Improved Dijkstra''s algorithm on the image segmentation problem

Research

IMPROVED DIJKSTRA'S ALGORITHM ON THE IMAGE
SEGMENTATION PROBLEM
Nguyen Nam Phuc1*, Le Tien Hung2, Nguyen Quoc Trung3, Ha Huu Huy4
Abstract: This paper presents the improved multiple-source Dijkstra algorithm
as an efficient solution to the multiple objects, interactive image segmentation
problem, which can be applied for precomputing process in many tasks of image
processing. Given an input image, we can build a sparse undirected graph of pixels
and pick some pixels with predefined labels as sources. By running the Dijkstra’s
algorithm on the whole graph, the closest source with the corresponding label from
all pixels can be determined efficiently in terms of both time performance and
accuracy. The main contributions include the idea of parallel precomputing the
local shortest distances in a neighborhood of each pixel to build an undirected
graph and combining label-correcting methods with the Dijkstra’s algorithm to
reduce the running time. Our experimental results on real-world images
demonstrate significant improvement to the existing image segmentation methods in
terms of both time performance and accuracy.
Keywords: Shortest distance; Dijkstra’s algorithm; Graph; Parallel; Image segmentation; Label correcting.

1. INTRODUCTION
In recent years, the task of image segmentation has attracted a much attention in
a variety of interactive applications [1]. Roughly speaking, the ultimate goal of the
task is to extract objects from images with as few user interactions as possible.
Many image segmentation algorithms have been proposed in the literature and
could be classified into different groups as edge detection [2], thresholding [3] and
clustering based [4] segmentations. However, designing an automated image
segmentation technique which can be applied fully in real-world applications is
still a challenging problem. The main reason is about the large requirement of
input information from user to define the ground truth of objects, especially when
the input images contain multiple objects with different colors or when users need

to change the input information frequently to archive satisfactory results. And for
the iris/vessel detection problems [5,6], it is difficult to enhance multiscale vessellike structures, especially on the images with low contrast or high homogeneity of
illumination conditions.
In contrast, the graph based approaches [7] have been broadly used in largescale image database applications and they have become a major trend in image
segmentation. Greig et al. [8] evaluated two-colour or binary images by using
Ford-Fulkerson algorithm for finding the maximum flow in a capacitated network.
Felzenszwalb determined the similarity of image regions by measuring the internal
variation between neighboring pixels [9]. Recently, Wei et al. [10] presented a
multi-source variant of the Dijkstra's algorithm which allows us to perform
interactive segmentation. User can predefine labels for a small number of pixels
(seeds) and then the closest seed from each unlabeled pixel is determined in an
accurately way (see Fig.1). Their computational approach is promising since it
requires the initiating settings from user only once.

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(a)

(b)

Figure 1. Seeds 1 and 2 represent two different labels 1 and 2 (a). By finding
the closest seeds, the whole pixels in the image can be properly marked (b). Figure
is borrowed from [10].
All of the existing graph based algorithms use only single thread to check the local
connectivity of the embedded graph before each propagation of the Dijkstra’s algorithm.

And because the Dijkstra’s algorithm performs at most one iteration per node, its required
extra overhead (e.g., to extract the node with minimum label) depends on the total number
of nodes in the whole graph. However, with the rapid development of scanning
technology, the input images for segmentation could have very large size nowadays. It
means the computational cost is expensive for the classical Dijkstra's algorithm and it may
not be executed interactively. The speed time of the Dijkstra's algorithm is more concerned
in the cases when segmentation must be performed multi time on sets of image databases
for analyzing purposes.
In this paper, we focus on improving the Dijkstra’s algorithm in term of time
performance. Our framework is based on the idea of partitioning an image into

regions by using multiple-source Dijkstra's algorithm for label checking, such that
all pixels in each region have the same label. The complete framework consists of
two stages. In the first stage, the input image is mapped into a closed undirected
graph where its local connectivity is precomputed parallelly. It allows us to break
down one of the most expensive part of the Dijkstra’s algorithm into sub-problems
which can be easily solved in multi-threading mode.
In the second stage, there are some pixel with individual label mapped to will be
set as seeds for the Dijkstra's algorithm. In each iteration, nodes with new distance
information are inserted into or extracted from a set/list using incremental sorting
operations. Once the algorithm ruminates, the label of all unprocessed pixels is
changed to the label of their closest seed by computing their shortest distances from
the seeds. To avoid using expensive internal sorting in linked set/list as common and
reduce the extra overhead, we adapted the label correcting strategies on priority
queue. Thanks for that the propagating speed is boosted up significantly and our
framework can be applied in interactive applications. We have evaluated our
framework on real-world images and observed promising results.
The rest of the paper is organized as follows. Section 2 reviews related work on
image segmentation and the Dijkstra's algorithm. Section 3 explains about the
graph generation and how to efficiently perform the Dijkstra's algorithm on it with

the label correcting strategies applied. Section 4 contains our experimental results
and discusses the merits and limitations of our work. Finally, Section 5 concludes
the paper.
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2. RELATED WORKS
2.1. Graph-based segmentation
Image segmentation is a wide range topic. Therefore, in this section we will
briefly discuss only some of the classical interactive and graph-based methods in
recent years.
For the graph-based algorithms, the image segmentation is treated as a graph
partitioning problem. In such approaches, a graph G=(V, E) needs to be prebuilt
where each node vi  V of the graph corresponds to a pixel pi of the input image .
Certain nodes vi, vj  V are connected by edges eij  E whose weights measure the
distance between the corresponding pixels pi, pj   in some feature space. There
are several weighting functions can be found in [1,11]. In our work, an exponential
weighting function is introduced for gray images as from [11]:
( Ii  I j )2

wij  e



(1)

1

where Ii indicates the value of intensity at pixel pi   and  is the sensitivity
coefficient. When the input image is given in RGB mode, Eq. (1) is modified as
below with Ci is the color vector at pi:
( Ci C j )( Ci C j )T

wij  e



1

(2)

In the graph-based approaches, each component (or region) in a segment S
corresponds to a connected subgraph G' = (V; E'), where E'  E. This means any
segmentation is induced by a subset E'  E. The quality of a segmentation can be
measured in different ways, but in general we prefer all the elements to be similar
in the same component to be dissimilar in different components. In other words,
the edges between two vertices should have relatively low weights in the same
component and higher weights in different components.
The earliest graph-based algorithms use fixed thresholds plus local measures for
segmenting. Zahn [13] proposed an algorithm based on the minimum spanning tree
of the graph. This algorithm has been extended for point clustering with the
weights are computed based on distances between points. Another algorithm
proposed in [14] uses a measure based only on the nearest neighbors of each node
in graph to decide which edges to break in partitioning. However, the local
variability around nodes alone may not be enough to get a reasonable measure of
image variability over arbitrarily large regions of the input image. In the graph cuts

based algorithms [1], the chosen foreground/background seeds are treated as
sources for max-flow/min-cut operations to returns a set of edges with minimum
total weight as boundary. One of the main issues in the graph cuts is that the
boundary may take a shortcut over a protruded section of the object in an attempt
to minimize boundary length. Moreover, the segmentation results are strongly
sensitive to noise and becomes fragmented on images with low quality. In the
random walks based algorithms [15], the weight of an edge eij was treated as the
probability of a particle at node vi traveling to its neighboring node vj. Given the

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seeds, one may compare the probability that a particle initiating at any pixel travels
first to the foreground or background seeds and then assign that pixel to the
corresponding label. A common problem of random walks based algorithms is the
tendency to produce segmentations that are overly likely to provide balance
between the foreground and background, which may make images hard to be
properly segmented.
2.2. Multi-source Dijkstra's algorithm
The graph based approaches naturally link the segmentation problem on an
image and the shortest path problem on a graph, since the label of each pixel pi is
set depends on the shortest distances between sources and the corresponding node
vi. Although the numerical techniques [16,17] have advantage in fast computing
the shortest distances, their accuracy and stability are still the common issues we
need to concerns about for segmentation of small or non-trivial shapes. In contrast,
the computational solutions, includes the Dijkstra's algorithm [18], are more

reliable and they are widely used for especially in iris/vessels detecting
applications. Recently, Wei et al. [10] adopted a multi-source variant of the
Dijkstra's algorithm to change the label on pixels by running once. Their idea is to
find the closest seed for each pixel but not the shortest distance from all of the
seeds. Suppose the certain weighted graph G = (V, E) is given and there exists two
kinds of seeds S1 = {f1, f2,…, fm} and S2 = {b1, b2,…, bn}, where fi denotes the
foreground seeds and bi is the marked background seeds. Firstly, all of the seeds
are initialized with their distance to zero, and labeled with 1 and 2 respectively
before putting them into a list Q. During the Dijkstra’s expansion, the pixel’s label
is marked as same as its closest seed’s label. Thus, the foreground can be finally
segmented from the background when all pixels in the image are marked with their
certain labels. Nowadays, with the rapid development of modern image
capture/scanning technology, the input images for segmentation are often very
large, so the embedded graphs could have millions of nodes or more. It means the
Dijkstra's algorithm could be the bottleneck of the partitioning process and hard to
be used in real-time applications. In addition, for the analysis purposes, the
partitioning process needs to be carried out multiple times until the results match
the user's requirements, so designing graphs with simple data structures for storing
is a good way to reduce computing resource and running time.
3. THE PROPOSED METHOD
3.1. Efficient graph construction
The graph construction is required only once before running the Dijkstra's
algorithm, then all information about the local connectivity is stored in
conventional incidence lists. Each node stores its incident edges and each edge
stores two incident nodes plus its length.
struct Node
{
unsigned int
id;
vector<unsigned int>

id_incident_edges;
}
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Research

struct Edge
{
unsigned int id_v1, id_v2; //index of the related start and end vertices
double length;
//edge's length
}
In graph based algorithms, the 4-connected and 8-connected neighborhood
structures are usually used to transfer the local connectivity from the input image
into its corresponding graph. Thus, only weights of the edges that connect two
adjacent nodes can be computed using Eq. (1) or (2). Additionally, we can increase
area of the local neighborhood to speed up the convergence of the Dijkstra's
algorithm, the trade-off is that some disconnected similar objects could be linked
because of the wrong labels marked on the pixels between these objects.
Our construction algorithm is shown in Algorithm 1. For each vertex vi  V, we
compute distances from vi in a local neighborhood (vi). Then all these distances
are taken to form the edges from vi. Computing the distances is the bottleneck of
the entire graph construction algorithm due to the expensive math functions. In our
implementation, the distances, i.e. the edges' length, can be computed in parallel on
the CPU/GPU: the graph data structure is accessible to all threads in the read-only
manner, and each thread maintains its own data. To avoid the written conflicts, the
thread for node vi produces directed edges originated from vi. The gain of the

constructing time strongly depends on the number of threads on CPU as it is shown
in Table 1.
Table 1. The total time for edges’ construction on images with different size.
Tsingle_thread is the time required in the classical Dijkstra’s algorithm which runs on
single thread. Ttwo_threads and Tfour_threads are our constructing time for edges on two
threads and four threads respectively.
Image size
640*480

Tsingle_thread (ms)
40

Ttwo_threads (ms)
17

Tfour_threads (ms)
16

1080*720

92
185
255
2428

54
123
173
1515

36
95
130
998

1920*1080
2560*1080
3000*4000

Algorithm 1: Parallel Graph Constructing Algorithm
Input: Image  .
Output: Connected graph G = (V, E).
1. initialize all nodes vi in V and edges ei in E
2. parallel for each pixel pi in 
//each loop runs on on separate
thread
3.
for each pixel pj in the neighborhood (pi) of pi
4.
Compute distance dij between pi and pj using Equation (1)
or (2)
5.
Update dij to edge eij
6.
end for
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7. end for
8. return graph G
3.2. Improving performance of the Dijkstra's algorithm
To compute the shortest distances from a source, say v1, to all of the other
nodes in graph G, the Dijkstra’s algorithm maintains a label vector (d1, . . . , d|V |)
and a set of nodes C, called the "candidate list", starting with d1 = 0, di = ∞ for all i
≠ 1 and C = {v1}. It iteratively processes the nodes from C and terminates only
when C is empty. Upon the termination, the shortest distance from v1 to node vi is
equal to label di. In each iteration, it takes the front node v* with the smallest label
from C to update distances to v*'s neighborhood. Different data structures of C (for
example, binary or Fibonacci heaps) can be used to determine the node with
smallest label in C. Since each node enters and exits C exactly once, the Dijkstra’s
algorithm needs |V| iterations to complete.
In our work, the label correcting methods are adopted to maintain C as a
priority queue with multiple entrances, from which nodes can enter or exit in
constant time O(1). Node insertion follows the small label to the front (SLF)
scheme [19] and node extraction follows the large label last (LLL) scheme [20]:
when a node vj enters the priority queue C, its label dj is compared with the label di
of the top node vi of C. Node vj is placed at the top of C if dj ≤ di; otherwise vj is
placed at the bottom of C. Node vi remains at the top of C while its label is less
than a threshold; otherwise it is sent to the bottom of C. The threshold is usually set
as the mean label value of all nodes in C. The Dijkstra’s algorithm performs |V|
iterations at most and requires some extra overhead to extract the node with
minimum label per iteration. In contrast, the SLF-LL methods requires more than
|V| iterations but the overhead per iteration is smaller. The performance of the
Dijkstra's algorithm and the SLF-LLL methods is evaluated on several images for
comparison. As Table 2 shows, the SLF-LLL methods can almost double the
runtime performance achieved by the Dijkstra’s algorithm for the same image.

Table 2. Performance of the Dijkstra's algorithm (TDijkstra), and our SLF/LLLenhanced method (TSLF/LLL) on same graphs.
|V|

TDijkstra

TSLF/LLL

200K

0.20 s

0.08 s

400K

0.37 s

0.18 s

548K

0.48 s

0.26 s

800K

0.70 s

0.33 s

1.28M

1.09 s

0.51 s

Algorithm 2: SLF/LLL-enhanced Dijkstra's algorithm
Input: Graph G=(V, E) with sources S={s1, s2, …, sc}.
Output: The shortest distance vector D = {d1, d2, ..., d|V|} and the label
vector L = {l1, l2, ..., l|V|}.
1. initialize D to ∞;
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Research

2. initialize L to 0;
3. initialize status vector P = {p1, p2, ..., p|V|} to false
4. initialize priority queue C as empty
5. for each source si in S
6.
initialize di to 0
7.
initialize li to i
8.
initialize pi to true
9.

add si to C
10. end for
11. initialize  to 0
12. while C is not empty
13.
while true
//extract minimum element from C using LLL policy
14.
extract minimum element vi from C
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.

if di greater than



then
size of C

add node vi to bottom of C

else
set  equal to ( - di)
set pi equal to false
break

end if
end while
for each node vj in the neighborhood (vi) of vi
if dj greater than (di+ eij) then
initialize dtemp equal to dj
set dj equal to (di+ eij)
set lj equal to li
if pj is false then
//add element to C using SLF
policy
29.
add node vj to C
30.
set pj equal to true
31.
set  equal to ( + dj)
32.
else
33.
set  equal to ( - dtemp)
34.
set  equal to ( + dj)
35.

end if
36.
end if
37.
end for
38. end while
39. return vectors D and L
4. EXPERIMENTAL RESULTS
We implemented our framework on a 64-bit PC with an Intel Xeon 4 cores CPU
and 32GB memory. The parallelized parts are implemented to run on CPU using

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multi-threading OpenMP interface. Our results are compared with [10] to illustrate
the time performance. We observed that our parallel implementation on a quadcore CPU was up to 3 times faster than when using a single-core, thus it allows
interactive manipulation on image with million pixels (see Table 3).
Table 3. Comparing results of graph based algorithms. TW is the running time of
Wei’s algorithm [10]. Our required memory is shown in the third column. The last
column reports executing time of our framework (Tours).
Image size

TW

Memor
y

Tours

640*480

0.30 s

14.7 MB

0.10 s

1080*720

0.65 s

37.3 MB

0.30 s

1920*1080

1.15 s

99.5 MB

0.57 s

2560*1080

2.60 s

132.7 MB

1.10 s

As we adopted the multi-source Dijkstra's algorithm, our segmentation results
should have similar quality in comparison with results of the method proposed in
[10]. The main advantage obtained is about the time efficiency which allows our
work be more interactive and suitable for real-time applications. As we divided the
construction of embedded graph from the main Dijkstra’s algorithm, this process
can be effectively solved on multi-cores CPUs. The precomputing process costs
less than 20% of total running time of the whole algorithm on our computer with 8
threads CPU. This ratio can be smaller if we have more thread available on CPUs,
and it means the expensive computational part is not the bottleneck of the
Dijkstra’s algorithm anymore. Thus, the main performance boost comes from the
SLF/LLL-enhanced strategy as Table 3 shows.

(a)

(b)

(c)

(d)

Figure 2. The procedure of our framework. (a) and (c) are the original RGB
and gray images from BSDS300 database. The corresponding segmentation results
are in (b, (d).
Our experiment is completed on different image datasets. Some of our segmentation
results are shown in Figures 2, 3 and 4. In the Fig. 2, there are 3 groups of pixels picked at

the background, red and yellow areas as the seeds for multiple objects segmentation. It is
noted that the user’s labeled seeds can be freely settled and the precomputing graph needs
to be processed only once, so the gain is more noticed in cases we want to modify the
previous seeds and rerun the algorithm until the results are satisfied. This is the main
advantage of our work, especially for interactive image segmentation problems.
Our proposed framework is also tested on DRIVE database images for vessels
detection problem. Fig. 3 shows the results in segmenting small vessels from the retinal

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images. Each pixel finds the nearest seed from itself, so the quality of segmentation
strongly depends on the connectivity of similar regions in the input images. It can be seen
that the small vessel pixels are segmented properly with high speed, but as part of the
trade-off, there could be some tiny areas appeared in background. For some complex
images with some physically unconnected pixels or noises, the segmentation results are
drastically influenced. Fortunately, we can change the seeds setting (see Fig. 3) or applied
some preprocessing technics to improve the quality of segmentation. In the Fig. 4, the
input image was preprocessed using noise filters from [21], and thanks to it, our
framework can produce satisfactory segmentation results.

(a)

(b)

(c)

(d)

Figure 3. The segmentation results for vessels detection problem. (a) is the
original image with 2 seeds given. The corresponding segmentation result is in (b).
With one more seed added in (c), the vessels’ background can be properly detected
(b).

(a)

(b)

Figure 4. Fast 3-labels Iris Segmentation result (b). The original image (a) is from the
present CASIA V1 iris database.

5. CONCLUSION
In this paper, we introduce an improved multi-source Dijkstra's algorithm for
image segmentation with high performance in both time and user's convenience
aspects. The first improvement is about designing an efficient way for graph
constructing, which can be carried out in a parallel fashion on CPUs. Instead of
solving the shortest distance problem multiple times, we describe this process
through a multi-source variant of the Dijkstra's algorithm and replace its
inserting/extracting policies by the SLF/LLL rules to reduce the overhead in its
propagation. Plus a comfortable seed selection for users, we show that these
improvements make our algorithm performs remarkably well on a variate range of
large images with multiple objects contained inside.

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Although our work can out-perform other graph based algorithms running on
CPUs with only a small additional memory usage, it is still challenging to detect
the tiny objects in noise images. In our algorithm, picking correct seeds is an
important task to guarantee the optimal performance, hence it is highly desirable to
pre-process the input images if they are strongly noised. For the common images
from modern scanning/capturing devices, our algorithm can efficiently produce
satisfactory segmentation results.
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TÓM TẮT
CẢI TIẾN THUẬT TOÁN DIJKSTRA ĐA NGUỒN
TRONG PHÂN VÙNG ẢNH
Bài báo này trình bày về thuật toán Dijkstra đa nguồn cải tiến nhằm giải
quyết một cách nhanh chóng, hiệu quả bài toán phân vùng ảnh có chứa
nhiều đối tượng và có thể áp dụng cho mục đích tiền tính toán trong các
nhiệm vụ xử lý ảnh. Từ bức ảnh đầu vào, nhóm tác giả đã xây dựng một đồ
thị đóng vô hướng từ các điểm ảnh và lựa chọn một số điểm ảnh nguồn được
gán nhãn cho trước. Thông qua thuật toán Dijkstra trên đồ thị, điểm nguồn
gần nhất của từng điểm ảnh cùng với nhãn tương ứng của nó sẽ được xác
định một cách hiệu quả và chính xác Trong phạm vi bài báo, tác giả đã đề
xuất một số cải tiến mới bao gồm việc song song hóa quá trình xây dựng đồ
thị từ các khoảng cách ngắn nhất cục bộ quanh từng điểm ảnh và áp dụng
các quy tắc hiệu chỉnh nhãn cho thuật toán Dijkstra trên đồ thị đó nhằm
giảm thời gian tính toán cần thiết. Kết quả thử nghiệm trên các bức ảnh thực
tế cho thấy thuật toán được đề xuất trong bài báo đã cải thiện đáng kể độ
chính xác và tốc độ tính toán trong phân vùng ảnh.
Từ khóa: Khoảng cách ngắn nhất; Thuật toán Dijkstra; Đồ thị; Tính toán song song; Phân vùng ảnh; Hiệu
chỉnh nhãn.

Received 13th March 2019
Revised 06th May 2019

Accepted 15th May 2019
Author affiliations:
1
Information Technology Department, Ministry of Public Security, Vietnam;
2
Military Technical Academy, Vietnam;
3
Hanoi University of Science and Technology, Vietnam;
4
Academy of Military science and technology, Vietnam.
* Corresponding author: .

Journal of Military Science and Technology, Special Issue, No.60A, 05 - 2019

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