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Texture Segmentation Using Filters with Optimized Energy Separation Trygve Randen and John H˚akon Husøy Fig. 1. Abstract— The design of filters for texture feature extraction is addressed. Based on a new feature extraction model, optimization approaches utilizing various feature (energy) separation criteria are developed. Both two- and multiple-texture problems are addressed. The approaches are assessed by supervised segmentation experiments. The experiments also include results from alternative filter optimization approaches. Index Terms— Digital filters, image classification, image processing, image recognition, image texture analysis.
I. INTRODUCTION
Experimental setup.
In Section I-A, an overview of previous filter optimization approaches is presented, in Section II, our approach is outlined, and in Section III, we present our feature extraction model. Various twotexture filter optimization approaches are presented in Section IV. In Section V, a multitexture approach is presented. The result is summarized in Section VI. Mathematical details and derivations are presented in the Appendices. A. Survey of Filter Optimization Approaches
Most natural surfaces exhibit texture. A texture recognition system will therefore be a natural part of many computer vision systems. Applications of texture analysis range from geophysical surveying through medical diagnosis to document image analysis. Several methods for texture discrimination have been proposed and an overview is given by Tuceryan and Jain [1]. One important category of texture discrimination techniques is the category of signal processing methods, where typically a filter and a response energy measure is used as feature extractor. A typical system setup is illustrated in Fig. 1. In multichannel approaches, one such system is commonly used for each channel. Often the filter, h(m; n), has been selected to be a Gabor filter or a filter derived from a wavelet transform, but other filters have also been employed [2]. Many approaches to feature extraction with filters use banks of filters (multichannel filters) which are more or less determined irrespective of the actual textures to be discriminated (see, e.g., [2] and [3]). Thus, there is no guarantee that the banks of filters will be able to work properly for a specific set of textures. Another unpleasant consequence of most filterbank approaches is that they are computationally demanding. For these reasons, the design of optimized filters for texture discrimination is gaining increased interest [4]–[16]. Optimization offers the potential of reduced feature dimensionality and thus reduced computational complexity and/or better feature separation. In this work, we develop and evaluate several approaches to the design of linear finite impulse response (FIR) filters with optimized energy separation. We present approaches for two-texture feature extraction with one filter and multitexture feature extraction with multiple filters. A necessary model for the feature mean and variance extracted by the system in Fig. 1 is developed, and this model is used for the filter optimization. The feature extraction techniques developed are advantageous in that they provide good feature separation along with low computational requirements. Examples on benchmark texture images are given, but the approach should be applicable in most texture applications with a priori class knowledge.
In filter optimization, the filter impulse response, h(m; n), or some other description of the filter, is determined. Several approaches to the design of Gabor filters optimized with respect to one texture representation [5], [6], discrimination of a pair of textures [10], [12], and discrimination of multiple textures [13], [14] have been presented. Optimization approaches for infinite impulse response (IIR) filters based on first-order allpass filter building blocks have also been presented [8]. However, both the Gabor and the IIR filters are filters with only a few free parameters, thus the search space is very restricted. A general form finite impulse response (FIR) [17] filter generally has many more free parameters than the above approaches. A few one-texture FIR filter optimization approaches have been presented [4], [15], [18], but design with respect to only one texture only guarantees optimized representation, not optimized discrimination. In the approaches by Cohen and You [7], [9] filters are optimized with respect to discrimination using random search. The search is accomplished by random draw of a filter, feature extraction with that filter, computing a criterion function, gradient estimation, semirandom draw of a new filter, etc. Jain and Karu [16] propose another approach to random search filter optimization using multilayer perceptron neural networks [19] for the filtering and local energy operations and the backpropagation training rule for the parameter optimization for multiple textures. The major drawbacks with these approaches are that they do not guarantee convergence to a global optimum in finite time and that the design complexity is likely to be very large for large filters. Mahalanobis and Singh [11] use yet another approach, determining the optimized filter by stochastic texture analysis with a closed-form solution. A related approach was presented by Unser [20] for design of an optimal transform for the two-texture classification problem. The ideas of Mahalanobis and Singh [11] and Unser [20] will be elaborated further in this article.
Manuscript received October 28, 1996; revised April 16, 1998. This work was supported in part by the Norwegian Research Council under Grant 110914/410. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Dmitry B. Goldgof. T. Randen is with Schlumberger Geco-Prakla, 4001 Stavanger, Norway (email:
[email protected]). J. H. Husøy is with Høgskolen i Stavanger, Stavanger, Norway. Publisher Item Identifier S 1057-7149(99)02452-5.
The feature extraction system used in this chapter is illustrated in Fig. 1. Basically, the purpose of the filter h(m; n) is extraction of local frequencies where some of the textures have low signal energy and other textures have high. If this is accomplished, a composite texture image can be segmented and the regions classified by analysis of their energies. Hence, the first block of our feature extraction system illustrated in Fig. 1 is the filter h(m; n). The output, y (m; n),
II. OVERVIEW
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is computed by
III. FEATURE EXTRACTION MODEL
y (m; n) = x(m; n) 3 h(m; n)
(1)
where 3 denotes two-dimensional (2-D) convolution. The energy of a signal, y , is defined as E fy 2 g and a local energy measure is obtained by smoothing y 2 . That is, the features are computed by nonlinearity:
z (m; n) = f (y(m; n)) = y2 (m; n)
(2)
v(m; n) = z (m; n) 3 w(m; n):
(3)
and smoothing:
A. Vector Formulation
A thresholding operation is applied for associating a class label with the extracted features. In Section III, we present a new model for the mean and variance of the features extracted with the system in Fig. 1. This model is used for optimization of a single filter with respect to the relative distance between the average feature values,
JU =
(v
0
v
v v
2 )
JMS =
(4)
0
(v v )2 2 v + v2
yi (m; n) = h(m; n) 3 xi (m; n) M
01
k=0
N
01
h(k; l)xi (m 0 k; n 0 l);
(7)
h(0; 0) .. .
h=
h(0; Nh 0 1) h(1; 0)
(8)
.. .
h(1; Nh 0 1) .. .
h(Mh 0 1; Nh 0 1) i.e., the vector formed by lexicographically ordering of the rows of
h(m; n), and
xi (m; n) .. .
xi (m; n) = (6)
where v2 and v2 are the feature variances. Now we need estimators for the feature means, v , and variances, v2 . We develop the framework for designing filters optimized with respect to this criterion in Section IV-B. However, we are not able to develop an exact closed form solution to this optimization problem, but an approximate closed-form solution is developed. Furthermore, the necessary mathematics for gradient search optimization is developed. In Section IV-C, common characteristics of the closed form two-texture solutions with respect to the different criteria are pointed out. A generalized solution is formulated on the basis of these common characteristics. Inspection problems, like defect versus nondefect detection, may in some cases be formulated as two-texture problems. However, except for inspection problems, the range of applications of twotexture approaches is limited. To extend the usefulness, we therefore develop a multitexture filterbank optimization scheme in Section V. We manage to develop an exact closed form solution based on optimization with respect to the mean values of the extracted features.
l=0
where h(m; n) is an Mh 2 Nh filter. This operation corresponds to the first block of Fig. 1. We define
(5)
was suggested by Mahalanobis and Singh [11]. However, tests included in this paper indicate that this is a considerably less powerful approach. The main drawback with the Unser criterion, JU , and the Mahalanobis–Singh criterion, JMS , is that they do not take the variances of the extracted features into account. Consequently, a filter found to be optimal may yield a considerable overlap between the features. A criterion well known in the pattern recognition literature that takes the feature variances into account is the Fisher criterion [21]:
JF (h) =
Denote texture number i by xi (m; n) where m and n are the spatial indices to the texture image. Filtering xi (m; n) by the filter h(m; n) yields
=
for two textures. Here, v and v are the model feature means. This criterion was originally suggested by Unser [20] for designing optimal texture transforms, but has not been applied for texture filter design earlier. An exact closed form optimal filter solution is detailed in Section IV-A. A related optimization approach with respect to the ratio between the average feature values
v v
In this section we develop estimates for the mean and variance of the extracted features using the feature extraction scheme illustrated in Fig. 1. The feature mean and variance models may be used for analyzing given filters or, as we will do later in this article, for designing optimized filters. In Section III-A, the feature extraction process is expressed in a matrix/vector notation, in Sections III-B and C, expressions for the feature mean and variance are developed. In Section III-D these expressions are assessed by experiments. We assume that the textures being filtered are wide sense stationary and that they are well described by their autocorrelation functions.
xi (m; n 0 Nh + 1) xi (m 0 1; n) .. .
:
(9)
xi (m 0 1; n 0 Nh + 1) .. .
xi (m 0 Mh + 1; n 0 Nh + 1) Then, the output at pixel (m; n) for texture i (7) may be rewritten as
yi (m; n) = hT xi (m; n):
(10)
Similarly, the smoothing operation (see Fig. 1) may be defined as
vi (m; n) = wT zi (m; n);
(11)
where w corresponds to the smoothing filter, w(m; n). This formulation may easily be extended to filter masks, h(m; n), of any shape, but for simplicity, we only consider filter masks with rectangular regions of support here. B. Feature Mean Assume that the input texture is wide sense stationary and that w(m; n) = 1. Then, using the vector formulation, it may m; n
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easily be shown that the expected feature value at the output of the filter, defined by v = E fvi (m; n)g, may be written as v = (hT Rx x h) (12) where
Rx x
=
E fxi (m; n)xTi (m; n)g:
(13)
By writing the elements of Rx x explicitly and assuming wide sense stationarity [22], it is readily seen that Rx x is constructed from the two-dimensional autocorrelation function of texture xi . The derivative of v is given by [23] @v @ (hT Rx x h) (14) = = 2Rx x h:
@h
@h
C. Feature Variance If we assume that yi (m; n) is Gaussian (assessed in Section III-D), the variance of the feature image vi (m; n) v2 = E f(vi (m; n) 0 v )2 g may be written M 01 N 01 M 01 N 01 v2 = 2 w(k1 ; l1 ) w(k2 ; l2 ) k =0 l =0 k =0 l =0 1 (hT Rx x (k1 0 k2 ; l1 0 l2 )h)2 0 24y (15)
Fig. 2. Filter response histograms from filters designed in the experiments illustrated in Fig. 7, i.e., closed-form optimization with respect to the Fisher criterion. The continuous curves illustrate the corresponding Gaussian distributions with the same means and variances.
where
Rx x (k; l) = E fxi (m; n)xTi (m + k; n + l)g; y = x 1T h4
f
g
(16) (17)
111
T x = E xi (m; n) , and 1 = [1 1 1] . The smoothing filter, w(m; n), is of size Mw Nw . The derivation is given in Appendix B, along with the expression for the partial derivative of v2 with respect to h.
2
Fig. 3. Estimated feature variance (vertical axis) versus true feature variance for a number of random drawn filters h(m; n) and the Brodatz texture D4.
D. Model Assessment The variance estimate above was based on the assumption that the output of the filter was Gaussian. This is an approximation which is generally not true. However, we tested distributions for several filters and several real-world textures from the Brodatz album [24] by plotting the histograms of yi (m; n). Examples of some histograms for yi (m; n) are shown in Fig. 2.1 How good are the total mean and variance models? In all experiments with the estimated feature mean, 1 : 1 matches between estimated and true were found. Slightly larger errors were encountered with the variance estimation. The true and estimated variances for a number of textures with several filters having Gaussian random drawn coefficients were tested. An example where this is apparent is shown in Fig. 3. In this experiment, filters of size 7 2 7 were designed. The smoothing filter, w(m; n), was Gaussian:
hG (m; n) =
p21 e0
(1=2)((m+n)
s
=
)
(18)
with s = 8 having a spatial extent of Mw 2 Nw = 33 2 33. The test image was the texture D4 [24]. As we see, the ratio between the true and modeled feature variance is approximately constant. A large number of textures were applied, and apparent linear relations were found in all cases, but the ratio was different for different textures. One possible explanation for this observation is that it is caused by problems with the estimates of the textures’ autocorrelation functions. Since the autocorrelation function is typically estimated from an image of finite size, there is a certain variance in the estimate 1 These distributions were generated by optimized filters of Section IV-D, Fig. 7.
depending on the image size. Furthermore, real-world textures are typically nonstationary, thus estimating the autocorrelation with the wide sense stationary assumption imposes some error in the autocorrelation estimate. Hence, reliable mean and variance estimates can only be expected if the autocorrelation function of the texture is estimated from a large stationary image. The estimate for the mean value of yi (m; n), y , was of a similar quality as the mean estimate for the feature images, vi (m; n). However, since the expression for v2 contains 4y [see (15)], even a small deviation in the estimate for y may yield a considerable error in the estimate of v2 . Due to the linearity of the filter, y is a linear function of the mean of the input texture. Hence, if the feature variance is small compared to the mean of the input texture, the variance estimate is likely to be unreliable. We do not consider this to be a major limitation because if the mean values of two homogeneous patterns differ significantly, they are easily discriminated by smoothing. In such a case, texture techniques are not necessary. Hence, we may subtract the mean prior to design and filtering in most cases where texture is of interest. IV. TWO-TEXTURE OPTIMIZATION APPROACHES In this section, several two-texture one-filter optimization approaches will be presented. In Section IV-A, an exact closed-form solution for a filter maximizing the relative average feature distance is developed and in Section IV-B an approximate closed form optimized solution with respect to the Fisher criterion is developed. In Section IV-C common characteristics are pointed out. Finally, in Section IV-D some experiments are presented.
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A. Optimization with Respect to Relative Average Feature Distance A classification algorithm is only successful if the features for the different classes are discriminable. Optimizing with respect to the relative distance between the average features
JU (h) =
0 v
(v
v v
2 )
(19)
is one way of attempting this. Optimization with respect to JU is easily developed and entails the solution of
@JU (h) @h
= 0:
The chain rule for differentiation yields
@JU (h) @h
@JU (h) @v @v @ h
=
+
(20)
@JU (h) @v : @v @ h
and their derivatives, (12), (14), (15), and (47), we have not been able to find a general closed-form solution. Then we have two alternatives, either to do some approximation and find an approximate closed form solution or to use some iterative search for the optimum. By approximating the textures as separable3 [25] autoregressive fields [22] of order one, AR(1),4 we are in Appendix C able to obtain the simplified variance expression
v2
2i (hT Rx x h)2 ;
(25)
h
where i is a scalar with empirically @i =@ 0. Substituting the mean and approximate variance expressions and the corresponding derivatives into @JF ( )=@ yields
h h h Rx x h h R h)(hT Rx x h))Rx x h T h h Rx x h)(hT Rx x h))Rx x h: = (1 (h R
T (2 (
(21)
2 T ) + 1 ( x x 2 T x x ) + 2 (
(26)
Combining (12), (14), and (21), we get the eigenvalue equation
R0x 1x Rx x )h = h
(
where
=
(22)
h
hT Rx x h) : (hT Rx x h)
(
(23)
R
h
h
B. Optimization with Respect to the Fisher Criterion A good feature extractor should not only yield a large distance between the mean values of the extracted features, but also yield low feature variances. This is exactly what is expressed by the Fisher criterion [21]
JF (h) =
0
(27)
We note that this solution is identical to the eigenproblem, 01 x x x x ) = , in the previous approach (22). The optimized filter coefficient vector is found by calculating the eigenvectors of the matrix ( x01x x x ) and selecting the vector maximizing the criterion JF ( ). This approach is similar to the optimization with respect to JU ( ) in that 1) the eigenvectors of ( x01x x x ) are found; and 2) the eigenvector yielding maximum criterion is selected. However, different criterion functions are used to select the eigenvectors in the two approaches. Hence, different eigenvectors may be selected. We note that the separable AR(1) approximation is a very rude approximation, but at least it enables us to find a closed form solution which experimentally proves to be very useful. An alternative to finding the closed-form solution to an approximate criterion function is to use some iterative approach optimizing a less simplified criterion function. A detailed discussion of iterative optimization techniques is beyond the scope of this article, but a brief discussion will be given. Gradient search is one potential optimization technique. The fact that we have expressions for the derivative of the criterion function is a distinct advantage with the gradient search technique. However, besides time consumption, the major drawback with the gradient search technique is that it is likely to converge to a local optimum. Tests with random start vectors indicated that this is indeed the case. We had, however, success using the closed-form solution as starting point and some results are given in Section IV-D. Other iterative techniques than gradient search are also possible and some experiments on simulated annealing, annealing evolution, and exact linear optimization have been reported [26], [27]. However, in order to find the global optimum, these techniques may require an infinite number of iterations and the experiments did not show any significant improvement over the alternative approaches.
R
(
R
h
h
R R h h
R
R
C. Unified Approach
(v v )2 2 v + v2
(24)
well known in pattern recognition. As with the previous approach, in order to find the optimized filter, we equate the partial derivative of the criterion, @JF ( )=@ , to zero. However, by inserting the expressions for the mean and the variance
h h
2 This
T Rx01x Rx x )h = ((hhT RRxx xx hh)) h:
(
The optimal filter coefficient vector is found as the eigenvector of 01 x x ) with the maximum corresponding criterion value, ( x x JU ( ). The proof that this is a valid solution to the optimization problem, i.e., that all eigenvectors will satisfy (23) where is the corresponding eigenvalue is given in Appendix A. We note that this approach yielded an exact closed-form optimized solution, guaranteeing that the global optimum is attained. However, the major conceptual drawback with this method is the fact that the variances are not included in the criterion. Consequently, the optimized filter may yield a large feature mean separation, but with large feature variances. Hence, the density functions for the features for the two textures may have a considerable overlap, yielding poor classification. Optimization with respect to the Mahalanobis–Singh criterion (5) proceeds similarly [11], but yields far less powerful features as indicated by the experiments in Section IV-D. The only plausible explanation we have found why the Unser criterion, JU , yields better features than the Mahalanobis–Singh criterion is obtained by rewriting the criterion as JU ( ) = v =v + v =v 0 2. Even if it is inherently difficult designing a filter yielding high feature values for texture 1 and low for texture 2,2 the Mahalanobis–Singh criterion will lead to the selection of a filter attempting to emphasize texture 1, i.e., maximize v =v . However, in this particular case, the Unser criterion may lead to the selection of a filter emphasizing texture 2 so that the term v =v is large.
R
This equation may be simplified to
may, for example, be the case if the energy of texture 1 is generally low compared to texture 2.
As we have seen, the closed form solutions presented thus far yield the same eigenproblem ( x01x x x ) = , where the optimized filter coefficient vector is the eigenvector with maximum criterion
R
R
h
h
3 A separable image may be expressed by separate one-dimensional generating equations in the vertical and horizontal dimensions. 4 A one-dimensional AR(1) function x(n) may be written x(n) = ax(n 1) + w (n), where w (n) is noise (often called innovation process) and a is a constant.
0
0
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Fig. 4.
575
(a)
(b)
(c)
(d)
Textures (a) D4-D84, (b) D21-D105, (c) D28-D29, and (d) D9-D24 from [24].
value. Indeed, the same eigenproblem is also the solution to the generalized criterion function
J
=
1 2v 1 2v
+ 2 v
v + 2 v v
2 2 2 + 3 v + 4 v + 5 v + 3 2v + 4 v2 + 5 v2
h
(28)
where the i s and i s are arbitrary scalar constants. All the criterion functions we have considered thus far are special cases of this formulation. Given the same assumptions and following the same steps as in Section IV-B, the closed form optimization yields the eigenproblem T R0x 1x Rx x )h = ((hhT RRxx xx hh)) h:
(
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(29)
As we see, this solution is independent of the i ’s and i ’s. The proof is straightforward, but tedious. We also note that if we omit the variance terms, the only assumption we have to make to obtain the closed-form solution is that the input textures are wide sense stationary. D. Experimental Results Several homogeneous texture pairs have been used as test images with the methods presented in this paper, and the ones selected for this article illustrate the performances well. The test images are shown in Fig. 4. In all experiments, the analyzed textures have zero mean and have been histogram equalized. We designed an Mh 2 Nh = 7 2 7 filter, h(m; n), letting the smoothing filter, w(m; n), be Gaussian, (18) with the empirical value s = 8 and a 33 2 33 region of support. The size of the smoothing filter is a trade-off between feature variations and edge accuracy. The results with the Unser criterion are shown in Fig. 5 and the Mahalanobis–Singh criterion in Fig. 6. The Fisher criterion closed form and gradient search optimized results are shown in Figs. 7 and 8, respectively. In the feature plots, the true texture border is indicated by a discontinuity in the surface. The class maps were made by thresholding with a threshold yielding minimum classification error. The results are summarized in Table I and a few feature histograms are shown in Fig. 9. The gradient search overall doubled the estimated Fisher criterion value over the closed-form solution. However, due to the generally marginal classification error, typically only moderate improvements were observed on the classification error. In one case the classification accuracy is even reduced.
Fig. 5. Features optimal with respect to JU , (4) for textures (a) D4-D84, (b) D21-D105, (e) D28-D29, and (f) D9-D24 and their corresponding class maps (c), (d), (g), and (h). The true texture boundaries are indicated by discontinuities in the feature energy surfaces and the horizontal and vertical axes of the texture images (Fig. 4) are indicated by “Hor.” and “Vert.” on the surface plots.
As we can easily see, the Mahalanobis–Singh criterion optimized features has considerable overlap in Fig. 6(e) and (f), while the Unser criterion optimized features have too much overlap in Fig. 5(f). Clearly, these optimization approaches yield features that are nondiscriminable for the D9-D24 texture.
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(a)
(b)
(a)
(b)
(c)
(d)
(c)
(d)
(e)
(f)
(e)
(f)
(g)
(h)
(g)
(h)
Fig. 6. Features optimal with respect to JMS (5) [11] for textures (a) D4-D84, (b) D21-D105, (e) D28-D29, and (f) D9-D24 and their corresponding class maps (c), (d), (g), and (h), respectively.
Fig. 7. Features optimized with the closed-form solution JF (24) for textures (a) D4-D84, (b) D21-D105, (e) D28-D29, and (f) D9-D24 and their corresponding class maps (c), (d), (g), and (h), respectively.
Not even the Fisher criterion is ideal for all cases, as is illustrated in the results for D21-D105, Fig. 7(d). For this image, we see that the features extracted according to the JU and JMS criteria are significantly better separated than the features extracted according to the JF criterion with closed-form solution. The reason for the poor performance of the Fisher criterion in this case is unclear. However, it is certain that the Fisher solution utilizes larger lags in the autocorrelation function than the other approaches. Larger lags are more susceptible to noise due to the size of the training data and due to inhomogeneities. This may yield inaccurate variance estimates, and consequently inaccurate criterion evaluation. It is evident from the presented results that none of the criteria presented thus far are better than the rest for selection of the eigensolution in all cases. An alternative approach may be to apply all 1 filters corresponding to the eigenvectors of ( 0 x x x x ) and select the one yielding maximum separation of the histograms. This solution is motivated by the unified solution of Section IV-C. The results using this approach are shown in Fig. 10 and in Table I. This approach will be equal to or better than any of the closed form experiments in all cases. However, filters optimized without the closed-form solution, e.g., gradient search, may be better. The texture pair D9-D24 is an example of this. Here, the gradient search yields 5.2% error, while the best closed-form solution yields 7.3%. To illustrate what the optimized filters may look like, we give examples of amplitude responses in Fig. 11. Two examples of amplitude spectra illustrated as gray-level images in conjunction with the amplitude spectra of the two corresponding textures are shown in Fig. 12. This texture set, D21-D105, was chosen due to the
TABLE I CLASSIFICATION ERRORS FOR ALL EXPERIMENTS REPORTED IN THIS ARTICLE
R
R
clarity of the spectral peaks. If we consider the spatial frequency properties of an optimized filter, we would intuitively expect it to emphasize frequencies discriminating the textures well. That is, frequencies where one texture has strong components and the other weak. If we study Fig. 12 carefully, we can see that this is the case for this example. For this texture set, the Fisher criterion did not yield very good features, as we intuitively understand from visual inspection—the Unser criterion is highlighting a more discriminative region of the frequency spectrum. For reference, we also include results for the same textures using the Gabor filter optimization approach by Dunn and Higgins [12] in Fig. 13. The Gabor filters are sine modulated Gaussians and in the experiments we designed and applied filters with Gaussians with widths given by = 8 pixels. The nonlinearity and the smoothing filter were the same as in the other experiments. As we can see from the figures, this approach was incapable of performing successful segmentation of the difficult images in Fig. 4(c) and (d).
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(a)
(b)
(a)
(b)
(c)
(d)
(c)
(d)
(e)
(f)
(e)
(f)
(g)
(h)
(g)
(h)
Fig. 8. Features optimized with gradient search for JF (24) using the closed-form solution as starting point for textures (a) D4-D84, (b) D21-D105, (e) D28-D29, and (f) D9-D24, and their corresponding class maps (c), (d), (g), and (h), respectively.
(a)
(c)
Fig. 10. Features corresponding to optimal filter selected yielding minimum histogram overlap for textures (a) D4-D84, (b) D21-D105, (e) D28-D29, and (f) D9-D24, and their corresponding class maps (c), (d), (g), and (h), respectively.
(b) (a)
(b)
(c)
(d)
(d)
Fig. 9. Feature histograms visualizing feature separation for texture pair D28-D29 for filters designed with respect to (a) JU , (b) JMS , (c) JF closed form, and (d) JF gradient search.
In an extensive experiment, we calculated the classification error for all approaches for 276 texture pairs. For computational reasons, the gradient optimized filters were excluded from this analysis. Assume that errors below 5% correspond with edge effects and are “acceptable.” Then we found that the Unser criterion did yield unacceptable features in 4% of the cases, Mahalanobis–Singh [11] in 29%, Fisher in 13%, the histogram-approach in 1%, and the Dunn [12] approach in 14%. The Mahalanobis–Singh [11] approach was
Fig. 11. Amplitude responses for the filters optimal with respect to JF (24) closed-form solution for the texture pairs (a) D4-D84, (b) D21-D105, (c) D28-D29, and (d) D9-D24.
clearly less robust than the other approaches. Some problems were also encountered with filters optimized with respect to the Fisher criterion and with the Dunn [12] approach. The most likely cause of the problems with the Fisher-optimized filters is the sensitivity of the variance estimate with respect to nonstationarities. The performances of the Unser- and Fisher-criterion optimized filters were generally
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(a)
(b)
(c)
(d)
Fig. 12. Amplitude spectrum (log-scale) (a) for texture D21 and (b) for texture D105 and amplitude responses of the filters optimized with respect to (c) JU (4) and (d) JF (24).
complementary. That is, when one failed, the other succeeded in nearly 98% of the cases. The Unser criterion was yielding best features in about 60% of the cases. The focus of this paper and the experiments herein have been on supervised segmentation of benchmark images. However, the introduced techniques have also been applied to other problems. An application to unsupervised texture segmentation was presented by Randen et al. [28]. A broad comparison of the techniques with techniques covering most approaches to filtering for texture separation is reported by Randen [29]. It turns out that these approaches are superior. Furthermore, Gulsrud et al. provide examples of these approaches successfully applied to detection of stellate lesions [30], circumscribed masses [30], and microcalcification [31] in mammograms.
V. MULTITEXTURE FILTERBANK OPTIMIZATION In the previous section, we saw some filter optimization approaches for two-texture discrimination. This may be interesting in some inspection applications for discriminating normal from abnormal. However, many inspection problems and the majority of texture segmentation and classification problems involve more than two textures. Consequently, extensions of the solutions to be applicable for multiple textures are desirable. To have reliable features for more than two textures, we generally believe that more than one filter is needed. We will address the design and application of multiple filters for multiple texture problems in this section.
The straightforward approach for generalizing the two-texture optimization framework to solutions for more than two textures is to design one filter for each possible pairing of two textures in the problem. This approach was, for instance, applied to the filter design method of Dunn et al. [10] by Randen and Husøy [2]. This is a viable approach as long as the number of textures is moderate. However, as the number of textures increases, the number of filters gets very large, e.g., ten textures give 45 filters and 20 textures give 190 filters. Hence, we need another way of handling such problems. Building on the two-texture approaches, we suggest a hierarchical feature extraction scheme where each filter separates one group of textures from another. The textures within the groups are separated by subsequent filters recursively. The problems that need to be solved are as follows. 1) How do we design a filter for optimized feature separation between two groups of textures? This issue is addressed in Section V-A. 2) How do we form texture groups allowing robust feature extractors to be constructed? This issue is addressed in Section V-B. Finally, the section is concluded in Section V-C with experimental results. A. Optimized Texture-Group Filters We recall from (12) that the mean value of the features extracted from texture xi (m; n) by the filter h(m; n) can be written as v = ( T x x ). If the textures x1 (m; n) 1 1 1 xI (m; n) are
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of the components. A consequence of (30) is that the feature mean estimated using the autocorrelation function for the entire texture is approximately equal to the average feature means of its components. The same is not true for the variance. B. Texture Grouping (a)
(b)
(c)
(d)
(e)
(f)
If the textures in one group share some characteristics that are less prevalent in another group, good filters may probably be designed. In order to make such a grouping, some measure of the similarity is needed. It can not be expected that very dissimilar textures may be grouped with success. Of course, filters corresponding to all possible group pairs of the textures may be designed and the expected feature statistics computed for each texture to determine which grouping is best. With many textures, such an approach becomes exhaustive. For instance, with 16 textures, there are 215 0 1 = 32 767 different group pairs. Hence, another way of determining the group pairs is desired. In order to form two texture groups that may be successfully separated, we must ensure that the maximum distance between any two textures within a group is less than the minimum distance between any two textures between the groups. More formally, assume that the I textures xi (m; n) 1 1 1 xi (m; n) are one texture group and the J textures xj (m; n) 1 1 1 xj (m; n) are another. The conditions for any group pair to yield successful filters are then
fd
g < minfd
g 8 k; l; p < I; q < J
(31)
fd
g < minfd
g 8 k; l; p < J; q < I:
(32)
max
and
max
(g)
(h)
Fig. 13. Features extracted with optimal Gabor filter [10] for textures (a) D4-D84, (b) D21-D105, (e) D28-D29, and (f) D9-D24, and their corresponding class maps (c), (d), (g), and (h).
filtered by the same filter, the overall mean feature value for all textures will be
v
=
1
I
(v +
111 +
v
)
h R h) + 1 1 1 + (h R h)) 1 =h + 111 + R )h (R I h (30) =h R where R is the average R . This expression is identical to the expression for the mean of one texture if we replace R for that texture with the average R ’s of the textures in the group, R . Hence, by replacing R and R with the R ’s for the =
1
I
((
T
T
T
x
T
x x
x x
x
x x
x x
i ;i
j ;j
i ;j
j ;i
The measures di ; i and dj ; j are intragroup distances and di ; j and dj ; i are intergroup distances. As similarity measure, we use the criterion value for the optimized filter for the two textures, e.g., if the filter is optimized with respect to the Unser criterion, we define d = 0JU . A disadvantage with the above conditions is that the textures are fragmented into a lot of groups, since typically only small groups satisfy conditions (31) and (32). To be able to separate two groups, one filter is needed and with many groups the number of group pairs will be large, hence a large number of filters must be applied. To reduce the number of filters, we therefore add textures to the groups as long as all the modeled feature means for one of the groups are above the means for the other. In this approach, no account is made for the variances, although this may easily be added.
x
x
x x
x x
x x
x x
x
x x
x
x
x
two texture groups in the approaches of Section IV-A, the same filter design approaches may be used. Including the exact closedform solution, this relation can easily be shown. We have not been able to derive a similarly simple expression for the variance of a group of textures, even with the assumptions we made in Appendix C. Consequently, a closed-form solution for the Fisher criterion for multiple textures is not found. Still we may use iterative search, but due to complexity issues this is not very practical and has not been tested. If a texture is nonstationary (nonhomogeneous), it may in some cases be considered as being composed of multiple different textures. Then the autocorrelation function estimated over the entire texture image will be approximately equal to the average of the autocorrelation functions of its components, at least for small lags. Hence x x for the entire texture is approximately equal to the average x x ’s
R R
C. Experimental Results A result from this approach with the Unser criterion for the textures D4, D9, D19, D21, D24, D28, D29, D36, D37, and D38 [24] is shown in Fig. 14. There is just a moderate amount of misclassified regions, although some of the textures are visually very similar. The texture grouping yielded 17 texture group pairs where four were duplicates, i.e., 13 filters were designed. The filtering order is practically hierarchical and is illustrated in Fig. 15. Note that due to the practically hierarchical nature of the solution, it is not necessary to compute filter responses for all image positions. It is, for instance, not necessary to apply the filter separating between classes 0 and 5 to regions of the image that are already recognized as belonging to one of the classes 1 or 6. As a consequence, the effective number of filters is less than 13 in this case, which is very low compared to many filter bank approaches, e.g., [2] and [3]. In an artificial mosaic of textures as the one in Fig. 14(a), the edges between textures are very discontinuous, i.e., having large highfrequency contents. The feature extraction filter may have highpass characteristics. Hence, it may have a boosted response at the edge
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(a)
(b) Fig. 14. (a) Ten-texture image and (b) segmentation by the multitexture approach, in this instance yielding 13 filters.
1) optimized exact closed form two-texture filter design with respect to the Unser criterion; 2) optimized approximate closed form two-texture filter design with respect to the Fisher criterion; 3) optimized approximate closed form two-texture filter design with respect to a generalized criterion, using feature histogram overlap as the filter selection criterion; 4) extensions to optimized multiple texture filterbank design In order to find closed-form solutions to the optimization problems incorporating the variance, we had to use an approximated variance model. The two-texture design techniques were thoroughly tested on several texture images. The performances of the design techniques based on the Unser and Fisher criteria were very good in most cases. Best performance was, however, achieved with the histogram-based technique resulting from the unified approach. It is important to keep in mind that none of the criteria are guaranteed to have a direct relation to the classification error, the ultimate feature goodness criterion. Issues that are not addressed in this work are how to determine the ideal filter size and the ideal smoothing filter size and type. These parameters have impact on the feature extraction results and should be addressed in future work. Generally, compared to using fixed filterbanks as in [2] and [3], the main disadvantages with an optimized approach are the need for better insight and a more complex design. The main advantages are the optimality, fewer filters and hence fewer features, and much simpler classification (thresholding with one feature is far less complex than applying a neural net or statistic classifier with multiple features). Furthermore, compared to the commonly used Gabor filters [3], [10], [12], [14], [32], successful results are obtained with significantly smaller, i.e., less computationally complex filters. VII. MATHEMATICAL DETAILS
Fig. 15. Texture grouping structure for the segmentation in Fig. 14. The numbers refer to texture number, where the textures in Fig. 14(a) are numbered zero to four from left to right in the first row and five to nine in the second.
instead of a smooth transition between the texture regions. In combination with the local energy function, this implies reduced edge accuracy. This effect is, e.g., evident around the second texture from the right in the top row of Fig. 14(b). Such a boosted response will probably not occur to a great extent in real-world texture mosaics since texture edges in real-world images are likely to be much smoother. However, if strongly discontinuous edges are present, this problem can be alleviated by edge modeling, Appendix D, and compensation as in [28]. Other alternatives are neighborhood examination [32] or application of smaller filters or smaller smoothing windows in the regions near the edges. In order to keep this paper at a manageable size, we do neither detail nor show any examples of this here. VI. SUMMARY
AND
CONCLUSIONS
In this work, we have developed estimators for the mean and variance of texture features extracted with a texture filter, a quadratic nonlinearity, and a smoothing filter. The mathematical models for these estimators formed the basis for the development of the following filter optimization procedures:
In the Appendices, we give the mathematical details for several of the derivations presented earlier in this work. In Appendix A, we give a validation of the existence of the eigensolutions found earlier, in Appendix B, we develop the details of the feature variance model, and in Appendix C we develop an approximation of the variance expression useful in the closed form solutions. Finally, in Appendix D, we discuss modeling of edge and boundary effects using the developed model. APPENDIX A VALIDATION OF EXISTENCE OF OPTIMIZED SOLUTIONS Are we guaranteed that all solutions to the eigenproblems (22), (27), and (29), are valid solutions? That is, are we guaranteed that the solution to
R0x 1x Rx x )h = h will yield a solution vector h such that T (h Rx x h) = (hT Rx x h) (
(33)
(34)
hR h hR h
where is the corresponding eigenvalue? Since ( T x x ) is a scalar, is unmodified by dividing and multiplying by ( T x x ), i.e., T T ( ) x x = T x x (35) = : ( ( T x x ) x x )
hR h hR h
hR hR
Combining this with (33), we get T 01 ( x x ( x x x x ) = T ( x x )
hR
R R hR h
h h
h = (hT Rx x h) (hT Rx x h)
(36)
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which is identical to the requirement of (34). Hence, all eigensolutions are valid solutions.
v
2
vi (m; n), is
f i 0 v g fi g 0 2E fvi(m; n)gv + 2v T T 2 = E fw zi (m; n)zi (m; n)wg 0 v T 2 = (w Rz z w) 0 v = E (v (m; n) 2 = E v (m; n)
where
Rz z
=
2 )
E fzi (m; n)zTi (m; n)g:
yielding
v2
(37)
@v2 @h
y2 y (0; 0) (40) where cy y (k; l) is the covariance function [22] of yi (m; n). That is cy y (k; l) = E f(yi (m; n) 0 y )(yi (m + k; n + l) 0 y )g 2 (41) = ry y (k; l) 0 y with ry y (k; l) being the autocorrelation function of yi (m; n). Hence, we have
rz z (k; l) = 2(ry y (k; l) 0 y 2 2 + 4(ry y (k; l) 0 y )y + ry y 2 4 2 = 2ry y (k; l) 0 2y + v : Furthermore, we have
where
Rx x (k; l) = E fxi (m; n)xTi (m + k; n + l)g:
v2
T Rz z w) 0 2v M 01 N 01
= (w =
=
w(k1 ; l1 )
M 01 N 01
k =0 l =0 k =0 l =0 1 rz z (k1 0 k2 ; l1 0 l2 ) 0 2v M 01 N 01 M 01 N 01 w(k1 ; l1 )
i
1 (0; 0),
where
M 01 N 01
M 01 N 01
w(k1 ; l1 )
M 01 N 01
k =0 l =0 k =0 l =0 1 rz z (k1 0 k2 ; l1 0 l2 ) 0 2v ;
w(k2 ; l2 )
w(k2 ; l2 ) (48)
r^z z (k; l) = i1 r^z z (k 0 1; l) 8 k > 0
(49)
r^z z (k; l) = i2 r^z z (k; l 0 1) 8 l > 0
(50)
and i2 are scalar parameters and r^z we may write
rz z (k; l) r^z z (k; l) = jik1j jil2j rz z
z (0; 0)
(0; 0):
=
(51)
We note that the approximations in (49) and (50) correspond to modeling the image as separable and autoregressive (AR) [22] of order one.5 With this approximation, we may write
(43)
(44)
M 01 N 01
w(k1 ; l1 )
M 01 N 01
w(k2 ; l2 )
k =0 l =0 k =0 l =0 1 jik1 0k jjil2 0l jrz z (0; 0) 0 2v M 01 N 01 M 01 N 01 = rz z (0; 0) w(k1 ; l1 ) k =0 l =0 k =0 l =0 w(k2 ; l2 )ijk1 0k j jil2 0l j 0 2v :
(52)
To keep expressions manageable, we define
i =
M 01 N 01
w(k1 ; l1 )
M 01 N 01
w(k2 ; l2 )
k =0 l =0 k =0 l =0 j k 0 k j j l 0 l j (53) 1 i1 i2 2 2 thus v i rz z (0; 0) 0 v . Applying (42) and assuming zero
mean input, we get
w(k2 ; l2 )
v2
w(k2 ; l2 )2
k =0 l =0 k =0 l =0 2 1 ry y (k1 0 k2 ; l1 0 l2 ) 0 24y + 2v 0 2v
=
and
v2
Note that from (13), we have Rx x = Rx x (0; 0). Combining (37), (38), (42), and (43), we are able to estimate the variance of the extracted features, v2 , given the autocorrelation function of the texture, rx x (m; n), the filter, h(m; n), and the smoothing filter w(m; n). Summarizing the derivation of this Appendix, we may write the expression for v2 explicitly as
w(k1 ; l1 )
but as described in Section IV-B, this is not found to yield a closed form solution. However, if we approximate rz z (k; l) by r^z z (k; l), where
(0; 0)
(42)
M 01 N 01
k =0 l =0 k =0 l =0 1 (hT Rx x (k; l)h)(Rx x (k; l)h + Rx x (k; l)T h) (47) 0 8(x 1T h)3 x 1:
v2
rz z
ry y (k; l) = E fyi (m; n)yi (m + k; n + l)g T T = E fh xi (m; n)xi (m + k; n + l)hg T = (h Rx x (k; l)h)
w(k2 ; l2 )
From Appendix B, (45) we know that it is possible to write the variance as
(39)
+r
2 ) 2
M 01 N 01
APPENDIX C APPROXIMATE VARIANCE EXPRESSION
We have not been able to find any useful variance model including this fourth order moment. However, if we assume that yi (m; n) is Gaussian, we may write this [22] as
2
=8
(38)
rz z (k; l) = E fzi (m; n)zi (m + k; n + l)g 2 2 = E fyi (m; n)yi (m + k; n + l)g:
w(k1 ; l1 )
differentiation rules of Searle [23], we get the partial derivative
The matrix Rz z is readily constructed from the autocorrelation function of the image zi (m; n), given by
rz z (k; l) = 2c2y y (k; l) + 4cy y (k; l)y2
M 01 N 01
=2
k =0 l =0 k =0 l =0 1 (hT Rx x (k1 0 k2 ; l1 0 l2 )h)2 0 2(x 1T h)4 (46) where 1 = [1 1 1 1 1 1]T and w(m; n) is the smoothing filter which is of size Mw 2 Nw . Consequently, using the matrix/vector
APPENDIX B VARIANCE EXPRESSIONS The variance of the feature image,
581
(45)
2i ry2 y (0; 0) + ry2 y (0; 0) 0 2v T 2 T 2 T 2 = 2i (h Rx x h) + (h Rx x h) 0 (h Rx x h) T 2 = 2i (h Rx x h) : (54)
5 We note that, generally, making these approximations (51) makes the estimate of r (k; l) less accurate. The approximations are, however, justified by the useful results they give.
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To be able to optimize, we need the partial derivative of this approximation of v2 with respect to . As we see, the approximation of v2 consists of two elements, ( T x x )2 and i which are both functions of . We tested the relative perturbations of ( T x x ) and i as functions of small perturbations of . These tests indicated that the relative perturbations of ( T x x ) were generally much larger than the relative perturbations of i . Hence, approximating i as constant will only perturbate the derivative of v2 minimally and we get the approximation
h hR h
h
h hR h
hR h
@v2 @
T (55) h 8i(h Rx x h)Rx x h: When substituted into @JF =@ h = 0, these approximate expressions
yield the approximate optimized closed form solution given in Section IV-B. APPENDIX D TEXTURE EDGE AND BOUNDARY MODEL
The estimate for the mean value of the extracted features given by (12) was based on the assumption that xi (m; n) was wide sense stationary (WSS). However, when computing the expected feature value somewhere on the edge between two textures or on the boundary of the image, the WSS assumption is violated. By writing the elements of x x (k; l) explicitly for a given edge location, it is readily seen that x x (k; l) may be created from the autoand cross-correlation functions of the two textures making up the edge and similarly for border regions. Using this, we can model and compensate for the feature response along edges of any desired orientation and shape. An application of this was given by Randen et al. [28].
R
R
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