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- Digital Image Processing: PIKS Inside, Third Edition. William K. Pratt
Copyright © 2001 John Wiley & Sons, Inc.
ISBNs: 0-471-37407-5 (Hardback); 0-471-22132-5 (Electronic)
16
IMAGE FEATURE EXTRACTION
An image feature is a distinguishing primitive characteristic or attribute of an image.
Some features are natural in the sense that such features are defined by the visual
appearance of an image, while other, artificial features result from specific manipu-
lations of an image. Natural features include the luminance of a region of pixels and
gray scale textural regions. Image amplitude histograms and spatial frequency spec-
tra are examples of artificial features.
Image features are of major importance in the isolation of regions of common
property within an image (image segmentation) and subsequent identification or
labeling of such regions (image classification). Image segmentation is discussed in
Chapter 16. References 1 to 4 provide information on image classification tech-
niques.
This chapter describes several types of image features that have been proposed
for image segmentation and classification. Before introducing them, however,
methods of evaluating their performance are discussed.
16.1. IMAGE FEATURE EVALUATION
There are two quantitative approaches to the evaluation of image features: prototype
performance and figure of merit. In the prototype performance approach for image
classification, a prototype image with regions (segments) that have been indepen-
dently categorized is classified by a classification procedure using various image
features to be evaluated. The classification error is then measured for each feature
set. The best set of features is, of course, that which results in the least classification
error. The prototype performance approach for image segmentation is similar in
nature. A prototype image with independently identified regions is segmented by a
509
- 510 IMAGE FEATURE EXTRACTION
segmentation procedure using a test set of features. Then, the detected segments are
compared to the known segments, and the segmentation error is evaluated. The
problems associated with the prototype performance methods of feature evaluation
are the integrity of the prototype data and the fact that the performance indication is
dependent not only on the quality of the features but also on the classification or seg-
mentation ability of the classifier or segmenter.
The figure-of-merit approach to feature evaluation involves the establishment of
some functional distance measurements between sets of image features such that a
large distance implies a low classification error, and vice versa. Faugeras and Pratt
(5) have utilized the Bhattacharyya distance (3) figure-of-merit for texture feature
evaluation. The method should be extensible for other features as well. The Bhatta-
charyya distance (B-distance for simplicity) is a scalar function of the probability
densities of features of a pair of classes defined as
1⁄2
B ( S 1, S 2 ) = – ln ∫ [ p ( x S 1 )p ( x S 2 ) ] dx (16.1-1)
where x denotes a vector containing individual image feature measurements with
conditional density p ( x S i ) . It can be shown (3) that the B-distance is related mono-
tonically to the Chernoff bound for the probability of classification error using a
Bayes classifier. The bound on the error probability is
1⁄2
P ≤ [ P ( S 1 )P ( S2 ) ] exp { – B ( S 1, S 2 ) } (16.1-2)
where P ( Si ) represents the a priori class probability. For future reference, the Cher-
noff error bound is tabulated in Table 16.1-1 as a function of B-distance for equally
likely feature classes.
For Gaussian densities, the B-distance becomes
1
T Σ1 + Σ2
–1 -- Σ 1 + Σ 2
-
B ( S 1, S 2 ) = 1 ( u 1 – u 2 ) ----------------- ( u 1 – u 2 ) + 1 ln --------------------------------- (16.1-3)
--
- - --
-
2
-
8 2 2
Σ1 1 ⁄ 2 Σ2 1 ⁄ 2
where ui and Σ i represent the feature mean vector and the feature covariance matrix
of the classes, respectively. Calculation of the B-distance for other densities is gener-
ally difficult. Consequently, the B-distance figure of merit is applicable only for
Gaussian-distributed feature data, which fortunately is the common case. In prac-
tice, features to be evaluated by Eq. 16.1-3 are measured in regions whose class has
been determined independently. Sufficient feature measurements need be taken so
that the feature mean vector and covariance can be estimated accurately.
- AMPLITUDE FEATURES 511
TABLE 16.1-1 Relationship of Bhattacharyya Distance
and Chernoff Error Bound
B ( S 1, S 2 ) Error Bound
1 1.84 × 10–1
2 6.77 × 10–2
4 9.16 × 10–3
6 1.24 × 10–3
8 1.68 × 10–4
10 2.27 × 10–5
12 2.07 × 10–6
16.2. AMPLITUDE FEATURES
The most basic of all image features is some measure of image amplitude in terms of
luminance, tristimulus value, spectral value, or other units. There are many degrees
of freedom in establishing image amplitude features. Image variables such as lumi-
nance or tristimulus values may be utilized directly, or alternatively, some linear,
nonlinear, or perhaps noninvertible transformation can be performed to generate
variables in a new amplitude space. Amplitude measurements may be made at spe-
cific image points, [e.g., the amplitude F ( j, k ) at pixel coordinate ( j, k ) , or over a
neighborhood centered at ( j, k ) ]. For example, the average or mean image amplitude
in a W × W pixel neighborhood is given by
w w
1
M ( j, k ) = ------
W
-
2 ∑ ∑ F ( j + m, k + n ) (16.2-1)
m = –w n = –w
where W = 2w + 1. An advantage of a neighborhood, as opposed to a point measure-
ment, is a diminishing of noise effects because of the averaging process. A disadvan-
tage is that object edges falling within the neighborhood can lead to erroneous
measurements.
The median of pixels within a W × W neighborhood can be used as an alternative
amplitude feature to the mean measurement of Eq. 16.2-1, or as an additional
feature. The median is defined to be that pixel amplitude in the window for which
one-half of the pixels are equal or smaller in amplitude, and one-half are equal or
greater in amplitude. Another useful image amplitude feature is the neighborhood
standard deviation, which can be computed as
w w 1⁄2
1 2
S ( j, k ) = ----
W
- ∑ ∑ [ F ( j + m, k + n ) – M ( j + m, k + n ) ] (16.2-2)
m = –w n = – w
- 512 IMAGE FEATURE EXTRACTION
(a) Original (b) 7 × 7 pyramid mean
(c) 7 × 7 standard deviation (d ) 7 × 7 plus median
FIGURE 16.2-1. Image amplitude features of the washington_ir image.
In the literature, the standard deviation image feature is sometimes called the image
dispersion. Figure 16.2-1 shows an original image and the mean, median, and stan-
dard deviation of the image computed over a small neighborhood.
The mean and standard deviation of Eqs. 16.2-1 and 16.2-2 can be computed
indirectly in terms of the histogram of image pixels within a neighborhood. This
leads to a class of image amplitude histogram features. Referring to Section 5.7, the
first-order probability distribution of the amplitude of a quantized image may be
defined as
P ( b ) = P R [ F ( j, k ) = r b ] (16.2-3)
where r b denotes the quantized amplitude level for 0 ≤ b ≤ L – 1 . The first-order his-
togram estimate of P(b) is simply
- AMPLITUDE FEATURES 513
N( b)
P ( b ) ≈ -----------
- (16.2-4)
M
where M represents the total number of pixels in a neighborhood window centered
about ( j, k ) , and N ( b ) is the number of pixels of amplitude r b in the same window.
The shape of an image histogram provides many clues as to the character of the
image. For example, a narrowly distributed histogram indicates a low-contrast
image. A bimodal histogram often suggests that the image contains an object with a
narrow amplitude range against a background of differing amplitude. The following
measures have been formulated as quantitative shape descriptions of a first-order
histogram (6).
Mean:
L–1
SM ≡ b = ∑ bP ( b ) (16.2-5)
b=0
Standard deviation:
L–1 1⁄2
2
SD ≡ σb = ∑ (b – b ) P( b) (16.2-6)
b=0
Skewness:
L–1
1 3
S S = -----
σb
-
3 ∑ ( b – b) P(b ) (16.2-7)
b=0
Kurtosis:
L–1
1- 4
S K = -----
4 ∑ ( b – b) P( b ) – 3 (16.2-8)
σb b=0
Energy:
L–1
2
SN = ∑ [P(b )] (16.2-9)
b=0
Entropy:
L–1
SE = – ∑ P ( b ) log 2 { P ( b ) } (16.2-10)
b=0
- 514 IMAGE FEATURE EXTRACTION
m,n
r q
j,k
FIGURE 16.2-2. Relationship of pixel pairs.
The factor of 3 inserted in the expression for the Kurtosis measure normalizes SK to
zero for a zero-mean, Gaussian-shaped histogram. Another useful histogram shape
measure is the histogram mode, which is the pixel amplitude corresponding to the
histogram peak (i.e., the most commonly occurring pixel amplitude in the window).
If the histogram peak is not unique, the pixel at the peak closest to the mean is usu-
ally chosen as the histogram shape descriptor.
Second-order histogram features are based on the definition of the joint proba-
bility distribution of pairs of pixels. Consider two pixels F ( j, k ) and F ( m, n ) that
are located at coordinates ( j, k ) and ( m, n ), respectively, and, as shown in Figure
16.2-2, are separated by r radial units at an angle θ with respect to the horizontal
axis. The joint distribution of image amplitude values is then expressed as
P ( a, b ) = P R [ F ( j, k ) = r a, F ( m, n ) = r b ] (16.2-11)
where r a and r b represent quantized pixel amplitude values. As a result of the dis-
crete rectilinear representation of an image, the separation parameters ( r, θ ) may
assume only certain discrete values. The histogram estimate of the second-order dis-
tribution is
N ( a, b )
P ( a, b ) ≈ -----------------
- (16.2-12)
M
where M is the total number of pixels in the measurement window and N ( a, b )
denotes the number of occurrences for which F ( j, k ) = r a and F ( m, n ) = r b .
If the pixel pairs within an image are highly correlated, the entries in P ( a, b ) will
be clustered along the diagonal of the array. Various measures, listed below, have
been proposed (6,7) as measures that specify the energy spread about the diagonal of
P ( a, b ).
Autocorrelation:
L–1 L–1
SA = ∑ ∑ abP ( a, b ) (16.2-13)
a=0 b=0
- AMPLITUDE FEATURES 515
Covariance:
L–1 L–1
SC = ∑ ∑ ( a – a ) ( b – b )P ( a, b ) (16.2-14a)
a=0 b=0
where
L–1 L–1
a = ∑ ∑ aP ( a, b ) (16.2-14b)
a=0 b=0
L–1 L–1
b = ∑ ∑ bP ( a, b ) (16.2-14c)
a=0 b=0
Inertia:
L–1 L–1
2
SI = ∑ ∑ (a – b) P ( a, b ) (16.2-15)
a=0 b=0
Absolute value:
L–1 L–1
SV = ∑ ∑ a – b P ( a, b ) (16.2-16)
a=0 b=0
Inverse difference:
L–1 L–1
P ( a, b )
SF = ∑ ∑ ----------------------------
1 + ( a – b)
2
(16.2-17)
a=0 b=0
Energy:
L–1 L–1
2
SG = ∑ ∑ [ P ( a, b ) ] (16.2-18)
a=0 b=0
Entropy:
L–1 L–1
ST = – ∑ ∑ P ( a, b ) log 2 { P ( a, b ) } (16.2-19)
a=0 b=0
The utilization of second-order histogram measures for texture analysis is consid-
ered in Section 16.6.
- 516 IMAGE FEATURE EXTRACTION
16.3. TRANSFORM COEFFICIENT FEATURES
The coefficients of a two-dimensional transform of a luminance image specify the
amplitude of the luminance patterns (two-dimensional basis functions) of a trans-
form such that the weighted sum of the luminance patterns is identical to the image.
By this characterization of a transform, the coefficients may be considered to indi-
cate the degree of correspondence of a particular luminance pattern with an image
field. If a basis pattern is of the same spatial form as a feature to be detected within
the image, image detection can be performed simply by monitoring the value of the
transform coefficient. The problem, in practice, is that objects to be detected within
an image are often of complex shape and luminance distribution, and hence do not
correspond closely to the more primitive luminance patterns of most image trans-
forms.
Lendaris and Stanley (8) have investigated the application of the continuous two-
dimensional Fourier transform of an image, obtained by a coherent optical proces-
sor, as a means of image feature extraction. The optical system produces an electric
field radiation pattern proportional to
∞ ∞
F ( ω x, ω y ) = ∫–∞ ∫–∞ F ( x, y ) exp { – i ( ωx x + ωy y ) } dx dy (16.3-1)
where ( ω x, ω y ) are the image spatial frequencies. An optical sensor produces an out-
put
2
M ( ω x, ω y ) = F ( ω x, ω y ) (16.3-2)
proportional to the intensity of the radiation pattern. It should be observed that
F ( ω x, ω y ) and F ( x, y ) are unique transform pairs, but M ( ω x, ω y ) is not uniquely
related to F ( x, y ) . For example, M ( ω x, ω y ) does not change if the origin of F ( x, y )
is shifted. In some applications, the translation invariance of M ( ω x, ω y ) may be a
benefit. Angular integration of M ( ω x, ω y ) over the spatial frequency plane produces
a spatial frequency feature that is invariant to translation and rotation. Representing
M ( ω x, ω y ) in polar form, this feature is defined as
2π
N (ρ) = ∫0 M ( ρ, θ ) dθ (16.3-3)
2 2 2
where θ = arc tan { ω x ⁄ ω y } and ρ = ω x + ω y . Invariance to changes in scale is an
attribute of the feature
∞
P(θ ) = ∫0 M ( ρ, θ ) dρ (16.3-4)
- TRANSFORM COEFFICIENT FEATURES 517
FIGURE 16.3-1. Fourier transform feature masks.
The Fourier domain intensity pattern M ( ω x, ω y ) is normally examined in specific
regions to isolate image features. As an example, Figure 16.3-1 defines regions for
the following Fourier features:
Horizontal slit:
∞ ωy (m + 1 )
S1( m ) = ∫– ∞ ∫ω ( m )
y
M ( ω x, ω y ) dω x dω y (16.3-5)
Vertical slit:
ωx (m + 1 ) ∞
S2 ( m ) = ∫ω ( m) ∫–∞ M ( ω x, ωy ) dωx dωy
x
(16.3-6)
Ring:
ρ ( m + 1 ) 2π
S3 ( m ) = ∫ρ ( m ) ∫0 M ( ρ, θ ) dρ dθ (16.3-7)
Sector:
∞ θ(m + 1 )
S4 ( m ) = ∫0 ∫ θ ( m ) M ( ρ, θ ) dρ dθ (16.3-8)
- 518 IMAGE FEATURE EXTRACTION
(a ) Rectangle (b ) Rectangle transform
(c ) Ellipse (d ) Ellipse transform
(e ) Triangle (f ) Triangle transform
FIGURE 16.3-2. Discrete Fourier spectra of objects; log magnitude displays.
For a discrete image array F ( j, k ) , the discrete Fourier transform
N–1 N–1
1 – 2πi
F ( u, v ) = ---
N
- ∑ ∑ F ( j, k ) exp ----------- ( ux + vy )
N
(16.3-9)
j=0 k=0
- TEXTURE DEFINITION 519
for u, v = 0, …, N – 1 can be examined directly for feature extraction purposes. Hor-
izontal slit, vertical slit, ring, and sector features can be defined analogous to
Eqs. 16.3-5 to 16.3-8. This concept can be extended to other unitary transforms,
such as the Hadamard and Haar transforms. Figure 16.3-2 presents discrete Fourier
transform log magnitude displays of several geometric shapes.
16.4. TEXTURE DEFINITION
Many portions of images of natural scenes are devoid of sharp edges over large
areas. In these areas, the scene can often be characterized as exhibiting a consistent
structure analogous to the texture of cloth. Image texture measurements can be used
to segment an image and classify its segments.
Several authors have attempted qualitatively to define texture. Pickett (9) states
that “texture is used to describe two dimensional arrays of variations... The ele-
ments and rules of spacing or arrangement may be arbitrarily manipulated, provided
a characteristic repetitiveness remains.” Hawkins (10) has provided a more detailed
description of texture: “The notion of texture appears to depend upon three ingredi-
ents: (1) some local 'order' is repeated over a region which is large in comparison to
the order's size, (2) the order consists in the nonrandom arrangement of elementary
parts and (3) the parts are roughly uniform entities having approximately the same
dimensions everywhere within the textured region.” Although these descriptions of
texture seem perceptually reasonably, they do not immediately lead to simple quan-
titative textural measures in the sense that the description of an edge discontinuity
leads to a quantitative description of an edge in terms of its location, slope angle,
and height.
Texture is often qualitatively described by its coarseness in the sense that a patch
of wool cloth is coarser than a patch of silk cloth under the same viewing conditions.
The coarseness index is related to the spatial repetition period of the local structure.
A large period implies a coarse texture; a small period implies a fine texture. This
perceptual coarseness index is clearly not sufficient as a quantitative texture mea-
sure, but can at least be used as a guide for the slope of texture measures; that is,
small numerical texture measures should imply fine texture, and large numerical
measures should indicate coarse texture. It should be recognized that texture is a
neighborhood property of an image point. Therefore, texture measures are inher-
ently dependent on the size of the observation neighborhood. Because texture is a
spatial property, measurements should be restricted to regions of relative uniformity.
Hence it is necessary to establish the boundary of a uniform textural region by some
form of image segmentation before attempting texture measurements.
Texture may be classified as being artificial or natural. Artificial textures consist of
arrangements of symbols, such as line segments, dots, and stars placed against a
neutral background. Several examples of artificial texture are presented in Figure
16.4-1 (9). As the name implies, natural textures are images of natural scenes con-
taining semirepetitive arrangements of pixels. Examples include photographs
of brick walls, terrazzo tile, sand, and grass. Brodatz (11) has published an album of
photographs of naturally occurring textures. Figure 16.4-2 shows several natural
texture examples obtained by digitizing photographs from the Brodatz album.
- 520 IMAGE FEATURE EXTRACTION
FIGURE 16.4-1. Artificial texture.
- VISUAL TEXTURE DISCRIMINATION 521
(a) Sand (b) Grass
(c) Wool (d) Raffia
FIGURE 16.4-2. Brodatz texture fields.
16.5. VISUAL TEXTURE DISCRIMINATION
A discrete stochastic field is an array of numbers that are randomly distributed in
amplitude and governed by some joint probability density (12). When converted to
light intensities, such fields can be made to approximate natural textures surpris-
ingly well by control of the generating probability density. This technique is useful
for generating realistic appearing artificial scenes for applications such as airplane
flight simulators. Stochastic texture fields are also an extremely useful tool for
investigating human perception of texture as a guide to the development of texture
feature extraction methods.
In the early 1960s, Julesz (13) attempted to determine the parameters of stochas-
tic texture fields of perceptual importance. This study was extended later by Julesz
et al. (14–16). Further extensions of Julesz’s work have been made by Pollack (17),
- 522 IMAGE FEATURE EXTRACTION
FIGURE 16.5-1. Stochastic texture field generation model.
Purks and Richards (18), and Pratt et al. (19). These studies have provided valuable
insight into the mechanism of human visual perception and have led to some useful
quantitative texture measurement methods.
Figure 16.5-1 is a model for stochastic texture generation. In this model, an array
of independent, identically distributed random variables W ( j, k ) passes through a
linear or nonlinear spatial operator O { · } to produce a stochastic texture array
F ( j, k ). By controlling the form of the generating probability density p ( W ) and the
spatial operator, it is possible to create texture fields with specified statistical proper-
ties. Consider a continuous amplitude pixel x 0 at some coordinate ( j, k ) in F ( j, k ) .
Let the set { z 1, z 2, …, z J } denote neighboring pixels but not necessarily nearest geo-
metric neighbors, raster scanned in a conventional top-to-bottom, left-to-right fash-
ion. The conditional probability density of x 0 conditioned on the state of its
neighbors is given by
p ( x 0, z 1, …, z J )
p ( x 0 z 1, …, z J ) = ------------------------------------- (16.5-1)
p ( z 1, …, z J )
The first-order density p ( x 0 ) employs no conditioning, the second-order density
p ( x 0 z 1 ) implies that J = 1, the third-order density implies that J = 2, and so on.
16.5.1. Julesz Texture Fields
In his pioneering texture discrimination experiments, Julesz utilized Markov process
state methods to create stochastic texture arrays independently along rows of the
array. The family of Julesz stochastic arrays are defined below.
1. Notation. Let x n = F ( j, k – n ) denote a row neighbor of pixel x 0 and let
P(m), for m = 1, 2,..., M, denote a desired probability generating function.
2. First-order process. Set x 0 = m for a desired probability function P(m).
The resulting pixel probability is
P ( x0 ) = P ( x0 = m ) = P ( m ) (16.5-2)
- VISUAL TEXTURE DISCRIMINATION 523
3. Second-order process. Set F ( j, 1 ) = m for P ( m ) = 1 ⁄ M , and set
x 0 = ( x 1 + m )MOD { M }, where the modulus function p MOD { q } ≡
p – [ q × ( p ÷ q ) ] for integers p and q. This gives a first-order probability
1-
P ( x 0 ) = ---- (16.5-3a)
M
and a transition probability
p ( x 0 x 1 ) = P [ x 0 = ( x 1 + m ) MOD { M } ] = P ( m ) (16.5-3b)
4. Third-order process. Set F ( j, 1 ) = m for P ( m ) = 1 ⁄ M , and set
F ( j, 2 ) = n for P ( n ) = 1 ⁄ M . Choose x 0 to satisfy 2x 0 = ( x 1 + x 2 + m )
MOD { M } . The governing probabilities then become
1
P ( x 0 ) = ----
- (16.5-4a)
M
1
p ( x 0 x 1 ) = ----
- (16.5-4b)
M
p ( x 0 x 1, x 2 ) = P [ 2x 0 = ( x 1 + x 2 + m ) MOD { M } ] = P ( m ) (16.5-4c)
This process has the interesting property that pixel pairs along a row are
independent, and consequently, the process is spatially uncorrelated.
Figure 16.5-2 contains several examples of Julesz texture field discrimination
tests performed by Pratt et al. (19). In these tests, the textures were generated
according to the presentation format of Figure 16.5-3. In these and subsequent
visual texture discrimination tests, the perceptual differences are often small. Proper
discrimination testing should be performed using high-quality photographic trans-
parencies, prints, or electronic displays. The following moments were used as sim-
ple indicators of differences between generating distributions and densities of the
stochastic fields.
η = E { x0 } (16.5-5a)
2 2
σ = E { [ x0 – η ] } (16.5-5b)
E { [ x0 – η ] [ x1 – η ] }
α = -------------------------------------------------
- (16.5-5c)
2
σ
E { [ x 0 – η ] [ x1 – η ] [ x2 – η ] }
θ = ---------------------------------------------------------------------
- (16.5-5d)
3
σ
- 524 IMAGE FEATURE EXTRACTION
(a) Different first order (b) Different second order
sA = 0.289, sB = 0.204 sA = 0.289, sB = 0.289
aA = 0.250, aB = − 0.250
(c) Different third order
sA = 0.289, sB = 0.289
aA = 0.000, aB = 0.000
qA = 0.058, qB = − 0.058
FIGURE 16.5-2. Field comparison of Julesz stochastic fields; η A = η B = 0.500 .
The examples of Figure 16.5-2a and b indicate that texture field pairs differing in
their first- and second-order distributions can be discriminated. The example of
Figure 16.5-2c supports the conjecture, attributed to Julesz, that differences in third-
order, and presumably, higher-order distribution texture fields cannot be perceived
provided that their first-order and second- distributions are pairwise identical.
- VISUAL TEXTURE DISCRIMINATION 525
FIGURE 16.5-3. Presentation format for visual texture discrimination experiments.
16.5.2. Pratt, Faugeras, and Gagalowicz Texture Fields
Pratt et al. (19) have extended the work of Julesz et al. (13–16) in an attempt to study
the discriminability of spatially correlated stochastic texture fields. A class of Gaus-
sian fields was generated according to the conditional probability density
–1 ⁄ 2
J+1 –1
exp – 1 ( v J + 1 – η J + 1 ) ( K J + 1 ) ( v J + 1 – η J + 1 )
T
( 2π ) KJ + 1 --
-
2
p ( x 0 z 1, …, z J ) = ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
-
–1 ⁄ 2
J 1 T –1
( 2π ) K J exp – -- ( v J – η J ) ( K J ) ( v J – η J )
-
2
(16.5-6a)
z1
where
vJ =
…
zJ
(16.5-6b)
x0
vJ + 1 = (16.5-6c)
vJ
The covariance matrix of Eq. 16.5-6a is of the parametric form
- 526 IMAGE FEATURE EXTRACTION
(a) Constrained second-order density
(b) Constrained third-order density
FIGURE 16.5-4. Row correlation factors for stochastic field generation. Dashed line, field
A; solid line, field B.
1 α β γ …
α
KJ + 1 = –2 (16.5-7)
β σ KJ
γ
…
where α, β, γ, … denote correlation lag terms. Figure 16.5-4 presents an example of
the row correlation functions used in the texture field comparison tests described
below.
Figures 16.5-5 and 16.5-6 contain examples of Gaussian texture field comparison
tests. In Figure 16.5-5, the first-order densities are set equal, but the second-order
nearest neighbor conditional densities differ according to the covariance function plot
of Figure 16.5-4a. Visual discrimination can be made in Figure 16.5-5, in which the
correlation parameter differs by 20%. Visual discrimination has been found to be
marginal when the correlation factor differs by less than 10% (19). The first- and
second-order densities of each field are fixed in Figure 16.5-6, and the third-order
- VISUAL TEXTURE DISCRIMINATION 527
(a) aA = 0.750, aB = 0.900 (b) aA = 0.500, aB = 0.600
FIGURE 16.5-5. Field comparison of Gaussian stochastic fields with different second-order
nearest neighbor densities; η A = η B = 0.500, σ A = σ B = 0.167.
conditional densities differ according to the plan of Figure 16.5-4b. Visual discrimi-
nation is possible. The test of Figure 16.5-6 seemingly provides a counterexample to
A B A B
the Julesz conjecture. In this test, [ p ( x 0 ) = p ( x 0 ) ] and p ( x 0, x 1 ) = p ( x 0, x 1 ), but
A B
p ( x 0, x 1, x 2 ) ≠ p ( x 0, x 1, x 2 ) . However, the general second-order density pairs
A B
p ( x 0, z j ) and p ( x 0, z j ) are not necessarily equal for an arbitrary neighbor z j , and
therefore the conditions necessary to disprove Julesz’s conjecture are violated.
To test the Julesz conjecture for realistically appearing texture fields, it is neces-
sary to generate a pair of fields with identical first-order densities, identical
Markovian type second-order densities, and differing third-order densities for every
(a) bA = 0.563, bB = 0.600 (b) bA = 0.563, bB = 0.400
FIGURE 16.5-6. Field comparison of Gaussian stochastic fields with different third-order
nearest neighbor densities; η A = η B = 0.500, σ A = σ B = 0.167, α A = α B = 0.750 .
- 528 IMAGE FEATURE EXTRACTION
hA = 0.500, hB = 0.500
sA = 0.167, sB = 0.167
aA = 0.850, aB = 0.850
qA = 0.040, qB = − 0.027
FIGURE 16.5-7. Field comparison of correlated Julesz stochastic fields with identical first-
and second-order densities, but different third-order densities.
pair of similar observation points in both fields. An example of such a pair of fields
is presented in Figure 16.5-7 for a non-Gaussian generating process (19). In this
example, the texture appears identical in both fields, thus supporting the Julesz
conjecture.
Gagalowicz has succeeded in generating a pair of texture fields that disprove the
Julesz conjecture (20). However, the counterexample, shown in Figure 16.5-8, is not
very realistic in appearance. Thus, it seems likely that if a statistically based texture
measure can be developed, it need not utilize statistics greater than second-order.
FIGURE 16.5-8. Gagalowicz counterexample.
nguon tai.lieu . vn
