Textures can be well modeled as stationary 2-D random processes. However, traditional methods characterize this 2-D random process only in terms of first and second order statistics and therefore cannot completely characterize non-Gaussian or asymmetric random processes. Here, we implement tests based on higher-order statistics (and spectra) to investigate the validity of modeling assumptions. We test the linearity, Gaussianity and spatial reversibility of textures using the tests developed in [1]. We find that textures are linear, non-Gaussian and spatially irreversible. Therefore, higher-order spectra are very useful in modeling textures.

We mainly use the bispectrum (the Fourier transform of the third-order cumulant) to test linearity, Gaussianity and spatial reversibility. These tests are based on the properties of the bispectrum of a linear 2-D random process. In order to understand the bispectrum of a 2-D linear random process, consider a 2-D process generated by passing zero-mean, independent, and identically distributed noise through a linear filter. Such a model is called a 2-D autoregressive moving average (ARMA) model.

The power spectrum of the output linear process, x(m,n), S_2x(u), is given by

and the bispectrum, S_3x(u,v), is given by

where s_2wis the variance of the input noise process, w(m,n), and c_3w is the third-order cumulant (which is equal to the third-order central moment).

Using the power spectral density and the bispectrum, we can define the bicoherence function, b(u,v), as

A process is linear if it is generated by a linear system as described above and it is Gaussian if the input noise process is Gaussian. A 2-D random process is spatially reversible if and only if the joint distrribution of a set of points indexed in 2-D by the set of indices {i} is the same as the joint distribution of the set of points indexed by -{i}.

The three tests are based on the following three properties.

1. For an input Gaussian process, the third-order cumulant is zero. Therefore, the bispectrum and the bicoherence function are identically zero for all u and v.
2. For a linear process, we can easily see from the expressions for the power spectrum and the bispectrum that b(u,v) is constant.
3. For a spatially reversible process, the imaginary part of the bispectrum is zero. See [1] for details.

For the Gaussianity test, we first make an estimate of b(u,v) from the given texture over the nonredundant region of the bispectrum [1,3]. Then we compute the sum of the magnitude squared values of the estimates over the nonredundant. This statistic is then compared to a threshold to check if the texture is Gaussian. The estimated statistic is asymptotically distributed as a chi-squared distribution. The number of degrees of freedom of this chi-squared distribution is equal to twice the number of nonredundant points in the computed bispectrum. Under the null hypothesis, the image is assumed to be Gaussian. Then, a Neyman-Pearson criterion is used to set the threshold given a predetermined probability of false alarm. Using this probability and a chi-squared distribution tail probability calculator [18], the threshold can be determined. If the computed statistic is greater than the threshold the Gaussianity hypothesis is rejected. In our case, we set a false alarm probability of 0.05 to compute the threshold.

Linearity Test

For the linearity test, we would like to measure how much the estimate of the bicoherence varies and compare that to a threshold. A statistical test based on the Neyman-Pearson criterion can be obtained in this case too. The variability in the bicoherence is estimated using the sample dispersion. The dispersion of a random variable, X, is measured as the interquantile range q₃ - q₁, where q₁ and q₃ are defined such that P[X < q₁] = 0.25 and P[X < q₃] = 0.75. The sample dispersion is asymptotically Gaussian distributed around the actual dispersion. The Neyman-Pearson criterion is used to set the threshold given a fixed false alarm probability. If the dispersion is too high the linearity hypothesis is rejected.

Spatial Reversibility Test

The spatial reversibility test is very similar to the Gaussianity test. The only difference is that we compare the imaginary part of the bicoherence to zero instead of the bicoherence. The statistic is obtained as the sum of the squares of the imaginary part of the estimated bicoherence over the nonredundant region. The number of degrees of freedom of the chi-squared distribution is equal to the number of points in the nonredundant region.

Results of the Statistical Tests

The statistical tests were performed on Brodatz textures obtained from [21]. Two textures, grass lawn and herringbone weave, and the results for them are shown below. The power spectrum and bispectrum of the image are estimated over blocks of 32 x32 and averaged to get a consistent estimator. A 256 x 256 image is considered. So, there are 64 blocks over which the averaging is done.

Grass Lawn

 

Results for Grass Lawn
 

Test	Threshold	Statistic	Conclusion
Gaussianity	118399	124020	Not Gaussian
Linearity	2.3409	2.3147	Linear
Spatial Reversibility	59365	61560	Spatially Irreversible

Herringbone Weave

Results for Herringbone Weave
 

Test	Threshold	Statistic	Conclusion
Gaussianity	118399	132490	Not Gaussian
Linearity	2.5020	2.4594	Linear
Spatial Reversibility	59365	61973	Spatially Irreversible

Finally, we show the results of just the Gaussianity and spatial reversibility test for an independent and identically distributed Gaussian noise texture. This texture is correctly determined to be Gaussian and spatially reversible. It is spatially reversible because it is independent and identically distributed. This can be easily verified.

Results for Gaussian noise texture
 

Test	Threshold	Statistic	Conclusion
Gaussianity	118399	117280	Gaussian
Spatial Reversibility	59365	58925	Spatially Reversible

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last updated on 5th May 2000