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# Multispectral texture characterization: application to computer aided diagnosis on prostatic tissue images

Riad Khelifi, Mouloud Adel, Salah Bourennane

► To cite this version:

Riad Khelifi, Mouloud Adel, Salah Bourennane. Multispectral texture characterization: application to computer aided diagnosis on prostatic tissue images. EURASIP Journal on Advances in Signal Processing, 2012, 10.1186/1687-6180-2012-118. [https://doi.org/10.1186/1687-6180-2012-118](https://doi.org/10.1186/1687-6180-2012-118)

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REVIEW

Open Access

Multispectral texture characterization: application
to computer aided diagnosis on prostatic tissue
images

Riad Khelifi, Mouloud Adel and Salah Bourennane*

Abstract

Various approaches have been proposed in the literature for texture characterization of images. Some of them are based on statistical properties, others on fractal measures and some more on multi-resolution analysis. Basically, these approaches have been applied on mono-band images. However, most of them have been extended by including the additional information between spectral bands to deal with multi-band texture images. In this article, we investigate the problem of texture characterization for multi-band images. Therefore, we aim to add spectral information to classical texture analysis methods that only treat gray-level spatial variations. To achieve this goal, we propose a spatial and spectral gray level dependence method (SSGLDM) in order to extend the concept of gray level co-occurrence matrix (GLCM) by assuming the presence of texture joint information between spectral bands. Thus, we propose new multi-dimensional functions for estimating the second-order joint conditional probability density of spectral vectors. These functions can be represented in structure form which can help us to compute the occurrences while keeping the corresponding components of spectral vectors. In addition, new texture features measurements related to (SSGLDM) which define the multi-spectral image properties are proposed. Extensive experiments have been carried out on 624 textured multi-spectral images for use in prostate cancer diagnosis and quantitative results showed the efficiency of this method compared to the GLCM. The results indicate a significant improvement in terms of global accuracy rate. Thus, the proposed approach can provide clinically useful information for discriminating pathological tissue from healthy tissue.

**Keywords:** texture characterization, multi-band texture images, spatial/spectral co-occurrence matrix, texture features, prostate cancer, classification accuracy

**1. Introduction**

Recently, spectral imaging technology has become a topic of growing interest in color-reproduction, remote-sensing, medical imaging, and other systems. These increased research efforts are likely to propagate into other application areas such as computer vision and pattern recognition. Usually, texture analysis is an efficient measure to estimate the structural orientation, roughness, smoothness or regularity differences of diverse regions in an image scene.

The traditional popular tool used for texture-based
segmentation and classification is the gray level depen-
dence method (GLCM) [1,2], but some other techniques

have been proposed with success: description through
wavelet coefficient statistics [3], the Markov random
field [4] or Markov chain [5] models. Nevertheless, tex-
ture analysis remains a difficult problem when applied
to color [6], multi or hyperspectral images; where image
pixel takes its values in a multidimensional space. For
this purpose, a great deal of work has been done for
modeling the multispectral and hyperspectral texture
analysis [7-12].

The main idea of extracting texture from hyperspec-
tral images is the use of combined spectral and spatial
information. Numerous approaches have been con-
ducted on the use of Gabor filters [13], co-occurrence
matrices [14-19], mathematical morphology [20-22].

In [23], the authors used the potential of the spectral/
textural approach to improve the classification accuracy

<figure><img src="image_4.png" /></figure>

<footnote>* Correspondence: salah.bourennane@fresnel.fr
Groupe GSM, Institut Fresnel, Ecole Centrale Marseille D. U. de Saint Jérôme
Av. Escadrille Normandie, 13397, Marseille, France</footnote>
of intra-urban land cover types; Claussi [24] studied the
effect of gray quantization on the ability of co-occur-
rence probability statistics; Kiema [25] examined the
gray-level co-occurrence based texture image fused to
thematic mapper (TM) imagery to expand the object
feature base to include both spectral and spatial features;
while Bau and Healey [26] used a bank of rotation scale
invariant Gabor feature vectors to represent the spec-
tral/spatial properties of a region.

Besides, color opponent features were first introduced
in color texture characterization with fairly good perfor-
mance [27] and later extended to deal with multi-band
texture images [28]. Other methods combine color and
texture information for the segmentation of color
images [29]. Moreover, several researches studied the
spatial interaction within each channel and interaction
between spectral channels, applying gray level texture
techniques to each channel independently [30], or using
3D colour histograms as a way to combine information
from all colour channels [31].

Recently, it has been shown that the concept of spatial
co-occurrence matrix could be generalized by assuming
texture joint information between spectral bands [17,32].

However, the main problem of texture analysis of
multi-band images is related to the high dimension of
the data and its high correlation.

Driven by classification or discrimination accuracy, one
would expect that, as the number of multi or hyperspectral
bands increases, the accuracy of classification should also
increase. Nonetheless, this is not the case in a model-
based analysis [33-35]. Redundancy in data can cause con-
vergence instability of models. Furthermore, variations due
to noise in redundant data propagate through a classifica-
tion or discrimination model. Thus, processing a large
number of multi or hyperspectral bands can result in
higher classification inaccuracy than processing a subset of
relevant bands without redundancy [34,35].

One way of overcoming this problem is to adopt a
proper selection band method before applying classifica-
tion task. The reason is that, in a selection band proce-
dure, the amount of data is reduced into a lower
dimensional subspace without practically losing relevant
information [36]. In addition, computational require-
ments for processing large hyperspectral data sets might
be prohibitive and a method for selecting a data subset
is therefore sought [33].

Several approaches have been investigated by looking
into how to remove information re-dundancy resulting
from highly correlated bands [33,37-40]. Most of the
methods usually involve two separate tasks: (a) selecting
the bands that can indicate the particular material well,
feature bands selection; and (b) removing the feature
bands contributing redundant information, redundancy
reduction [41]. Recently, Information theory has also

been used in feature bands selection [42,43], it consists
of analyzing the amount of information in a subset of
features (bands), measuring the degree of independence
between image bands as a relevance criterion.

In this article, we investigate the problem of the analy-
sis of multispectral image textures allowing a joint spec-
tral and spatial analysis of texture. Therefore, we aim to
add spectral information to classical texture analysis
methods that only treat gray-level spatial variations. We
try to apply the proposed method on prostate cancer
textured multispectral images. These images have been
chosen to reflect different grades of malignancy in pro-
static tissues, and they correspond to different structural
patterns as well as apparent textures.

The remainder of this article is organized as follows:
Section 2 describes a new method for multi and hyper-
spectral texture analysis, based on joint spatial and spec-
tral information; while Section 3 is concerned with
some comparative results. Finally, Section 4 gives a con-
clusion of the article.

2. Spatial and spectral gray level dependence method (SSGLDM)

2.1. N-mode flattening matrix of a tensor

A multi-band image can be represented as a 3-D data
array also called third order tensor X ∈ ℝ^I₁×I₂×I₃, where
the three entries are related to pixel localization and
spectral band, and each element of X could be arranged
as xᵢ₁ᵢ₂ᵢ₃, i₁ = 1, ..., I₁, i₂ = 1, ..., I₂, i₃ = 1, ..., I₃; ℝ is the
real manifold.

A tensor can be transformed into a *n*-mode matrix.
The *n*-mode flattened matrix X_n of tensor X ∈ ℝ^I_1×I_2×I_3
is a *I*_n × *M*_n matrix where:

$$
M_n = I_p \times I_q, \text{ with } p, q \neq n.
$$

By definition, X_n is generated by the column vectors
of the *n*-mode flattened matrix. columns are *I*_n dimen-
sional vectors obtained from *X* by varying the index *i*_n
and keeping the other indices fixed [44]. These vectors
are called “*n*-mode vectors”. The three *n*-mode flatten-
ing of a third-order tensor are shown in Figure 1.

2.2. SSGLDM algorithm

The idea of the SSGLDM is based on the estimation of
the second-order joint conditional probability density
multi-dimensional functions P (V, V<sub>Δ</sub>|d, θ).

Each P (V, V<sub>Δ</sub> | d, θ) is the probability that two spectral
vectors V and V<sub>Δ</sub> with components (i<sub>1</sub>, ..., i<sub>k</sub>, ..., i<sub>n</sub>), and
(j<sub>1</sub>, ..., j<sub>b</sub>, ..., j<sub>n</sub>) respectively, occur for a given distance d
and direction θ (see Figure 2), where n is the number of
spectral bands, and (i<sub>(k)</sub>, j<sub>(l)</sub>) ∈ [1, N<sub>g</sub>]<sup>2</sup>, N<sub>g</sub> is the number
of gray levels.

In the whole article, we consider multi-spectral data as
a third-order tensor denoted by X, in which the entries
are accessed via three index.
<figure><img src="image_5.png" /><figcaption>Figure 1 Flattening matrices of a third order tensor [52].</figcaption></figure>

Unlike the spatial gray level dependence method in which the estimated second-order joint conditional density functions can be described in a matrix, we propose to represent P (V, V<sub>Δ</sub>|d, θ) in a structure form of an array I<sub>occu</sub> which is the vector of occurrences and a matrix M<sub>V,V<sub>Δ</sub></sub> which is the matrix of V and V<sub>Δ</sub> components. Each row in the matrix M<sub>V,V<sub>Δ</sub></sub> corresponds to V and V<sub>Δ</sub> components. The use of this structure can help us to compute the occurrences while keeping the corresponding components vectors of V and V<sub>Δ</sub>. For example if we take the first value of I<sub>occu</sub> (i.e I<sub>occu</sub>(1)), the corresponding components vectors of V and V<sub>Δ</sub> are in the first row of M<sub>V,V<sub>Δ</sub></sub> (see Figure 3), so I<sub>occu</sub>(1|d, θ) = P(V<sup>(1)</sup>, V<sub>Δ</sub><sup>(1)</sup>|d, θ) where V<sup>(1)</sup> = M<sub>V,V<sub>Δ</sub></sub>(1, 1 : n) and V<sub>Δ</sub><sup>(1)</sup> = M<sub>V,V<sub>Δ</sub></sub>(1, n + 1 : 2 × n). (1 : n): designs columns 1 to n.

In order to compute the matrix M<sub>V,V<sub>Δ</sub></sub>, we define two sub-tensors A, B extracted from the tensor data X for each direction θ as shown in Figure 4. Using the 3-mode flattening matrix of A, B, we obtain respectively

<figure><img src="image_6.png" /><figcaption>Figure 2 3D representation of the two vectors V and V<sub>Δ</sub> for a given distance d = 1 and θ = 45°.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 3 Structure form representation of P (V, V<sub>Δ</sub>|d, θ).</figcaption></figure>

A<sub>3</sub>, B<sub>3</sub>. Let T denotes the matrix that horizontally concatenates A<sub>3</sub><sup>t</sup> and B<sub>3</sub><sup>t</sup>, and vertically concatenates B<sub>3</sub><sup>t</sup> and A<sub>3</sub><sup>t</sup>, where A<sub>3</sub><sup>t</sup>, B<sub>3</sub><sup>t</sup> denote the transposed matrices of A<sub>3</sub>, B<sub>3</sub> respectively. T can be represented as follows:

$$
T = \begin{pmatrix} A_3^t & B_3^t \\ B_3^t & A_3^t \end{pmatrix} \qquad (1)
$$

The use of this procedure yields us to calculate the
spatial and spectral gray level dependence for both θ
and (θ + π).

Finally, the matrix M<sub>V,V<sub>Δ</sub></sub> is obtained by keeping only the rows of T with no repetitions, however, the number of repetitions for each row is represented in an array of occurrences called I<sub>occu</sub>. At the end of the process, I<sub>occu</sub> is normalized with respect to the size of M<sub>V,V<sub>Δ</sub></sub> so that each component represents the probability of occurrence of a given combination (V, V<sub>Δ</sub>). The spatial and spectral gray level dependence algorithm can be given as follows:

STEP 1: For a given distance *d* and angle *θ*, extract the two sub-tensors *A* and *B* from the tensor data *X*.

STEP 2: Compute the 3-mode flattening matrices A<sub>3</sub>, B<sub>3</sub>.

STEP 3: Build the matrix T.

STEP 4: Compute M<sub>V,V<sub>Δ</sub></sub> by keeping only the rows T with no repetitions.

STEP 5: Compute I<sub>occu</sub>.

## 2.3. Proposed generalized textures features related to SSGLDM

Haralick texture features comprise 14 features as summarized in [1]. However, the most used in literature are energy, entropy, inertia, local homogeneity and correlation. In this article we propose seven new generalized texture features.

Suppose that $V^{(k)} = (i_1^{(k)}, i_2^{(k)}, \dots, i_n^{(k)})$, and
$V_{\Delta}^{(k)} = (j_1^{(k)}, j_2^{(k)}, \dots, j_n^{(k)})$, where $1 \le k \le m$, and
$1 \le m \le N_g^{2n}$ is the length of the vector of occurrences,
$N_g$ is the number of gray levels and $n$ is the number of
bands.
<figure><img src="image_8.png" /></figure>

For convenience, $P(V^{(k)}, V_{\Delta}^{(k)}|d, \theta)$ will be written as $P(V^{(k)}, V_{\Delta}^{(k)})$ in the following equations. Energy:

$$H = \sum_{k=1}^{k=m} P^2(V^{(k)}, V_{\Delta}^{(k)}) \qquad (2)$$

This descriptor is also known as angular second moment or uniformity. In our case, it measures the spatial/spectral image homogeneity.

Entropy:

$$E = - \sum_{k=1}^{k=m} P(V^{(k)}, V_{\Delta}^{(k)}) \log(P(V^{(k)}, V_{\Delta}^{(k)})) \qquad (3)$$

The entropy is the counterpart measure of energy. It measures randomness and is smaller for a smooth image than for a coarse image.
Inertia:

$$
\left\{
\begin{array}{l}
I = \sum_{k=1}^{k=m} |V^{(k)} - V_{\Delta}^{(k)}|^2 P(V^{(k)}, V_{\Delta}^{(k)}) \\
\text{where : } [V^{(k)} - V_{\Delta}^{(k)}]^2 = [(i_1^{(k)} - j_1^{(k)})^2 + (i_2^{(k)} - j_2^{(k)})^2 + \dots + (i_n^{(k)} - j_n^{(k)})^2]
\end{array}
\right.
\quad (4)
$$

This descriptor is also known as contrast or difference
moment. It is a measure of image intensity contrast or
the spatial/spectral variations present in an image to
show the texture fineness.

The inertia value is high when similar spectral vectors
are adjacent to each other in the input image and pro-
vides a measure of coarseness (similarity).

Local homogeneity:

$$
In = \sum_{k=1}^{k=m} \frac{1}{1 + [V^{(k)} - V_{\Delta}^{(k)}]^2} P(V^{(k)}, V_{\Delta}^{(k)}) \quad (5)
$$

The local homogeneity or inverse difference moment
is a counterpart measure to the contrast descriptor.

Asymmetry:

$$
I = \sum_{k=1}^{k=m} \left[ (V^{(k)} - V_x) + (V_{\Delta}^{(k)} - V_{\Delta y}) \right]^3 P(V^{(k)}, V_{\Delta}^{(k)}) \quad (6)
$$

where:

$$
\begin{align*}
(V^{(k)} - V_x) &= \sqrt{(i_1^{(k)} - x_1)^2 + \dots + (i_n^{(k)} - x_n)^2} \\
(V^{(k)} - V_{\Delta y}) &= \sqrt{(j_1^{(k)} - \gamma_1)^2 + \dots + (j_n^{(k)} - \gamma_n)^2} \\
V_x &= (x_1, x_2, \dots, x_n) = \sum_{k=1}^{k=m} V^{(k)} P(V^{(k)}, V_{\Delta}^{(k)}) \\
V_{\Delta y} &= (\gamma_1, \gamma_2, \dots, \gamma_n) = \sum_{k=1}^{k=m} V_{\Delta}^{(k)} P(V^{(k)}, V_{\Delta}^{(k)})
\end{align*}
$$

This descriptor measures the distribution of spectral
vectors values around the vectors average using third
order statistics.

Proeminence:

$$
I = \sum_{k=1}^{k=m} \left[ (V^{(k)} - V_x) + (V_{\Delta}^{(k)} - V_{\Delta y}) \right]^4 P(V^{(k)}, V_{\Delta}^{(k)}) \quad (7)
$$

This descriptor measures the distribution of spectral
vectors values around the vectors average using fourth
order statistics.

Correlation:

$$
I = \frac{1}{\sigma_i \sigma_j} \sum_{k=1}^{k=m} (V^{(k)} - V_x)(V_{\Delta}^{(k)} - V_{\Delta y})P(V^{(k)}, V_{\Delta}^{(k)}) \quad (8)
$$

where:

$$
\begin{align*}
\sigma_i^2 &= \sum_{k=1}^{k=m} (V^{(k)} - V_x)^2 P(V^{(k)}, V_{\Delta}^{(k)}) \\
\sigma_j^2 &= \sum_{k=1}^{k=m} (V_{\Delta}^{(k)} - V_{\Delta y})^2 P(V^{(k)}, V_{\Delta}^{(k)})
\end{align*}
$$

The correlation descriptor measures the linear depen-
dency of spectral vectors in the co-occurrence matrix.

In practice, as the number of bands increases, the joint probability P(V<sup>(k)</sup>, V<sub>Δ</sub><sup>(k)</sup>) decreases. Consequently, the values of P(V<sup>(k)</sup>, V<sub>Δ</sub><sup>(k)</sup>) become nonsignificant which implies a poor overall classification [17,32].

To overcome this issue, we suggest to apply the proposed SSGLDM on a subset of bands in the multi-spectral image, instead of processing the whole set of bands. The subset of bands can be chosen to remove information redundancy resulting from highly correlated bands.

2.4. Band selection by minimization of dependent
information

In [43], the extraction of selected subsets of spectral
image bands can be obtained by means of a criterion
based on the minimization of the dependent information
(MDI), which consists of a relation between the joint
entropy and the union of the conditional entropies of
the considered set of image bands. This criterion could
be defined by the following expression:

$$
\Theta_{DI} = H(A_1, \dots, A_n) - \sum_{i_1=1}^{n} H(A_{i_1} | A_{i_2}, \dots, A_{i_n}) \quad (9)
$$

Where Aᵢ₁ is a random variable representing the image band i₁. Aᵢ₂, ..., Aᵢₙ are the complementary variables of Aᵢ₁; H(A₁, ..., Aₙ) is the joint entropy, which represents the total amount of joint information of set of variables A₁, ..., Aₙ; and H(Aᵢ₁|Aᵢ₂, ..., Aᵢₙ) is the conditional entropy that represents the amount of independent information in image band Aᵢ₁ having measured the rest image bands Aᵢ₂, ..., Aᵢₙ.

This technique behaves as an unsupervised feature selection criterion, providing very satisfactory results with respect to classification accuracy when using the selected bands, even outperforming the other supervised methods used in the comparison in most situations [43]. These performances justify our choice of using MDI [43] for selecting the best relevant bands in the second part of our experiments.

**3. Experimental results**

The first part of this section is concerned with multi-
spectral texture characterization. For this purpose, an
extensive experiments are carried out on many multi-
spectral images for use in prostate cancer diagnosis.
Furthermore, the results of proposed method
(SSLGMD) are compared with GLCM, followed by per-
formance comparative study and discussion. The second
part deals with the curse-of-dimensionality problem.
Thus, we extract the best relevant bands of multispectral
prostate cancer using MDI technique proposed by
Sotoca et al. [43]. Thereafter, we provide some experi-
mental results, in order, to see the impact of the
selected bands using MDI criterion on the classification
accuracy performance using proposed method and
GLCM respectively.

**3.1. Data set**

Over the last decade, multi-spectral imagery in pro-
tate cancer has become a very useful tool for analyzing
and diagnosing pathologies. The prostate cancer is the
second most common cancer in men after the skin
cancer, and it is also the second in the list of causes
for cancer death after the lung cancer. However, the
most known method for prostate cancer diagnosis is
the prostate-specific antigen (PSA) blood test. In the
case of a positive diagnosis, the urologist will often
advise a needle biopsy, in which a tiny piece of a tissue
is taken from the prostate for analysis [45]. From the
analysis, different grades of malignancy correspond to
different structural patterns as well as to apparent tex-
tures. After the analysis, an experienced pathologist
selects the images to reflect the different patterns in
prostatic tissues, four major groups usually should be
identified (see Figure 5).

• Stroma: STR (normal muscular tissue)

* Benign prostatic hyperplasia: BPH (a benign condition)

* Prostatic intraepithelial neoplasia: PIN (a precursor state for cancer).

* Prostatic carcinoma: PCa (abnormal tissue development corresponding to cancer).

The data set consists of 624 textured multi-spectral
images, of 128 by 128, with each image taken with 16
spectral channels (from 500 to 650 nm) with a 5 nm
step [45]. The images were captured using a classical
microscope and CCD camera. A liquid crystal tunable
filter (LCTF) was inserted in the optical path
between the light source and the chilled CCD camera.
The LCTF has a bandwidth accuracy of 5 nm.

The captured images have been chosen to reflect differ-
ent grades of malignancy in prostatic tissues and they are
labeled into four classes: 176 cases of cancer (Pca), 160
cases of BPH, 144 cases of PIN and 144 cases of Stroma
(STR).ᵃ The samples were routinely viewed at prostatic
section seen at low power (40x objective magnification)
by two highly experimented independent pathologists.
The pathologist initially views slides at low power,
thereby enabling the location of potentially abnormal
regions. Subsequent analysis of these regions at high
power enables the histological grading of these areas.

The X-Y resolution depends on the magnification cho-
sen, which is usually high for visualization purposes. In
this study, the images of 128 by 128 pixels were cap-
tured at an X-Y resolution of 0.12 µ/pixel.

Usually, when a biopsy is submitted for analysis, it is
very rare that the pathologist finds that the sample is per-
fectly normal. There must be at least some benign condi-
tion that would explain the high levels of PSA that
usually justify needle biopsy. So the main issue is to iden-
tify benign from malignant and premalignant conditions.

For this purpose, Figure 6 summarizes the different
steps used in this article for classification process. The
proposed methodology is very important task in prostate
cancer diagnosis and could be viewed as a computer-
aided system to automatically classify pathological pros-
tate images, since each image can be classified into an
appropriate class of prostatic tissue.

**3.2. Features extraction (Step 1)**

The goal of this step of analysis is to characterize the
four different groups of multi-spectral images by

<figure><img src="image_9.png" /><figcaption>Figure 5 Images showing representative samples of the four classes. (a) Stroma, (b) BPH, (c) PIN, (d) PCa.</figcaption></figure>
<figure><img src="image_10.png" /></figure>

extracting features in order to make classification
possible.

All generalized and traditional features mentioned in
Sections 3.4 and 3.5 have been used for our experi-
ments, and the best performances have been achieved
by choosing (d = 1) and the four principal directions (θ
= 0°, θ = 45°, θ = 90°, θ = 135°).

We note that, the problem of spatial selection rela-
tionships (i.e d and θ) for defining co-occurrence
matrices is addressed in [46]. These matrices are maxi-
mally reflecting the structure of the underlying texture.

**3.3. Features selection for prediction procedure (Step 2)**

Feature selection technique is applied to reduce the
number of features before applying the classification
task. Irrelevant features may have negative effects on a
prediction procedure. Moreover, the computational
complexity of a classification algorithm may suffer from
the curse of dimensionality caused by several features.
Features can be selected in many different ways. One

scheme is to select features that correlate strongest to
the classification variable.

This has been called maximum-relevance selection.
Many heuristic algorithms can be used, such as the
sequential forward, backward, or floating selections [42].

On the other hand, features can be selected to be
mutually far away from each other, while they still have
high correlation to the classification variable. This
scheme, termed as minimum-redundancy-maximum-
relevance selection [42], has been found to be more
powerful than the maximum relevance selection. This
justifies the use of this technique in our experiments.

We note that, this technique is not applied when the
number of feature is small.

**3.4. Classification process (Step 3)**

A classification process usually involves *training* and *testing data* which consist of some data instances. Each instance in the training set contains one “target value” (class labels) and several “attributes” (features).
### 3.4.1. Using SVM for classification

In [47], several neural networks were compared to the SVM for the classification of hyper-spectral data. The robustness of SVMs was demonstrated and the best results were obtained using a non-linear SVM. In addition, [47] studied Radial Basis Functions (RBF) classifiers and SVM and they noted the favorable behavior of the SVM, from both a theoretical and a practical point of view (see Appendix).

Visually, the images illustrated in Figure 5 are very similar, so the use of SVM for classification issue could be suitable for our applications.

### 3.5. Results and discussion

The assessment of the classification performance was made using 3-fold cross-validation. Thus, *data* were randomly split into 3-sets of a roughly equal size. Splitting was carried out such that the proportion of samples per class was roughly equal across the sets. Each run of the 3-fold cross-validation algorithm consisted of a classifier designed on two data subsets (training) while testing was performed on the remaining subset; this is repeated three times. The SVM optimization was implemented using LIBSVM library through its Matlab interface [48]. The penalty term *C* of Gaussian kernel, was fixed to 200 and $\sigma^2$ was selected by using a fivefold cross validation [48]. So that, fivefold cross validation was applied on *training data* in order to estimate $\sigma^2$ which gives the highest classification accuracy rate.

Firstly, we compared the performance of SSGLDM with GLCM by the overall classification rate as a criterion of the comparison, with reference to a set of images manually classified by experts. Like in the GLCM, in which the co-occurrence matrix is computed for each band, we can apply SSGLDM for each subset of bands in the image. Let *S* be a subset of *α* bands in the image (*α* ≤ *N*<sub>b</sub>) where *N*<sub>b</sub> is the number of image bands. For example, if *S* = (*B*<sub>l</sub>, *B*<sub>l+1</sub>) is a subset of two bands; the SSGLDM can be applied *N*<sub>b</sub> − 1 times in the image by varying *l* from 1 to *N*<sub>b</sub> − 1. Thus, GLCM can be seen as a special case of SSGLDM when *S* contains only one band.

The number of features is computed for each band in the GLCM, so the total number of features is 7 × (4 directions) × 16 bands = 448. However in the case of SSGLDM, the number of features depends on the choice of subset of bands (*S*). For example, if *S* contains two bands, the number of features is 7 × (4 directions) × (*N*<sub>b</sub> − 1) bands = 420, where *N*<sub>b</sub> = 16. The minimum redundancy-maximum relevance selection algorithm [42] was used to ensure good classification accuracy by selecting 40 better features from all features generated by SSGLDM.

Table 1 shows the classification results obtained by using SSGLDM, for different subsets of bands (*S*) for both 32 and 64 gray levels quantification.

It can be clearly observed from Table 1 that the SSGLDM yields better results than the GLCM when *S* is less than five bands. Best results are obtained with *S* = 3 bands, for both 32 and 64 gray levels. However, with 16 bands we obtained a classification accuracy rate around 93%. This may derive from the high correlation of the data. Moreover, as the set of bands increases, the joint probability *P* (*V*, *V*<sub>Δ</sub>|*d*, *θ*) decreases. Consequently, the values of *P* (*V*, *V*<sub>Δ</sub>|*d*, *θ*) become nonsignificant which implies a poor overall classification.

To illustrate the classification of different classes of prostate cancer, we show the confusion matrices obtained by using SSGLDM for different choices of *S*. Tables 2a,b depict the results using GLCM (*S* = 1 band) where Tables 2c,d give the corresponding results using SSGLDM for (*S* = 16 bands). As can be seen from these tables, the use of SSGLDM with (*S* = 16) yields worst results in terms of global accuracy rate. However, Tables 2e,f show the achieved improvements when using SSGLDM with (*S* = 3 bands). Note that in all the cases, BPH and Stroma classes present the highest error rate in terms of classification due to the similarities between two classes; however the use of SSGLDM with (*S* = 3 bands) reduces significantly the error rate in these classes.

On the other hand, tests of the computation time were performed for the both SSGLDM and GLCM. For this purpose, we used a PC Intel® Core<sup>(TM)</sup> 2 CPU, 2.66 GHz and 3.58 GB Ram. The two methods were implemented using Matlab 7.1. Times are given for the computation of the co-occurrence matrices and all the features of each method (SSGLDM and GLCM), on images 128 × 128 mentioned in Section 3.1. As shown in on Table 3 The execution time using SSGLDM with (*S* = 2) is almost the same as GLCM.

Note also that the time keeps growing, when increasing the set of bands during the process. However, it decreases for (*S* = 16), because the SSGLDM was applied just one time for the whole set of spectral bands. Thus, one can conclude that the use SSGLDM with (*S* = 2) is a quit good compromise between classification results and computation.

<table><caption>Table 1 Classification results obtained by using the SSGLDM</caption><thead><tr><th>Subset of bands (S)</th><th>1(GLCM)</th><th>2</th><th>3</th><th>4</th><th>5</th><th>16</th></tr></thead><tbody><tr><td>Overall classification (%) (32 gray levels)</td><td>94.87</td><td>97.44</td><td>97.76</td><td>97.11</td><td>96.31</td><td>93.91</td></tr><tr><td>Overall classification (%) (64 gray levels)</td><td>95.51</td><td>97.12</td><td>97.44</td><td>96.96</td><td>96.79</td><td>93.27</td></tr></tbody></table>
<table><caption>**Table 2 Confusion matrices obtained by using (a) GLCM (32 gray levels), (b) GLCM (64 gray level), (c) SSGLDM (S = 16 bands, 32 gray level), (d) SSGLDM (S = 16 bands, 64 gray level), (e) SSGLDM (S = 3 bands, 32 gray levels), (f) SSGLDM (S = 3 bands, 64 gray levels)**</caption><thead><tr><th>Classified as:</th><th>BPH</th><th>PCa</th><th>PIN</th><th>Stroma</th><th>Error (%)</th></tr></thead><tbody><tr><td>(a)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>BPH</td><td>135</td><td>1</td><td>24</td><td>0</td><td>15.62</td></tr><tr><td>PCa</td><td>0</td><td>175</td><td>0</td><td>1</td><td>0.57</td></tr><tr><td>PIN</td><td>2</td><td>0</td><td>141</td><td>1</td><td>2.08</td></tr><tr><td>Stroma</td><td>0</td><td>0</td><td>3</td><td>141</td><td>2.08</td></tr><tr><td>Overall</td><td></td><td></td><td></td><td></td><td>5.13</td></tr><tr><td>(b)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>BPH</td><td>138</td><td>0</td><td>22</td><td>0</td><td>13.75</td></tr><tr><td>PCa</td><td>0</td><td>176</td><td>0</td><td>0</td><td>0</td></tr><tr><td>PIN</td><td>3</td><td>0</td><td>141</td><td>0</td><td>2.08</td></tr><tr><td>Stroma</td><td>0</td><td>0</td><td>3</td><td>141</td><td>2.08</td></tr><tr><td>Overall</td><td></td><td></td><td></td><td></td><td>4.49</td></tr><tr><td>(c)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>BPH</td><td>149</td><td>0</td><td>11</td><td>0</td><td>6.88</td></tr><tr><td>PCa</td><td>0</td><td>173</td><td>0</td><td>3</td><td>1.70</td></tr><tr><td>PIN</td><td>0</td><td>0</td><td>136</td><td>8</td><td>5.56</td></tr><tr><td>Stroma</td><td>0</td><td>11</td><td>5</td><td>128</td><td>11.11</td></tr><tr><td>Overall</td><td></td><td></td><td></td><td></td><td>6.09</td></tr><tr><td>(d)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>BPH</td><td>145</td><td>0</td><td>15</td><td>0</td><td>9.38</td></tr><tr><td>PCa</td><td>0</td><td>174</td><td>0</td><td>2</td><td>1.14</td></tr><tr><td>PIN</td><td>0</td><td>0</td><td>135</td><td>9</td><td>6.25</td></tr><tr><td>Stroma</td><td>5</td><td>4</td><td>7</td><td>128</td><td>11.11</td></tr><tr><td>Overall</td><td></td><td></td><td></td><td></td><td>6.73</td></tr><tr><td>(e)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>BPH</td><td>147</td><td>0</td><td>13</td><td>0</td><td>8.13</td></tr><tr><td>PCa</td><td>0</td><td>175</td><td>0</td><td>1</td><td>0.57</td></tr><tr><td>PIN</td><td>0</td><td>0</td><td>144</td><td>0</td><td>0</td></tr><tr><td>Stroma</td><td>0</td><td>0</td><td>0</td><td>144</td><td>0</td></tr><tr><td>Overall</td><td></td><td></td><td></td><td></td><td>2.24</td></tr><tr><td>(f)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>BPH</td><td>147</td><td>0</td><td>13</td><td>0</td><td>7.5</td></tr><tr><td>PCa</td><td>0</td><td>175</td><td>0</td><td>1</td><td>0.57</td></tr><tr><td>PIN</td><td>0</td><td>0</td><td>143</td><td>1</td><td>0.69</td></tr><tr><td>Stroma</td><td>0</td><td>0</td><td>1</td><td>143</td><td>0.69</td></tr><tr><td>Overall</td><td></td><td></td><td></td><td></td><td>2.56</td></tr></tbody></table>

### 3.5.1. Receiver operating characteristic (ROC)

In order to plot ROC curves, the classifier should be tested using different parameters resulting from different values for false alarm (false positive rate FPR) and sensitivity (true positive rate TPR) rates.

<table><caption>**Table 3 Consuming time**</caption><thead><tr><th>Subset of bands</th><th>1 (GLCM)</th><th>2</th><th>3</th><th>4</th><th>5</th><th>16</th></tr></thead><tbody><tr><td>Time per image (sec)</td><td>0.63</td><td>0.67</td><td>1.3</td><td>1.62</td><td>3.13</td><td>1.56</td></tr></tbody></table>

Let $\pi_i$ be the prior probabilities in each class $i$. In the case of uniform prior probabilities, $\pi_i$ can be written as: $\forall i \pi_i = \frac{1}{N}$, where: $N$ is the number of classes.

Let us suppose a two-class prediction problem (binary classification), in which the outcomes are labeled either as positive ($p$) or negative ($n$) class. This is achieved by considering that BPH as the negative diagnosis while PIN and PCa together form the positive diagnosis outcome (the classification of Stroma is relatively simple because of its homogeneous nature at low resolution) [49,50].

Since there are two classes, prior probabilities are linked by the relation:

$$ \pi_{\text{positive}} = 1 - \pi_{\text{negative}} \quad (10) $$

ROC curves could be plotted by varying the positive values $\pi_{\text{positive}}$.

Figure 7 shows a comparative study of the ROC curves using GLCM and SSGLDM respectively, for both 32 and 64 gray levels. Dark, blue and red curves indicate the ROC curves of classified image using SSGLDM ($S = 3$ bands), GLCM and SSGLDM ($S = 16$ bands) respectively. These results indicate that for all possible values of prior probability $\pi_{\text{positive}}$ the SSGLDM features with ($S = 3$) perform better when compared with that derived from GLCM for both 32 and 64 gray levels. The results demonstrate also that our new proposed technique improves ability to distinguish cancer prostate tissues from healthy ones.

However, one can note, that the major issue arise from our proposed method is related to the high dimension of the data and its high correlation. This fact is clearly seen from Table 1 and 2 where classification accuracy is degraded while using SSGLDM with 16 bands. Thus, the main question to be solved is: does the band selection procedure improve the overall classification rate when using SSGLDM? How would one obtain the optimal number that maximizes classification accuracy and minimize computational requirements?

In the remainder of this section, we show the influence of selected band method using MDI criterion (Section 2.4) on the overall classification rate by applying the SSGLDM.

### 3.6. SSGLDM using band selection

In order to exploit inter-band correlation for reducing the multi-spectral band representation, the unsupervised band selection technique by Sotoca et al. [43] has been used to extract the best relevant bands of multispectral prostate cancer database.

All classification rates shown in this section were computed by using 3-fold cross-validation, such as described before.
<figure><img src="image_11.png" /><figcaption>**Figure 7** The comparative ROC curves. (a) 32 gray levels (b) 64 gray levels.</figcaption></figure>

Note that the same classification procedure used in Section 5.4 was applied for each data cube constructed by the selected bands. However, in the case of SSGLDM the features selection technique [42] was not used because of the small number of features (Figure 6).

To see the impact of the selected bands using MDI criterion on the classification accuracy performance, we tested the SSGLDM with a variable number of selected bands of multispectral images. Therefore, the plot of Figure 8 shows the result when using 2, 3, 4, 5, 6 and 16 selected bands as input data. A clear improvement of the classification accuracy can be observed by using five selected bands. However, the SSGLDM yields worst results, when more than six selected bands are used. Thus, processing a large number of multispectral bands can result in higher classification inaccuracy than processing a subset of relevant bands without redundancy.

On the other hand, the classical GLCM provides lower classification accuracy than SSGLDM. However, when more bands selected are used, GLCM performs better. This is mainly due to the use of minimum redundancy-maximum-relevance selection [42] that reduces the number of features and ensure a good classification accuracy.

<figure><img src="image_12.png" /><figcaption>**Figure 8** Behavior of the overall classification accuracy versus the selected bands. (a) 32 gray levels (b) 64 gray levels.</figcaption></figure>

To gain an insight into the classification of different classes of prostate cancer, the confusion matrices of both SSGLDM and GLCM using five selected bands are also given in Table 4. As can be seen from this table, the use of band selection procedure before applying SSGLDM reduces significantly the error rate in classes and insures a good discrimination power between different types of tissues.

Figure 9 illustrates the ROC curves obtained with the two methods GLCM and SSGLDM using five selected bands as input data. This clearly demonstrates that our new proposed SSGLDM results in an improved ability to distinguish cancer prostate tissues from healthy ones.

Finally, Table 5 summarizes the processing time for both GLCM and SSGLDM (implemented using Matlab 7.1) versus selected bands. This experiment is conducted on an Intel® Core™ 2 CPU, 2.66 GHz and 3.58 GB
<table><caption>**Table 4 Confusion matrices obtained by using (a) GLCM (5 selected bands, 32 gray levels), (b) SSGLDM (5 selected bands, 32 gray levels), (c) GLCM (5 selected bands, 64 gray levels), (d) SSGLDM (5 selected bands, 64 gray levels)**</caption><thead><tr><th>Classified as:</th><th>BPH</th><th>PCa</th><th>PIN</th><th>Stroma</th><th>Error (%)</th></tr></thead><tbody><tr><td>(a)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>BPH</td><td>147</td><td>0</td><td>13</td><td>0</td><td>8.13</td></tr><tr><td>PCa</td><td>0</td><td>174</td><td>2</td><td>0</td><td>1.14</td></tr><tr><td>PIN</td><td>2</td><td>12</td><td>129</td><td>1</td><td>10.42</td></tr><tr><td>Stroma</td><td>9</td><td>0</td><td>1</td><td>134</td><td>6.94</td></tr><tr><td>Overall</td><td></td><td></td><td></td><td></td><td>6.41</td></tr><tr><td>(b)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>BPH</td><td>148</td><td>0</td><td>12</td><td>0</td><td>7.5</td></tr><tr><td>PCa</td><td>0</td><td>174</td><td>2</td><td>0</td><td>1.14</td></tr><tr><td>PIN</td><td>2</td><td>11</td><td>130</td><td>1</td><td>9.72</td></tr><tr><td>Stroma</td><td>1</td><td>0</td><td>1</td><td>142</td><td>1.39</td></tr><tr><td>Overall</td><td></td><td></td><td></td><td></td><td>4.81</td></tr><tr><td>(c)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>BPH</td><td>141</td><td>0</td><td>15</td><td>4</td><td>11.88</td></tr><tr><td>PCa</td><td>0</td><td>175</td><td>1</td><td>0</td><td>0.57</td></tr><tr><td>PIN</td><td>0</td><td>13</td><td>130</td><td>1</td><td>9.72</td></tr><tr><td>Stroma</td><td>17</td><td>0</td><td>1</td><td>126</td><td>12.5</td></tr><tr><td>Overall</td><td></td><td></td><td></td><td></td><td>8.33</td></tr><tr><td>(d)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>BPH</td><td>149</td><td>0</td><td>11</td><td>0</td><td>6.88</td></tr><tr><td>PCa</td><td>0</td><td>176</td><td>0</td><td>0</td><td>0</td></tr><tr><td>PIN</td><td>2</td><td>10</td><td>131</td><td>1</td><td>9.03</td></tr><tr><td>Stroma</td><td>0</td><td>0</td><td>1</td><td>143</td><td>0.69</td></tr><tr><td>Overall</td><td></td><td></td><td></td><td></td><td>4.01</td></tr></tbody></table>

Ram. As mentioned before, times are given for the computation of co-occurrence matrices and all features for each method.

It's clear from the table that the use of band selection technique reduces significantly the time computation. On the other hand, comparing the results of two methods, SSGLDM ran much faster than the GLCM mainly because SSGLDM features are extracted from the whole data cube of selected bands, unlike the GLDM features which are computed from each selected band.

## 4. Conclusion

This article describes a new method to generalize the concept of spatial gray level dependence method by assuming the presence of texture joint information between spectral bands. Two ways have been suggested to implement the proposed spatial and spectral gray level dependence method (SSGLDM): (a) applying SSGLDM for each subset of bands in the multi-spectral image; and (b) making a connection between band selection and SSGLDM by using MDI criterion before applying SSGLDM. Extensive experiments have been

carried out on many multi-spectral images for use in prostate cancer diagnosis and quantitative results showed the efficiency of this method compared to the Gray GLCM. SSGLDM has also provided better performances in terms of classification accuracy and computational complexity. Finally, due to the aspect of this area of research, many issues could be suggested. Open problems that can be investigated in the future include the following:

1. The new texture characterization method described in this article focuses on second order statistics. Therefore, the way forward could be to investigate alternative methods using higher-order statistics.

2. In this work, the most time consuming task was, by far, the computation of the generalized co-occurrence matrix, which mainly depends on spectral vector-pairs distances and a large numbers of spectral bands. The

<table><caption>**Table 5 Computation time**</caption><thead><tr><th>Number of selected bands</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th></tr></thead><tbody><tr><td>GLCM (Time per image (sec))</td><td>0.08</td><td>0.12</td><td>0.16</td><td>0.2</td><td>0.24</td></tr><tr><td>SSGLDM (Time per image (sec))</td><td>0.05</td><td>0.09</td><td>0.12</td><td>0.17</td><td>0.21</td></tr></tbody></table>

<figure><img src="image_13.png" /><figcaption>**Figure 9 The comparative ROC curves using five selected bands as input data. (a) 32 gray levels (b) 64 gray levels.**</figcaption></figure>
nature of the calculation makes it suitable for parallel processing because the same calculations are performed on successive image blocks.

(3) The generalized multi-band texture method is proposed in this article to solve the characterization of multi-band texture images problem. Medical data sets that use multi-spectral data have been used to evaluate our proposed algorithm. In future, to apply our proposed algorithms to other applications such as hyperspectral satellite imagery or skin cancer detection.

## Endnote

ªThe data set used in this study were provided from Pathology department team at Queen's university of Belfast under the direction of Prof. Hamilton.

## Appendix

### Support vector machines (SVM)

The aim of SVM is to produce a model (based on the training data) that predicts target value of data instances in the testing set which are given only by the attributes. Given a labeled training data set {($x_i$, $y_i$), ... ($x_n$, $y_n$)}, where $x_i \in \mathbb{R}^n$ and $y_i \in \{-1, 1\}$, the SVM [47,51] require the solution of the following optimization problem:

$$ \min_{w, \xi_i, b} \left\{ \frac{1}{2} \|w\|^2 + C \sum_i \xi_i \right\} \quad (1) $$

constrained to:

$$ \begin{cases} y_i(\langle \phi(x_i), w \rangle + b) \geq 1 - \xi_i \quad \forall i = 1, \dots, n \\ \xi_i \geq 0 \quad \forall i = 1, \dots, n \end{cases} \quad (2) $$

where $\langle \cdot, \cdot \rangle$: is the inner product.

In the case of a nonlinear classification of samples, the training vectors $x_i$ are mapped into a higher (maybe infinite) dimensional space by the function $\phi$, $w$ is the vector of hyperplane coefficients (orientation), $b$ is a bias term. The regularization parameter $C$ controls the generalization capabilities of the classifier and it must be selected by the user, and $\xi_i$ are positive slack variables enabling to deal with permitted errors. The decision function is found by solving the convex optimization problem

$$ \max_{\alpha_i} \left\{ \sum_i \alpha_i - \frac{1}{2} \sum_{i,j} \alpha_i \alpha_j y_i y_j \langle \phi(x_i), \phi(x_j) \rangle \right\} \quad (3) $$

subject to:

$$ \begin{cases} 0 \le \alpha_i \le C \\ \sum_i \alpha_i y_i = 0 \quad i = 1, \dots, n \end{cases} \quad (4) $$

where $\alpha_i$ are the Lagrange coefficients. It is worth noting that all $\phi$ mappings used in the SVM learning occur in the form of inner product. This allows us to define the classical Gaussian kernel $K_\sigma$ given by this formula:

$$ \begin{aligned} k_\sigma(x_i, x_j) &= \langle \phi(x_i), \phi(x_j) \rangle \\ &= \left( \exp \left( - \frac{\|x_i - x_j\|^2}{2\sigma^2} \right) \right) \end{aligned} \quad (5) $$

where the norm is the Euclidean norm and $\sigma \in \mathbb{R}^+$ tunes the flexibility of the kernel. The classification of a sample $\times$ is achieved by looking to which side of the hyperplane it belongs

$$ f(x) = \operatorname{sgn} \left( \sum_{i=1}^{n} y_i \alpha_i k_\sigma(x_i, x) + b \right) \quad (6) $$

Where $b$ can be used for computing $\alpha_i$.

The SVMs are mainly a nonparametric method, yet some parameters need to be tuned before the optimization. In the case of Gaussian kernel, there are two parameters: $C$, which is the penalty term, and $\sigma$, which is the width of the exponential.

## Acknowledgements

I would to acknowledge contribution from Dr. MA Roula and the Pathology department team at the Queen's university of Belfast under the direction of Prof. Hamilton for kindly providing maging data used in this study.

## Competing interests

The authors declare that they have no competing interests.

Received: 8 September 2011 Accepted: 29 May 2012
Published: 29 May 2012

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