Coarse Image Edge Detection

using Self-adjusting Resistive-fuse Networks

Haichao Liang and Takashi Morie

Graduate School of Life Science and Systems Engineering, Kyushu Institute of Technology

Kitakyushu, Fukuoka, 808-0196, Japan

Abstract. We propose a model of coarse edge detection using self-adjusting

resistive-fuse networks. The resistive-fuse network model is known as a non-

linear image processing model, which can detect coarse edges from images by

smoothing noise and small regions. However, this model is hardly used in real

environment because of the sensitive dependence on the parameters and the com-

plexity of the annealing process. In this paper, we ﬁrst introduce self-adjusting

parameters to reduce the number of parameters to be controlled. Then, we pro-

pose a heating-and-cooling sequence for fast and robust edge detection. The pro-

posed model can detect edges more correctly than the original one, even if an

input image includes a gradation.

1 Introduction

Edge detection is a common task in image processing. It can not only serve for pattern

recognition, such as image retrieval [1] and text extraction [2], but can also reduce the

image information to accelerate conventional image processing [3,4].

In some applications, coarse edge detection is desired to reduce the complexity of

edge images. The resistive-fuse network model provides such a function [5]. This model

smoothes image noise and small regions, and detects discontinuities as coarse edges by

using iterative operations to minimize an energy function. This model can converge

faster than stochastic methods [6] or coupled MRF models [7]. Therefore, we have

used this model for coarse edge detection and have implemented it in an FPGA [8].

However, the resistive-fuse network model has two limitations: sensitive depen-

dence on the parameters, and complexity of the annealing process, which is necessary

to avoid local minimum. These two limitations greatly restrict the use of this model in

real applications.

In this paper, we propose a new resistive-fuse network model. We ﬁrst introduce a

self-adjusting element to reduce the number of parameters to be controlled. Then, we

propose a new annealing sequence for fast and robust edge detection.

2 Original Resistive-fuse Network Model

Resistive-fuse networks (hereafter, RFN) are used to implement an image reconstruc-

tion model, which performs energy minimization for coarse edge detection [5]. The

energy function is given by

Liang H. and Morie T.

Coarse Image Edge Detection using Self-adjusting Resistive-fuse Networks.

DOI: 10.5220/0003015600430052

In Proceedings of the 10th International Workshop on Pattern Recognition in Information Systems (ICEIS 2010), page

ISBN: 978-989-8425-14-0

E(f) = σ

− d

)

+ λ

− f

i+1

)

(1 − l

) + α

, (1)

where f

, d

and l

are the values of the pixel state f, input data d and binary line

processes l at location i, respectively, and σ, λ and α are free parameters. The ﬁrst

term forces the pixel state f to be close to the input data d, while the second term

smoothes that. The third term, line processes l, functions as follows.

(

1, λ(f

− f

i+1

)

> α,

0, otherwise.

(2)

So far, many methods have been proposed to solve the local minimum problem in an

MRF model, such as stochastic methods or using coupled MRF model. However, a con-

tinuation method, such as the graduated non-convexity (GNC) algorithm [9], can make

the model converge fast and is therefore suitable for real-time image processing. Such

an method eliminates the line processes l, and gradually changes the energy function

from a convex shape to the ﬁnal desirable one.

RFN circuit implements the model using the continuation method. The original

RFN circuit is composed of voltage sources, linear resistances and resistive-fuse ele-

ments, as shown in Fig. 1 (a). Each node corresponds to a pixel state f

i,j

, when the

value of each voltage source represents the image input d

i,j

at pixel (i , j). Each node

and voltage source are connected with a linear resistance, whose conductance is σ. The

adjacent nodes are connected with resistive-fuse elements, whose conductance function

is given by

I = G(V ) =

λV

1 + e xp{η(V

− δ

)}

, (3)

where λ is the conductance of a resistive-fuse, η is a parameter for changing the con-

ductance function, and δ is the threshold value of the resistive-fuse elements. Equa-

tion (3) is derived by using a deterministic mean ﬁeld approximation to the MRF model

of piecewise smooth surface interpolation [10], and the parameter η in this equation

corresponds to the inverse of the temperature T for simulated annealing [6].

Fig.1. Original analog resistive-fuse network (a), and I-V characteristics (b) of a resistive-fuse

element.

The energy function of this circuit is given by

E(f) = σ

i,j

− d

i,j

)

i,j

m,n∈N

G(v)dv, (4)

where N represents the neighboring nodes of (i, j), and V = f

i,j

− f

m,n

When η is close to zero, i.e. temperature T is very high, the conductance is almost

linear, as shown in green color in Fig. 1 (b). In contrast, when η is equal to or larger than

unity, i.e. temperature T is very low, the conductance becomes an ideal resistive-fuse

characteristic, which is similar to the line process function, as shown in red color in

Fig. 1 (b).

As described above, the original RFN model uses a continuation method to avoid

the local minimum problem, by gradually changing the parameter η from near zero to

unity or larger than that. This is based on the assumption that the unique minimum of

the convex energy function is close to the global minimum of the ﬁnal function shape.

In practice, we calculate the pixel state f by using discrete-time dynamics with an

equation derived from the Kirchhoff’s Current Law (KCL). The dynamics is given by

i,j

(t + 1) = f

i,j

(t) − ξ{

m,n∈N

G(V ) + 2σ(f

i,j

− d

i,j

)}, (5)

where ξ is a step size parameter of gradient descent. The change of η is called an

annealing process, and the annealing schedule is usually as follows:

η(t + 1) = kη(t), (6)

where t is the index of iterations, and k is a real number which fulﬁlls k ∈ (1, 2] [6,9].

3 Limitations of Original RFN Model

The RFN model mainly has two limitations in actual applications, and each of them is

described in a following subsection.

3.1 Dependence on the Parameters

The original RFN model has four parameters for controlling the energy function: λ, σ,

η and δ. Each of them affects the results of edge detection. λ and σ control the strength

of smoothing, while η and δ control the range of that, i.e. the balance of smoothing and

segmentation. Therefore, it is necessary to make clear the relationships between these

parameters for the control of them.

First of all, let us focus our attention on the relationship between λ and σ. It is

obvious that the value of λ/σ determines the constraint of the input image and the

strength of smoothing. Therefore, the results of edge detection not only depend on the

threshold δ, but also sensitively depend on λ/σ, as shown in Fig. 2 (a). The larger the

ratio is, the stronger the strength of smoothing becomes.

Next, we discuss the relationship between η and δ. It is veriﬁed from Fig. 2 (b)

that the shapes of the energy function and conductance function G(·) vary according

to both η and δ. This means that the balance of smoothing and segmentation will vary

accordingly when using the same annealing schedule and different δ. This makes it

difﬁcult to select the threshold δ.

Fig.2. (a) Edge detection results using RFN model, when η(0) = 0.00055, k = 1.1 and iterations

= 80 in Eq. (6), and (b) Variations of the shape of conductance function, when η = 0.01.

3.2 Complexity of Annealing Process

We use a simple circuit as shown in Fig. 3 (a) to discuss the annealing process. In this

case, the energy function is given by

E(f) = σ(f

− d

)

G(v)dv + σ(f

i+1

− d

i+1

)

−V

G(v)dv, (7)

where V

= f

− f

i+1

When the energy function E(f ) reaches a minimum, we can obtain the following

equations:

∂E

∂f

= 2{σ(f

− d

) + G(V

)} = 0. (8)

∂E

∂f

i+1

= 2{σ(f

i+1

− d

i+1

) + G(−V

)} = 0. (9)

These two equations mean that E

reaches a minimum when the sum of the currents

at each node is zero, i.e. the circuit is in a stable state. Subtracting Eq. (9) from Eq. (8)

gives

G(V

) = σ

ddi

− V

, (10)

where V

ddi

= d

− d

i+1

. The right-hand side of Eq. (10) is therefore identical to the

current in the linear resistance between f

and d

. In Fig. 3 (b), the black lines and the

red lines represent I

(η = 1) and I

, where I

and I

denote the left-hand and

right-hand side of Eq. (10), respectively. It is clear that I

follows a linear function

of V

because V

ddi

is given by the input. Therefore, the intersection points of them

represent the stable states of the circuit.

Figure 3 (b) and Table 1 show that the number of stable states depends on the value

of V

ddi

, which is the intersection point of the red lines and the V

-axis. There is only one

Fig.3. A simple circuit (a) for discussion, and the relationship (b) between I

and V

where

= f

− f

i+1

. (c) and (d) show the deﬁnitions of A and B using relationship between I

and V

and that between P

and P

(η = 1), respectively.

Table 1. Local minimum problem concerned with v.

Value of |V

ddi

| E

min

G U Deﬁnitions

ddi

| ∈ (0, δ) single Region S Region S A = δ

1 +

2λ

, B = δ(1 +

), C = δ(1 +

2λ

)

ddi

| ∈ (δ, A) multiple Region S Region S G Global E

min

when η ≧ 1

ddi

| ∈ (A, B) multiple Region L Region S U Unique E

min

when η ≈ 0

ddi

| ∈ (B, C) multiple Region L Region L Region S the region where |V

| < δ

ddi

| > C single Region L Region L Region L the region where |V

| > δ

stable state if V

ddi

is smaller than δ or is large enough. However, there are two stable

states, which are denoted by P

and P

, if V

ddi

∈ (A, B) where A and B are deﬁned

as shown in Fig. 3 and Table 1.

Note that the series of the conductance function intersect at a point (δ, λδ/2), which

can be calculated from Eq. (3) and is shown as P

in Fig. 3 (c). Thus, A, B and C in

Table 1 are deﬁned as follows:

– If V

ddi

= A, E(P

) = E(P

– If V

ddi

= B, the red line moves across P

– If V

ddi

= C, the red line moves across (δ, λδ).

As described in Sec. 2, the annealing process in RFN model relies on the assumption

that the unique minimum of the convex energy function is close to the global minimum

of the ﬁnal desired one at each node. Here, “close” can be deﬁned as in the same region

where |V

| < δ or |V

| > δ, i.e. in the region S or region L.

It is clear from Table 1 that such assumption is not valid when V

ddi

∈ (A, B),

which we called a gray zone. Note that if |V

| > δ, the two nodes will be smoothed

from the beginning when using an annealing sequence, such as the annealing schedule

shown in Eq. (6) or a simpliﬁed three-stage sequence proposed in [8]. Because there

was no algorithm to prevent the nodes moving into the gray zone, such an annealing

sequence can not always give the global minimum solution at each node, and might

cause over-smoothing and lose some edges. Therefore, the annealing schedule should

be redesigned by considering the relationship between δ, η and λ/σ.

4 Improved RFN Model

In order to improve the RFN model, we ﬁrst introduce self-adjusting functions of λ

and η to reduce the number of parameters to be controlled. Then, we propose a new

annealing sequence for fast and robust edge detection.

4.1 Self-adjusting of λ

We ﬁrst propose a self-adjusting function of λ as shown in Eq. (11), which has the

similar effect as the annealing process.

m,n

(t + 1) =

(

m,n

(t)(1 + h

), |V | < δ,

m,n

(t)(1 − h

), otherwi se.

(11)

Here, h

is a small positive number and can be determined by the number of iterations

and the range of adjustment.

By using this self-adjusting function of λ, we can set σ and the initial value of λ

as constants, and let λ adjust itself to change the value of λ/σ according to the local

voltage difference. Thus, the manual control of λ and σ becomes unnecessary. Further-

more, this self-adjusting can also improve the robustness of the model because it can

reduce the length of gray zone by decreasing λ/σ when |V | > δ.

4.2 Self-adjusting of η

In order to keep the balance of smoothing and segmentation, we obtain Eq. (12) for the

scaling of conductance function when parameters except η and δ remain identical.

(V )|

V =kδ

: G

(V )|

V =kδ

= δ

: δ

, (12)

where k is a certain real number. Thus, we obtain the condition as follows by substitut-

ing Eq. (3) into Eq. (12).

= η

(13)

This means that η can be adjusted by δ and the desired shape of the conductance func-

tion.

Summarizing the above, the number of parameters need to be controlled can be

reduced to one, i.e. only the threshold δ, by using the self-adjusting functions of λ and

η. This result signiﬁcantly simpliﬁes the control of the parameters, and makes it easier

to select the appropriate threshold δ.

4.3 New Annealing Sequence

We propose a heating-and-cooling annealing sequence, and employ a “shaping” pro-

cess, which is named after that in the die making, to reduce the complexity and ambi-

guity of the input image for robust edge detection. The details of the proposed annealing

sequence are as follows. If the deﬁned threshold value is δ

1. “Sharpening” stage: δ = δ

/2 and η = 1.

2. “Shaping” stage: δ = δ

and η = 1.

3. “Heating” stage: δ = δ

and η = 0.1/δ

4. “Cooling” stage: δ = δ

and η = 5/δ

In the ﬁrst stage, we use a small δ to sharpen blurred boundaries. During this stage,

the interactions between two nodes where |V | > δ

/2 will be cut off and blurred bound-

aries will be sharpened by the smoothing in a small range.

In the second stage, we set δ = δ

and η = 1 to calculate the rough results which

obviously include the local minimum and small regions.

In the third stage, we use a small η, i.e. high temperature to smooth small regions

and to lead to global minimum.

In the ﬁnal stage, we use a large η, i.e. low temperature to “cool” the pixel states

down for coarse edge detection. Note that η is a little smaller than unity in the ﬁnal

stage. This is because that the binary resistive-fuse characteristic is known as lack of

robustness [11]. Therefore, we use η = 5/δ

in the ﬁnal stage, whose characteristic is

close to that when η = 1, but more convex around where |V | = δ. Therefore, it may

lead to less local minimum in the interactions with neighboring eight nodes.

In summary, we propose a heating-and-cooling annealing sequence, and employ

“sharpening” and “shaping” processes to improve the robustness of the model. Ideal an-

nealing sequences have to smooth noise and small regions at the same time, while keep-

ing the boundaries of large regions. However, simple cooling-down annealing cannot

achieve these functions due to the uncertainty of the input image, and may cause over-

smoothing. In contrast, the proposed annealing sequence can sharpen blurred bound-

aries and reduce noise in the input image before the “heating” stage for smoothing the

small regions. Therefore, the proposed annealing sequence can achieve more robust

detection than simple cooling-down sequences.

5 Simulation Results and Discussion

Figure 4 (a) shows processing results obtained by applying the self-adjusting function

of λ to the original RFN model. It is veriﬁed that these results are less sensitive to the

initial value of λ/σ compared with Fig. 2 (a). Figure 4 (b) shows the scaling of the

shape of conductance function, by keeping ηδ

constant.

We also evaluated the similarity between the results where (initial value of ) λ/ σ

is two and is four as shown in Table 2. The similarity is deﬁned as the ratio of the

count of identical edge pixels to that of all the edge pixels. In the simulation, 15 images

are selected from 4.1.01 to 4.2.07 in the USC-SIPI Image Database [12]. By comparing

3rd-column with 4th-column in Table 2, it is veriﬁed that our self-adjusting function can

signiﬁcantly weaken the dependence of the model on λ/σ, so that it can increase the

similarities of edge detection results when changing λ/σ. Furthermore, the similarities

of edge detection results come to more than 90% by additionally using our new anneal-

ing sequence as shown in the 5th-column in Table 2. Therefore, the manual control of

λ and σ is unnecessary any more.

Figure 5 shows processing results for images with gradation. The rectangular region

in the input image has a gradation of 10 levels per pixel in 256-level grayscale. These

Fig.4. (a) Edge detection results using self-adjusting of λ. The horizontal axis represents the

initial value of λ/σ. (b) Scaling of the shape of conductance function G(·), by using ηδ

= 1.

Fig.5. Edge detection results in the case of gradation. Input image (a) with a gradation, and

edge detection results using original RFN model (b), a simpliﬁed RFN model (c) in [8] using a

three-stage-cooling annealing sequence, and our proposed RFN model (d).

Table 2. Similarity Evaluation of Edge Detection Results.

Original Adjusting Proposed

model of λ only model

Lena

12 55.09% 69.13% 91.44%

10 58.14% 74.36% 93.32%

8 56.46% 77.59% 93.64%

Average

12 56.65% 69.43% 92.94%

10 57.41% 73.85% 93.61%

8 57.12% 75.67% 94.18%

results show that the original RFN model with an annealing process described in Eq. (6)

can only detect coarse edges, and the model with a simpliﬁed three-stage-cooling an-

nealing sequence [8] failed to detect correct edges in the gradation region. In contrast,

the proposed model in this paper can detect coarse edges when δ > 10, and can detect

ﬁne edges when δ ≤ 10.

Fig.6. Edge detection results using one-dimension input data with Gaussian white noise. Input

data (a), and edge detection results using original RFN model (b) and proposed model (c). Here,

red color shows the pixel state, and green color shows the ﬁrst derivation of that.

Figure 6 shows some results using input data shown in black color in Fig. 6 (a),

which is obtained by adding Gaussian white noise (sigma = 4) to the data shown in

blue color. In this simulation, λ/σ in the original RFN model and its initial value in

the proposed model were set to three. It is veriﬁed that original RFN model can detect

almost no edges because the boundaries are over-smoothed, as shown in Fig. 6 (b). In

contrast, the proposed model can control the level of edge detection, as shown in Fig. 6

(c), when λ

min

= 2 and λ

max

= 4.

6 Conclusions

We analyzed the resistive-fuse network model for coarse edge detection and clariﬁed

its limitations. First, we proposed self-adjusting functions to simplify the parameter

control. We concluded that only the threshold δ has to be controlled, and the local-

adjusting of λ improves the robustness of edge detection. Then, we proposed a new

annealing sequence for fast and robust detection. It was veriﬁed that the proposed model

can detect edges more correctly than the original one, even if an input image includes a

gradation.

Acknowledgements

This work was partly supported by fund from MEXT, Japan, via a Knowledge Cluster

Initiative Second Stage program, granted to Fukuoka Cluster for Advanced System LSI

Technology Development.

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