TEXTURED IMAGE SEGMENTATION BASED ON LOCAL

SPECTRAL HISTOGRAM AND ACTIVE CONTOUR

Xianghua Xie

Department of Computer Science, University of Wales Swansea, Swansea, U.K.

Keywords:

Active contour, Local spectral histogram, Level set method, Texture segmentation, Wasserstein distance.

Abstract:

In this paper, we propose a novel level set based active contour model to segment textured images. The pro-

posed methods is based on the assumption that local histograms of ﬁltering responses between foreground and

background regions are statistically separable. In order to be able to handle texture non-uniformities, which

often occur in real world images, we use rotation invariant ﬁltering features and local spectral histograms as

image feature to drive the snake segmentation. Automatic histogram bin size selection is carried out so that

its underlying distribution can be best represented. Experimental results on both synthetic and real data show

promising results and signiﬁcant improvements compared to direct modeling of ﬁltering responses.

1 INTRODUCTION

Active contours have been increasingly used in ana-

lyzing textured images, e.g. (Sandberg et al., 2002;

Ni et al., 2007; Houhou and Thiran, 2008; Savelonas

et al., 2008). Despite recent advances in edge based

approaches, e.g. (Paragios et al., 2004; Xie and

Mirmehdi, 2008), region based approaches have some

obvious advantages when analyzing heavily texture

images in that edge based boundary description can

easily be compromised by texture patterns. Region

based approach deforms initial contours towards the

region/object boundaries of interest by minimizing an

energy function, whose minimum ideally collocates

with those boundaries. Thus, it is vitally important

to use robust features and appropriate region indica-

tion/separation functional.

Various features have been investigated in the con-

tour segmentation framework, such as co-occurrence

matrices (Pujol and Radeva, 2004), structure ten-

sor (Rousson et al., 2004), and local binary patterns

(Savelonas et al., 2008). However, ﬁltering responses

are among the most popular approaches, e.g. (Para-

gios and Deriche, 2002; Sandberg et al., 2002; Au-

jol et al., 2003; He et al., 2004; Sagiv et al., 2006).

In (Sandberg et al., 2002) the authors decompose the

image using Gabor ﬁlters. The collected ﬁltering re-

sponses at each pixel are used to measure the differ-

ence between pixels in a piecewise constant model.

However, it largely ignores the spatial distribution

among local ﬁltering coefﬁcients and this direct com-

parison of ﬁlter responses is error prone since the re-

sponses can be misaligned due to the anisotropic na-

ture of most of the ﬁlters. Wavelet packet transform

is used in (Aujol et al., 2003) and the energy distribu-

tions in sub-bands are used to characterize textures.

One of the main difﬁculties in dealing with ﬁltering

responses is their large dimensionality. It is also chal-

lenging to handle textural variations within regions of

interest due to, for example, rotation or view point

changes, since most of the ﬁlters are orientation sen-

sitive.

Once the features are derived, one also needs to

decide how to model their distributions so that cor-

rect features are included in describing the object of

interest. In other words, this modeling provides a re-

gion indication or separation functional to drive the

active contours. Modeling based on global distribu-

tion is a popular approach. For example, in (Paragios

and Deriche, 2002; He et al., 2004) Mixture of Gaus-

sians are used to model the image features. Another

powerful approach is based on the piecewise constant

assumption (Chan and Vese, 2001). It also has been

recently adopted in texture segmentation, e.g. (Sand-

berg et al., 2002; Sagiv et al., 2006; Ni et al., 2007).

However, how to cope with texture inhomogeneity is

a major challenge.

217

Xie X. (2009).

TEXTURED IMAGE SEGMENTATION BASED ON LOCAL SPECTRAL HISTOGRAM AND ACTIVE CONTOUR.

In Proceedings of the Fourth International Conference on Computer Vision Theory and Applications, pages 217-225

DOI: 10.5220/0001805502170225

Copyright

c

SciTePress

2 PROPOSED APPROACH

In this paper, we propose a novel region based ac-

tive contour model, which is based on the assump-

tion that local histograms of ﬁltering responses be-

tween object of interest and background regions are

statistically separable. Brieﬂy, we ﬁrst apply a bank

of ﬁlters to the image, from which we have a set of

ﬁlter responses at different scales and orientations.

These responses are then grouped and condensed so

that it can handle textural non-uniformity which may

occur in real world images. Reduced, invariant fea-

tures are thus obtained. This process also effectively

decreases the dimensionality of ﬁlter feature space,

which is beneﬁcial for single image segmentation.

We then collect local distributions of these features

at each pixels, known as local spectral histograms.

These local histograms contains not only directly ﬁl-

tering responses but also their spatial distributions in

their local neighborhoods. The optimal bin size for

these histograms are obtained by minimizing a mean

integrated square error based cost function. An en-

ergy minimization problem is thus formulated by ﬁt-

ting two spectral histograms, one of which is used to

approximate the foreground region and the other for

the background. We will show that this approach is

effective to handle texture inhomogeneity, compared

to, for example, direct modeling of ﬁltering responses

(Sandberg et al., 2002) or local intensity distributions

(Ni et al., 2007).

Next, Section 2.1 describes the ﬁlter bank and ro-

tation invariant feature selection. Local spectral his-

togram extraction is presented in Section 2.2 and au-

tomatic optimal histogram bin size computation is

given in Section 2.3. Finally, Section 2.4 introduces

the level set based snake model using these invariant

features for image segmentation.

2.1 Filters and Feature Selection

Texture provides important information for recogni-

tion and interpolation. Numerous techniques have

been reported in the literature to carry out texture

analysis. They can be generally categorized in four

ways: statistical approaches which measure the spa-

tial distribution of pixel values, structural approaches

that are based on analyzing texture primitives and the

spatial arrangement of these primitives, ﬁlter based

approaches which analyze local pixel dependencies

using a bank of ﬁlters, and model based approaches

which often use derived model parameters as texture

features. Filter bank based approaches have been pop-

ular since they can analyze textures in arbitrary orien-

tations and scales and have been strongly motivated

Figure 1: The ﬁlter bank consists 38 ﬁlters in total, which

include one Gaussian ﬁlter, one Laplacian of Gaussian ﬁl-

ter, and 36 edge and bar ﬁlters across 6 orientations and 3

scales.

Figure 2: An example testing image.

by psychological studies of human vision system.

However, ﬁlter bank based methods often result

in high dimensional feature space which can be difﬁ-

cult to handle for certain applications. Unlike image

classiﬁcation, in snake based image segmentation, we

may not have enough features extracted from a sin-

gle image to populate the high dimensional feature

space in order to accurately estimate the underlying

feature distributions. Moreover, there are usually sig-

niﬁcant amount of redundant information among the

ﬁltering responses. For example, a set of anisotropic

ﬁlters will get the same responses from isotropic im-

age regions. Fig. 1 shows a bank of ﬁlters which

has been used in (Varma and Zisserman, 2002) for

image classiﬁcation. It contains two isotropic ﬁlters

and thirty six anisotropic ﬁlters. The two isotropic

ﬁlters are Gaussian and Laplacian of Gaussian both

with σ = 10. Those thrifty six anisotropic ﬁlters

come from two families, edges and bars, each of

which consists ﬁlters at three progressive scales, i.e.

(σ

x

,σ

y

) = {(1, 3), (2, 6),(4,12)}, and six uniformly

VISAPP 2009 - International Conference on Computer Vision Theory and Applications

218

Figure 3: Filter responses.

spaced different orientations. This moderate size ﬁl-

ter bank will produce a thirty eight dimensional fea-

ture space, which is considerably large for features

extracted from a single image to populate. Fig. 3

gives the ﬁlter responses of the example image shown

in Fig. 2. It is evidently clear that there are certain

correlations among these ﬁlter responses and not all

the channels are effectively revealing the image struc-

tures. Thus, it is natural to condense the feature space,

which is particularly desirable for our application.

It is also worth noting that object in the scene may

have inhomogeneous textures due to, for example,

perspective projection. This inhomogeneity will ex-

hibit nonuniform responses after applying directional

ﬁlters, e.g. animal stripe texture (see zebra exam-

ple in Fig. 7) and brick wall texture (see Fig. 8).

Rotation invariance is thus desirable in such circum-

stance. We follow (Varma and Zisserman, 2002) to

condense the ﬁlter responses by collecting only the

maximum ﬁlter response across all the six orienta-

tions, i.e. those thirty six directional ﬁlter responses

are reduced to six. Alternative methods, such as steer-

able ﬁlters (Jacob and Unser, 2004) can also be used.

Thus, this not only reduces the dimensionality of the

feature space but also simultaneously achieves rota-

tional invariancy.

Instead of applying convolution operators, the re-

cursive technique (Geusebroek et al., 2003) is used to

efﬁcientlyﬁlter the images. Fig. 4 showsthe collected

maximum responses from those thirty eight ﬁlter co-

efﬁcients. Note the isotropic ﬁlter responses are re-

main unchanged since they are inherently rotationally

invariant.

2.2 Local Spectral Histogram

The ﬁltering responses can be directly used to drive

the active contours as in, for example, (Sandberg

et al., 2002). However, we can further incorpo-

rate local spatial dependency of ﬁltering responses

by computing the marginal distributions of ﬁlter re-

sponses over a local window. Thus, it captures lo-

cal pixel dependency through ﬁltering and global pat-

terns through histograms. Local spectral histogram

has been found useful, for example, texture classiﬁ-

cation (Liu and Wang, 2003). The maximum ﬁlter

responses are largely local dominant features, such as

edges and bars (e.g. see 4). Their spatial distribu-

TEXTURED IMAGE SEGMENTATION BASED ON LOCAL SPECTRAL HISTOGRAM AND ACTIVE CONTOUR

219

Figure 4: Maximum ﬁlter responses - The ﬁrst row shows

the ﬁlter responses from the isotropic Gaussian and Lapla-

cian of Gaussian are kept the same. The rest six ﬁlter re-

sponses are collected from the 36 directional (isotropic)

ﬁlter responses. Each of them contains the maximum re-

sponses across 6 different orientations (i.e. the six rows of

the directional ﬁlter responses in Fig. 3 are collapsed into

six rotational invariant ﬁlter responses).

tion conveys important information regarding the na-

ture of the texture. Misaligning of ﬁlter responses due

to inhomogeneity of ﬁlter responses can be a serious

problem for direct approaches. Using local spectral

histogram further enhances our model in dealing with

texture inhomogeneity and helps to produce more co-

herent segmentation. Fig. 8 provides an example

where directly using ﬁlter response and without tak-

ing into texture inhomogeneity resulted in a very poor

segmentation, while as the proposed method correctly

segmented the foreground object from the texturally

nonuniform background.

Let W denote a local window and W

(α)

(x) a max-

imum ﬁlter response patch centered at x, where α =

1,2,...,8. Thus, for W

(α)

the histogram is deﬁned as

(Liu and Wang, 2006):

P

(α)

W

(z

1

,z

2

) =

∑

x∈W

Z

z

2

z

1

δ(z− W

(α)

(x))dz, (1)

where z

1

and z

2

specify the range of the bin. The spec-

tral histogram is then deﬁned as:

P

W

=

1

W

P

(1)

W

,P

(2)

W

,...,P

(8)

W

,

. (2)

Example spectral histograms extracted from the test-

ing image can be found in Fig. 6.

Very recently in (Ni et al., 2007), local image in-

tensity histogram was used in the Chan-Vese model

(Chan and Vese, 2001). However, this method may

have difﬁculties in dealing with highly textured im-

ages where intensity alone is not sufﬁcient to describe

the texture. Intensity variation, for example, due to il-

lumination changes can also cause severe problems.

A comparative example is given in 9 where the best

result reported in (Ni et al., 2007) is still signiﬁcantly

less accurate than the proposed approach.

2.3 Deducing Optimal Bin Size

Although histogram based methods have been rou-

tinely used in various image processing tasks, the im-

portance of automatically selecting appropriate his-

togram bin size has been largely ignored. However,

if a too small bin size is selected, the frequency value

at each bin will suffer from signiﬁcant large ﬂuctua-

tion due to the paucity of samples in each bin. On

the other hand, if the bin size is chosen too large, the

histogram will not be a good representation of the un-

derlying distribution. Thus, it is necessary to select

an optimal bin size. It also avoids the practical issues

associated with manual parameter tunning.

We follow the method in (Shimazaki and Shi-

nomoto, 2007) to estimate the optimal bin size. Let

us consider a histogram as a bar graph. Also, let ∆ de-

note the bin size. Theexpected frequencyfor s ∈ [0, ∆]

is:

θ =

1

∆

Z

∆

0

λ

s

ds, (3)

where λ

s

is the underlying true frequency which is not

known. The goodness of ﬁt of the estimated

ˆ

λ

s

to λ

s

is

measured according to mean integrated squared error

(MISE):

MISE =

1

∆

Z

∆

0

hE(

ˆ

θ− λ

s

)

2

ids, (4)

where E denotes expectation and the empirical bar

height

ˆ

θ

i

≡ k

i

/∆ (k

i

is the frequency count for ith bin).

The associated cost function is then deﬁned as:

O (∆) = MISE−

1

∆

Z

∆

0

h(λ

s

− hθi)

2

ids. (5)

The second term represents a mean squared ﬂuctua-

tion. By assuming the number of events counted in

each bin obeys a Poisson distribution, the cost func-

tion can be written as:

O (∆) =

2

∆

hE

ˆ

θi − hE(

ˆ

θ− hE

ˆ

θi)

2

i. (6)

VISAPP 2009 - International Conference on Computer Vision Theory and Applications

220

−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5

0

200

400

600

800

1000

1200

Coefficient

Frequency

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

−8.5

−8

−7.5

−7

−6.5

−6

−5.5

−5

x 10

7

Bin size

Cost

Figure 5: Optimal bin size selection - left: A typical spectral histogram for a single maximum response ﬁlter; right: The plot

shows the relationship between the MISE based cost function and bin size (the red cross indicates the optimal bin size with

the lowest MISE value).

The optimal bin size thus is obtained by minimizing

the above cost function, i.e.

ˆ

∆ = argmin

∆

O (∆). (7)

Thus, the testing image is ﬁrst ﬁltered through

the bank of isotropic and anisotropic ﬁlters and their

responses are condensed into eight channels. Local

spectral histograms at each pixel position are com-

puted and histograms from the same channel are put

together to produce a global histogram, i.e. mean lo-

cal spectral histogram. Then, this optimal bin size

selection is taken place for each of these eight mean

local spectral histograms. Finally, all the local spec-

tral histograms are ﬁnalized using the derived optimal

bin size values. Fig. 5 gives an example of optimal

bin size computation.

2.4 Active Contour based on

Wasserstein Distance

The snake based segmentation can be viewed as a

foreground-backgroundpartition problem (in the case

of bi-phase). The snake evolves in the image domain,

attempting to minimizing the feature similarity for

those inside and outside the contours. Meanwhile, it

tries to minimize the feature difference for those that

are belong to the same region. Thus, we can formulate

our snake based on the piece-wise constant assump-

tion (Chan and Vese, 2001; Ni et al., 2007). However,

since we are using invariant image features and local

spectral histograms, the proposed method can cope

with texture inhomogeneitymuch better than the orig-

inal piece-wise constant model (see Figs. 8 and 9 as

comparative examples).

Let Ω be the image domain, Λ

+

denote the re-

gions inside the snake (foreground)and Λ

−

those out-

side the snake (background). The snake segmentation

can be achieved by solving the following energy min-

imization problem:

inf

Λ

+

E (Λ

+

) =αL (Λ

+

)

+

Z

Λ

+

D (P(x),P

+

)dx (8)

+

Z

Λ

−

D (P(x),P

−

)dx,

where α is a constant, L denote length, D is the met-

ric which measures the difference between two his-

tograms, and P

+

and P

−

are the foreground and back-

ground mean local spectral histograms to be deter-

mined. The ﬁrst term is the length minimization term

which regularize the contour. The next two terms are

data ﬁtting terms, which carry out the binary segmen-

tation.

Among many other candidates, such as χ

2

dis-

tance and normalized cross correlation, The Wasser-

stein distance (also known as the earth mover’s dis-

tance) (Rubner et al., 1998) is used to compute the

distance between two normalized spectral histograms.

since it is a true metric (unlike χ

2

distance) and has

been found very useful in various applications, e.g.

image retrieval (Rubner et al., 1998). Let H

a

(y) and

H

b

(y) be two normalized spectral histograms. The

Wasserstein distance between these two histograms is

deﬁned as:

D (P

a

,P

b

) =

Z

T

|F

a

(y) − F

b

(y)|dy, (9)

where T denoted the range of the histogram bins, and

F

a

and F

b

are cumulative distributions of P

a

and P

b

,

respectively.

The level set method is implemented to solve

this energy minimization problem so that topological

changes, such as merging and splitting, can be effec-

tively handled. Let φ denote the level set function.

The foreground is identiﬁed as Λ

+

= { x ∈ Ω : φ(x) >

TEXTURED IMAGE SEGMENTATION BASED ON LOCAL SPECTRAL HISTOGRAM AND ACTIVE CONTOUR

221

0 100 200 300 400 500 600 700 800 900 1000

0

0.02

0.04

0.06

0.08

Histogram bin

Frequency

outside

inside

0 100 200 300 400 500 600 700 800 900 1000

0

0.02

0.04

0.06

0.08

0.1

Histogram bin

Frequency

outside

inside

Figure 6: The average local spectral histogram inside and outside the snake - top: These two histograms are largely over-

lapping each other; bottom: It clearly shows the difference between the histograms when the snake converged to the object

boundaries.

0}, which can be computed using the Heaviside func-

tion, i.e.

R

Ω

H (φ)dx where H is the Heaviside func-

tion. The level set formulation can be expressed as:

inf

Λ

+

E (Λ

+

) =α

Z

Ω

|∇H (φ)|dx

+

Z

Ω

D (P(x),P

+

)H (φ)dx (10)

+

Z

Ω

D (P(x),P

−

)(1− H )(φ)dx

The regularized Heaviside function proposed in

(Chan and Vese, 2001) is used to allow larger support

in the vicinity of the zero level set so that the contours

can be initialized anywhere across the image (e.g. see

Fig. 7):

H

ε

(z) =

1

2

1+

2

π

arctan(

z

ε

)

. (11)

Thus, minimizing E with respect to φ gives us the

following partial differential equation:

∂φ

∂t

= δ

ε

(φ)

h

α∇·

∇φ

|∇φ|

− (D (P(x),P

+

) − D (P(x),P

−

))

i

= δ

ε

(φ)

h

α∇·

∇φ

|∇φ|

−

Z

T

|F

x

(y) − F

+

(y)|dy

+

Z

T

|F

x

(y) − F

−

(y)|dy

i

, (12)

where δ

ε

(x) =

d

dx

H

ε

(x), F

+

and F

−

are the spectral

cumulative histogram inside and outside the contours,

respectively. The minimization process thus moves

the contours towards object boundaries through com-

peting pixels by measuring the similarity of local cu-

mulativespectral histogram with those inside and out-

side current foreground.

Fig. 6 shows an example of spectral histogram

changes between the initial stage and the stabilized

result. The corresponding segmentation result can be

found in the ﬁrst row of Fig. 7.

3 RESULTS

The proposed method have been tested on both syn-

thetic and real world images. Fig. 7 shows some

typical examples results obtained using the proposed

method. The ﬁrst row shows the result of the run-

ning example given earlier. Good segmentation was

achieved despite the large variations in the body re-

gion. In the second example, reasonable result was

obtained, missing some very ﬁne and thin structures.

In the third example, there are clearly texture orienta-

tion variations. In the last two rows, the initial snakes

were placed outside the objects of interest but still

managed to localize them. Particularly, in the last ex-

ample, there are signiﬁcant texture variations both in

foreground and background regions, which made it

very difﬁcult to segment.

In Figs. 8 and 9, we mainly compare our work

with two extensions of the piece-wise constant model,

which is also our fundamental model. Fig. 8 demon-

strates when dealing with inhomogeneous textures,

the proposed method performs signiﬁcantly better

than that directly using ﬁlter responses (Sandberg

et al., 2002). The proposed method also showed im-

provements against a very recent method based on lo-

VISAPP 2009 - International Conference on Computer Vision Theory and Applications

222

Figure 7: Examples results of the proposed method - from left: initial snake, intermediate stages, and stabilized results.

cal histograms (Ni et al., 2007). It illustrates the ef-

fectiveness of using invariant ﬁltering technique. Fig.

9 also gives example results obtained from geodesic

snake and generalized GVF snake (Xu and Prince,

1998). It is expected that these edge based techniques

are not appropriate when dealing with highly textured

images.

The proposed method requires very little parame-

ter tunning. All the images given in this paper are us-

ing a ﬁxed set of parameters. The parameters used to

generate the ﬁlter bank are given in Section 2.1. The

local window used to collect the spectral histogram

is empirically ﬁxed as 19. For a too small window

size, the local spectral histogram may have difﬁculties

in reﬂecting underlying distribution and can result in

isolated regions. For a too large window, the segmen-

tation can be less accurate around object boundaries.

We found that a window size of 19 is a good trade-

off, however, we attempt to automatically select the

window size as part of our future work. The parame-

ter α controls the smoothness of the contour and very

rarely needs adjustment. As for computational com-

plexity, the proposed method is very similar to (Sand-

berg et al., 2002).

4 CONCLUSIONS

In this paper, we introduced a novel region based

snake method which is based on the assumption that

foreground and background local ﬁltering response

distributions are statistically separable. Maximum re-

TEXTURED IMAGE SEGMENTATION BASED ON LOCAL SPECTRAL HISTOGRAM AND ACTIVE CONTOUR

223

Figure 8: From left: a synthetic texture collage which contains an inhomogeneous background due to orientation and scale

changes; a ﬁlter response to a particular orientation; the maximum response derived across different orientations which

highlights edge features in various directions, including vertical; segmentation result obtained using the Chan-Vese model

based on Gabor features (Sandberg et al., 2002) (result reported in(Sagiv et al., 2006)); segmentation result obtained using the

proposed method.

Figure 9: Comparative analysis - ﬁrst two images: results obtained using edges based methods, namely geodesic snake and

generalized GVF snake (Xu and Prince, 1998); third image: best result on the testing image reported in (Ni et al., 2007) using

a region based approach; last image: result obtained using the proposed method.

sponses ﬁlters were used to achieve rotational invari-

ancy and their local spectral histograms were used as

image features to drive the snake. The experimental

studies showed some very promising results. As part

of our future work, we will further investigate optimal

ﬁlter selection and automatic local spectral histogram

window selection.

ACKNOWLEDGEMENTS

The recursive ﬁltering is based on the library provided

by (Geusebroek et al., 2003).

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