of components, because a ﬁnal skeleton should have

one huge component and very few small ones.

The last column in Table 1 counts the total num-

ber of vertices including degree 2 vertices (bends

of polygonal lines). The straightening of polygonal

edges in Stage 3 substantially reduces these degree 2

vertices while the average intensity is only slightly in-

creased to 130.2 and remained smaller than for all

other past algorithms. If we remove small compo-

nents in outputs of (Zhang and Suen, 1984), (Guo and

Hall, 1989) so that their average numbers 177 and 112

of components drop, there will be hundreds (of 433

and 954 on average) degree 1 vertices in Figure 16.

In addition to Table 1, starting from 10 our outputs

on real images, we have generated 625 times more

synthetic ground truth images as follows:

• increased the thickness of skeletons by scaling each

black pixel by a factor k = 1,2,3,4,5.

• applied Gaussian blur for kernel sizes 3,5,7,9,11;

• added Gaussian noise with variances 1,2, 3, 4, 5;

• made 5%,10%,15%,20%,25% random pixels black.

Since (Arganda-Carreras et al., 2010) and

(Delgado-Friedrichs et al., 2014) don’t run for many

images, Table 2 compares our meshes with the

OpenCV algorithms (Zhang and Suen, 1984), (Guo

and Hall, 1989) over 6250 synthetic images.

Pixels in the ground truth and output skeletons are

considered close if they are in 3 × 3 neighborhoods of

each other. The true positives (TP or hits) are pixels in

the output close to ground truth. False positives (TP

or false alarms) are the pixels in the output that are

Polygonal Meshes of Highly Noisy Images based on a New Symmetric Thinning Algorithm with Theoretical Guarantees

143

Table 1: Objective comparison of 6 thinning algorithms (with variations) on more than 100 highly noisy real images.

algorithms / quality measures average intensity connected components deg 1 vertices all vertices

Ground truth 123.7 3 0 819

(Zhang and Suen, 1984): original

Otsu thresholding

135.3 956 1,577 13,927

(Zhang and Suen, 1984): threshold-

ing from section 3

137.6 177 433 30,525

(Guo and Hall, 1989): original Otsu

thresholding

135.7 966 2,417 12,129

(Guo and Hall, 1989): thresholding

from section 3

130.8 112 954 29,927

(Kamani et al., 2018): original Otsu

thresholding

156.7 5 2,653 25,364

(Kamani et al., 2018): thresholding

from section 3

161.3 5 451 61,355

ImageJ (Arganda-Carreras et al.,

2010)

139.2 52 268 30,462

Morse-Smale (Delgado-Friedrichs

et al., 2014)

162.4 3 34 27,612

section 4: without symmetry 145.4 3 0 28,563

section 5: with symmetry 126.7 3 0 23,449

our output after straightening 130.2 3 0 824

Table 2: Accuracy over 6250 synthetic ground truths.

algorithms precision recall F

score

Zhang-Suen

(Zhang and

Suen, 1984)

0.068 0.862 0.126

Guo-Hall (Guo

and Hall, 1989)

0.072 0.859 0.133

our method 0.933 0.963 0.948

not close to the ground truth. The false negatives (FN

or misses) are the pixels in the ground truth that are

not close to the output. The accuracy measures in Ta-

ble 2 are precision =

T P

T P + FP

, recall =

T P

T P + FN

and F

score = 2

precision × recall

precision + recall

The complexity of the algorithm is linear in the

number of pixels, because all steps in sections 3-5 in-

volve small neighborhoods of pixels. The time is less

than 1 sec per image on a laptop with 16GB RAM.

7 DISCUSSION AND

CONCLUSIONS

According to Table 1, the Morse-Smale complex

(Delgado-Friedrichs et al., 2014) is closest to our

vortex mesh in the total number of components and

degree 1 vertices, but has the worst average inten-

sity. The thinning algorithms (Zhang and Suen,

1984), (Guo and Hall, 1989) offered by OpenCV

produce well-centered skeletons (close to our vortex

meshes), but contain too many small components and

degree 1 vertices that are not ﬁltered out. The new

thresholding in section 3 makes these skeletons bet-

ter connected. The performance of AnalyzeSkeleton

(Arganda-Carreras et al., 2010) is roughly between

the Morse-Smale and OpenCV skeletons.

The deep learning papers (Asghari and Jalali,

2015), (Shen et al., 2017) provided their implemen-

tations and trained models. Since our images were

not in their training data, these algorithms performed

poorly on microscopic images because of the inten-

sity variations and noise. An extended version will

include these outputs on images similar to Figure 1.

• The new erosion algorithm in section 4 guarantees

a 1-pixel wide skeleton with a correct homotopy type

(stronger than a graph connectivity) by Theorems 6

and 8. The erosion rules are simpler than those in

(Zhang and Suen, 1984) and (Guo and Hall, 1989).

• The ﬁrst 6 rows of Table 1 show that the new adap-

tive thresholding in section 3 improves the continuity

of past skeletons by reducing degree 1 vertices.

• The new symmetric thinning in section 5 outper-

forms the past thinning by reducing the average in-

tensity over the mesh to 126.7, which goes up only

slightly after straightening polygonal lines.

• Tables 1, 2 and Figures 17, 18 show that our mesh

is robust under high noise and has better accuracy in

comparison with several past algorithms on more than

100 real and 6250 ground truth images.

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Figure 17: Our ﬁnal polygonal meshes on more challenging microscopic images with high noise.

Figure 18: 1st: one of 6250 synthetic images with noise over a ground truth skeleton (easy to skeletonize for humans), 2nd:

(Zhang and Suen, 1984), 3rd: (Guo and Hall, 1989), 4th: our output. True Positives are black. False Positives are red. False

Negatives are blue. The OpenCV algorithms (Zhang and Suen, 1984), (Guo and Hall, 1989) output too many false positives.

Polygonal Meshes of Highly Noisy Images based on a New Symmetric Thinning Algorithm with Theoretical Guarantees

145

REFERENCES

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