Regular Texture Fault Detection by Wold-Like Decomposition Sriram Subramanian , Kap Luk Chan , and Xiaoran Mo 1
1
2
2
Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore, India 2 School of Electrical & Electronics Engineering, Nanyang Technological University, Singapore e-mail:
[email protected]
Abstract: Detecting faults in regular textures typically involves identi cation of local structures, such as knots, worm holes, etc., that destroys the normally uniform regular textured pattern. In this paper, we present a texture fault detection algorithm based on the Wold decomposition model. According to this model a texture can be decomposed into a harmonic part and a stochastic part. We postulates that the fault in the regular texture is mainly encapsulated in the stochastic part. Thus by proper exploitation of the harmonic and the stochastic part we are able to locate the fault in a texture. Keywords: Texture Fault detection, Wold Texture Model.
1 Introduction Texture is an important element of vision and has been analyzed in its own right for the last two decades by researchers in psychophysics as well as in computer vision. Texture, being a contextual property, seems to defy any attempt at de nition since image context can be of in nite variety. As a result interpretation of what constitutes a texture varies widely. Textures play a critical role in inspecting surfaces that are produced at various stages in all types of manufacturing, such as semiconductor wafer inspection [1] and lumber processing. Defects in wood are rich in texture content and typically involve the identi cation of knots, worm holes, etc. example a
example b
example c
example d
Figure 1: Some examples of textured patterns with defects Figure 1 shows some regular texture with defects/faults. A quick glance at these gures is enough to locate the faults. Essentially what we locate are regions where the structure of the texture pattern is destroyed. Thus heuristically a fault(or defect) in a texture can be de ned as that region of the texture where the structure of the texture is destroyed. Such a de nition of the fault suggests that if we can split the texture into a structured part and the unstructured part then the structured part should contain the homogeneous texture with no faults and the unstructured part should contain regions with the fault. Here we use a 2-D Wold-Like decomposition to accomplish the above.
The paper is organized as follows. Section 2 describes the concept and theory involved with the Wold Model and gives a Spectral Decomposition Algorithm. Section 3 describes the Wold based Fault Detection Scheme used and discusses the Results obtained. Section 4 has the conclusion.
2 2-D Wold Decomposition Model and Spectral Decomposition Algorithm A 2-D Wold-like decomposition is a generalization of the 1-D Wold decomposition [2] and can be stated as follows.
Theorem 1 If fy(m; n)g is a 2-D regular and homogeneous random eld, it can be uniquely represented by the orthogonal decomposition where
y(m; n) = w(m; n) + v(m; n)
w(m; n) =
X
;
k;l)
a(k; l)u(m ? k; n ? l)
(1) (2)
(0 0) (
and E [v(m; n)] = E [u(m; n)] = 0. Symbol E [] denotes the expected value. Field fv(m; n)g is deterministic and eld fw(m; n)g is regular and purely indeterministic. The innovation eld fu(m; n)g is white, i.e., E [u(m; n)u(s; t)] = 0; 8(m; n) 6= (s; t). Fields fv(m; n)g and fu(m; n)g are orthogonal, i.e., E [v(m; n)u(s; t)] = 0; 8(m; n) and (s; t) 2 Z 2. Thus elds fv(m; n)g and fw(m; n)g are also orthogonal.
This theorem gives the basic decomposition of a regular eld into its deterministic and purely indeterministic components. The deterministic component can be further decomposed into the harmonic and the evanescent component. For the purpose of this work it was found sucient to just decompose the texture to the deterministic part and the indeterministic part. The e ectiveness of an algorithm must be gauged from the context of their application. For defect detection it is important that the form of the features extracted facilitates easy location of the defect. Towards this end a periodogram thresholding scheme [5] is used to decompose the image. The main reason for choosing this algorithm over the more robust and ecient spectral estimation algorithm of [3] has been the computational simplicity of this procedure. To make the thresholding adaptive to the distribution of the Fourier Magnitude Spectrum (FMS) rather than the maximum value in the FMS, the histogram of the FMS is taken and mirrored to form a Guassian like distribution. The standard deviation, , of the distribution is used and the threshold is set at 3. A smaller threshold will include more weak harmonic components and a high threshold will reduce the number of harmonic components selected to represent the deterministic parts of a regular texture. The spatial and frequency samples are indexed by (m; n) and (k; l) respectively, where m and k are the row indices and n and l are the column indices. The samples are de ned in the 2-D region , D = f({; |):0 { N-1, 0 | N ? 1g To detect the harmonic peaks, the image is rst zero-meaned. Gaussian tapering was not performed in contrary to common practice as the aliasing e ect normally leads to high frequency components which are normally weak. Fourier transform is performed on the image and the local maxima in the FMS are found by searching a small neighborhood of 3 3 region. By applying thresholding and retaining only those values above the threshold, we get the stronger Harmonics and Fundamentals of the image. This method though simplistic is found to be sucient for the purpose of fault detection as evident from the discussions and results that follow. A region of support for these spectral peaks is also found within a 3 3 window around the peaks. If we denote these frequencies as D! , the DFT of the random eld is then decomposed into the deterministic component
V (k; l) = Y (k; l) 8 (k; l) 2 D!
(3)
and the indeterministic component
W (k; l) = Y (k; l) 8 (k; l) 62 D!
(4)
Thus a texture eld is decomposed into a sum of two mutually orthogonal components, a deterministic component V (k; l) which results in the structural attributes of the texture and a purely indeterministic W (k; l) which results in the structureless, "random looking" component of the texture eld.
3 Fault Detection using Wold Decomposition Model Based on the Wold decomposition model, the deterministic part is composed of harmonics and evanescent components. In terms of energy distribution, the harmonics components dominate. In fact, by using only the harmonic components, one can reconstruct the dominant regular texture elements. Hence such reconstructed image does not contain the fault and the region is lled with the homogeneous texture primitive. In the indeterministic part of the image the fault stands out as an abnormal region. By further processing the deterministic part in comparison with the original image we can determine if The original texture has any fault or not and by processing the Stochastic part we can locate the fault if any. The rst part of the defect detection problem can be solved by looking at the harmonic part of the original texture. The ratio of total energy of the Texture to the energy of the harmonic part can be used to decide if the texture has a fault or not. The stochastic part of the texture is found to have a Gaussian distribution. The tail ends of the histogram of the stochastic components is the region where the fault in the texture is postulated. The tail end of the stochastic part contributing to 1% of the energy of the stochastic part may be considered to be due to defects. By considering the energy within a local window of suitable size ( in the e.g. we used 7 7) the fault can be localized.
4 Experimental results and discussion Experiments have been performed on several regular textures containing defects. Some of the results are shown in Figures 2 - 8. The top left is the original image containing defects. The top right is the FMS. The middle left is the selected local peaks and the middle right shows the reconstruction of the image using these local peaks. The bottom left shows the reconstruction from the stochastic part and the bottom right shows the defects that are detected and highlighted. It can be seen from the results that using the selected harmonic components, the normal uniform regular textured pattern can be reconstructed. The defect is encapsulated in the weaker stochastic part as some strong energy region. This is because the defects are themselves a pattern but does not conform to the majority repetitive texture elements. Hence, they are not captured by the thresholded local peaks. If local peak threshold is set too low, the local peaks that correspond to the defects may also be captured in the harmonics part. Hence, a proper choice of this value is crucial. The size of the support region must also be chosen properly. A large support region may cause the merger of several harmonic peaks.
5 Conclusions In this paper we have presented a texture fault detection method based on the 2-D Wold-Like decomposition. We have demonstrated that the Wold like decomposition is capable to reconstruct the dominant uniform regular texture patterns and the defect is encapsulated in the stochastic components of the texture. The results show that good localization of defect is possible. But a few crucial parameters have to be selected. Future work can look into more systematic, robust and ecient algorithm such as that used by [3].
original image
FMS of original image
Deterministc(harmonic) part of FMS
Reconstruction from deterministic part
Reconstruction from Stochastic part
Located defects
Figure 2: Synthetic image containing a defect original image
FMS of original image
Deterministc(harmonic) part of FMS
Reconstruction from deterministic part
Reconstruction from Stochastic part
Located defects
Figure 3: Real image containing a defect (stocking) original image
FMS of original image
Deterministc(harmonic) part of FMS
Reconstruction from deterministic part
Reconstruction from Stochastic part
Located defects
Figure 4: Real image containing a defect (towel)
original image
FMS of original image
Deterministc(harmonic) part of FMS
Reconstruction from deterministic part
Reconstruction from Stochastic part
Located defects
Figure 5: Real image containing a defect (gride) original image
FMS of original image
Deterministc(harmonic) part of FMS
Reconstruction from deterministic part
Reconstruction from Stochastic part
Located defects
Figure 6: Real image containing a defect (cloth) original image
FMS of original image
Deterministc(harmonic) part of FMS
Reconstruction from deterministic part
Reconstruction from Stochastic part
Located defects
Figure 7: Real image containing a defect (D77)
original image
FMS of original image
Deterministc(harmonic) part of FMS
Reconstruction from deterministic part
Reconstruction from Stochastic part
Located defects
Figure 8: Real image containing a defect (D77)
References [1] Ravishankar Rao,\A Taxonomy for Texture Description and Idnti cation,"Springer Verlag, 1990. [2] B. Picinbono,\Random Signals and Systems," Prentice Hall International, Inc. pp 516-526, 1993. [3] F. Liu and R. W. Picard, \Periodicity, directionality and randomness: Wold features for image modeling and retrieval," IEEE Transactions on Pattern Analysis and Machine Intelligence, pp. 722-733, July 1996. [4] J. M. Francos, A. Narasimhan, and J. W. Woods, \ Maximum Likelihood Parameter Estimation of Textures Using a Wold-Decomposition Based Model," IEEE Transactions on Image Processing, pp. 1655-1666, Dec. 1995. [5] J. M. Francos, A. Z. Meiri, and B. Porat, \A Wold-Like Decomposition of Two-Dimensional Descrete Homogeneous Random Fields," The Annals of Applied Probability, Vol. 5,No. 1, pp. 248-260,Feb. 1995. [6] J. Malik and P. Perona, \Preattentive texture discrimination with early vision mechanisms," J. Opt. Soc. Am. A, Vol. 7, No. 5, pp. 923-932, May 1990. [7] R. M. Haralick, \Statistical and structural approaches to texture," Proceedings of the IEEE, Vol. 67, pp. 786-804, May 1979.
