Abstract

Tooth bleaching is becoming increasingly popular among patients and dentists since it is a relatively noninvasive approach for whitening and lightening teeth. Instruments and visual assessment with respect to commercial shade guides are currently used to evaluate tooth color. However, the association between these procedures is imprecise and the degree of color change after tooth bleaching is known to vary substantially between studies; there are currently no objective guidelines to predict the effectiveness of a tooth-bleaching treatment. We propose a new methodology based on fuzzy logic as a natural means of representing the imprecision present when modeling the color change produced by a tooth-bleaching treatment on the basis of a tooth’s initial chromatic values. This system has the advantage of producing a set of interpretable fuzzy rules that can subsequently be used by scientists and dental practitioners. The fuzzy system obtained has the special characteristic whereby the rule antecedents correspond to prebleaching shades of the well-known Vita commercial shade guide. Additionally, the rule consequents directly correspond with the expected CIELAB postbleaching values for each Vita shade, thanks to a modification of the system’s inference structure. Finally, the values of these postbleaching CIELAB coordinates have been associated with Vita shades by evaluating their respective mem bership functions, thereby approximating which posttreatment Vita shades are to be expected for each prebleaching shade.

© 2010 Optical Society of America

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References

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  1. Council on Dental Therapeutics, “Guidelines for acceptance of peroxide containing oral hygiene products,” J. Am. Dent. Assoc. 125, 1140-1142 (1994).
  2. A. Joiner, “The bleaching of teeth: a review of the literature,” J. Dent. Res. 34, 412-419 (2006).
    [CrossRef]
  3. A. Joiner, “Tooth color: a review of the literature,” J. Dent. Res. 32, 3-12 (2004).
    [CrossRef]
  4. J. Preston, “Current status of shade selection and color matching,” Quintessence Int. 16, 47-58 (1985).
  5. Q. Li, H. Yu, and Y. Wang, “In vivo spectroradiometric evaluation of colour matching errors among five shade guides,” J. Oral Rehabil. 36, 65-70 (2009).
  6. S. Ishikawa-Nagai, “Prediction of optical efficacy of vital tooth bleaching using regression analysis,” Color Res. Appl. 29, 390-394 (2004).
    [CrossRef]
  7. CIE, “Colorimetry technical report15,” 3rd ed. (CIE Central Bureau, 2004).
  8. R. Douglas, “Precision of in vivo colorimetric assessments of teeth,” J. Prosthet. Dent. 77, 464-470 (1997).
    [CrossRef]
  9. K. Shimada, Y. Kakehashi, H. Matsumura, and N. Tanoue, “In vivo quantitative evaluation of tooth color with hand-held colorimeter and custom template,” J. Prosthet. Dent. 91, 389-391 (2004).
    [CrossRef]
  10. L. Zadeh, “Fuzzy sets,” Inform. Contr. 8, 338-353 (1965).
    [CrossRef]
  11. Z. Zalevsky, E. Gur, and D. Mendlovic, “Fuzzy-logic optical optimization of mainframe cpu and memory,” Appl. Opt. 45, 4647-4651 (2006).
    [CrossRef]
  12. L. Wang, Adaptive Fuzzy Systems and Control. Design and Stability Analysis (Prentice Hall, 1994).
  13. H. Pomares, I. Rojas, J. González, and A. Prieto, “Structure identification in complete rule-based fuzzy systems,” IEEE Trans. Fuzzy Syst. 10, 349-359 (2002).
    [CrossRef]
  14. L. Herrera, H. Pomares, I. Rojas, O. Valenzuela, and A. Prieto, “Tase, a Taylor series based fuzzy system model that combines interpretability and accuracy,” Fuzzy Sets Syst. 153, 403-427 (2005).
  15. R. Benavente, M. Vanrell, and R. Baldrich, “A data set for fuzzy colour naming,” Color Res. Appl. 31, 48-56(2006).
    [CrossRef]
  16. G. Menegaz, A. Le Troter, J. Sequeira, and J. M. Boi, “A discrete model for color naming,” EURASIP J. Adv. Signal Process. 2007, 1-10 (2007).
  17. J. Strackeljan, D. Behr, and T. Kocher, “Fuzzy-pattern recognition for automatic detection of different teeth substances,” Fuzzy Sets Syst. 85, 275-286 (1997).
    [CrossRef]
  18. J. Kang, L. Min, Q. Luan, X. Li, and J. Liu, “Novel modified fuzzy c-means algorithm with applications,” Digit. Signal Process. 19, 309-319 (2009).
    [CrossRef]
  19. R. Kruse, J. Gebhardt, and F. Klawonn, Foundations of Fuzzy Systems (Wiley, 1994).
  20. D. Driankov, H. Hellendoorn, and M. Reinfrank, An Introduction to Fuzzy Control (Springer-Verlag, 1996).
  21. L. J. Herrera, H. Pomares, I. Rojas, A. Guillén, G. Rubio, and J. Urquiza, “Global and local modelling in radial basis functions networks,” Lect. Notes Comput. Sci. 5517, 49-56 (2009).
    [CrossRef]
  22. T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modelling and control,” IEEE Trans. Syst. Man Cybern. 15, 116-132 (1985).
  23. J. Moody and C. Darken, “Fast learning in networks of locally-tuned processing units,” Neural Comput. 1, 281-294 (1989).
    [CrossRef]
  24. G. Golub and C. Loan, Matrix Computations (Johns Hopkins U. Press, 1961).
  25. T. Johansen and R. Babuska, “Multiobjective identification of Takagi-Sugeno fuzzy models,” IEEE Trans. Fuzzy Syst. 11, 847-860 (2003).
    [CrossRef]
  26. M. Bikdash, “A highly interpretable form of sugeno inference systems,” IEEE Trans. Fuzzy Syst. 7, 686-696 (1999).
    [CrossRef]

2009 (3)

Q. Li, H. Yu, and Y. Wang, “In vivo spectroradiometric evaluation of colour matching errors among five shade guides,” J. Oral Rehabil. 36, 65-70 (2009).

J. Kang, L. Min, Q. Luan, X. Li, and J. Liu, “Novel modified fuzzy c-means algorithm with applications,” Digit. Signal Process. 19, 309-319 (2009).
[CrossRef]

L. J. Herrera, H. Pomares, I. Rojas, A. Guillén, G. Rubio, and J. Urquiza, “Global and local modelling in radial basis functions networks,” Lect. Notes Comput. Sci. 5517, 49-56 (2009).
[CrossRef]

2007 (1)

G. Menegaz, A. Le Troter, J. Sequeira, and J. M. Boi, “A discrete model for color naming,” EURASIP J. Adv. Signal Process. 2007, 1-10 (2007).

2006 (3)

R. Benavente, M. Vanrell, and R. Baldrich, “A data set for fuzzy colour naming,” Color Res. Appl. 31, 48-56(2006).
[CrossRef]

A. Joiner, “The bleaching of teeth: a review of the literature,” J. Dent. Res. 34, 412-419 (2006).
[CrossRef]

Z. Zalevsky, E. Gur, and D. Mendlovic, “Fuzzy-logic optical optimization of mainframe cpu and memory,” Appl. Opt. 45, 4647-4651 (2006).
[CrossRef]

2005 (1)

L. Herrera, H. Pomares, I. Rojas, O. Valenzuela, and A. Prieto, “Tase, a Taylor series based fuzzy system model that combines interpretability and accuracy,” Fuzzy Sets Syst. 153, 403-427 (2005).

2004 (3)

K. Shimada, Y. Kakehashi, H. Matsumura, and N. Tanoue, “In vivo quantitative evaluation of tooth color with hand-held colorimeter and custom template,” J. Prosthet. Dent. 91, 389-391 (2004).
[CrossRef]

A. Joiner, “Tooth color: a review of the literature,” J. Dent. Res. 32, 3-12 (2004).
[CrossRef]

S. Ishikawa-Nagai, “Prediction of optical efficacy of vital tooth bleaching using regression analysis,” Color Res. Appl. 29, 390-394 (2004).
[CrossRef]

2003 (1)

T. Johansen and R. Babuska, “Multiobjective identification of Takagi-Sugeno fuzzy models,” IEEE Trans. Fuzzy Syst. 11, 847-860 (2003).
[CrossRef]

2002 (1)

H. Pomares, I. Rojas, J. González, and A. Prieto, “Structure identification in complete rule-based fuzzy systems,” IEEE Trans. Fuzzy Syst. 10, 349-359 (2002).
[CrossRef]

1999 (1)

M. Bikdash, “A highly interpretable form of sugeno inference systems,” IEEE Trans. Fuzzy Syst. 7, 686-696 (1999).
[CrossRef]

1997 (2)

R. Douglas, “Precision of in vivo colorimetric assessments of teeth,” J. Prosthet. Dent. 77, 464-470 (1997).
[CrossRef]

J. Strackeljan, D. Behr, and T. Kocher, “Fuzzy-pattern recognition for automatic detection of different teeth substances,” Fuzzy Sets Syst. 85, 275-286 (1997).
[CrossRef]

1994 (1)

Council on Dental Therapeutics, “Guidelines for acceptance of peroxide containing oral hygiene products,” J. Am. Dent. Assoc. 125, 1140-1142 (1994).

1989 (1)

J. Moody and C. Darken, “Fast learning in networks of locally-tuned processing units,” Neural Comput. 1, 281-294 (1989).
[CrossRef]

1985 (2)

T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modelling and control,” IEEE Trans. Syst. Man Cybern. 15, 116-132 (1985).

J. Preston, “Current status of shade selection and color matching,” Quintessence Int. 16, 47-58 (1985).

1965 (1)

L. Zadeh, “Fuzzy sets,” Inform. Contr. 8, 338-353 (1965).
[CrossRef]

Babuska, R.

T. Johansen and R. Babuska, “Multiobjective identification of Takagi-Sugeno fuzzy models,” IEEE Trans. Fuzzy Syst. 11, 847-860 (2003).
[CrossRef]

Baldrich, R.

R. Benavente, M. Vanrell, and R. Baldrich, “A data set for fuzzy colour naming,” Color Res. Appl. 31, 48-56(2006).
[CrossRef]

Behr, D.

J. Strackeljan, D. Behr, and T. Kocher, “Fuzzy-pattern recognition for automatic detection of different teeth substances,” Fuzzy Sets Syst. 85, 275-286 (1997).
[CrossRef]

Benavente, R.

R. Benavente, M. Vanrell, and R. Baldrich, “A data set for fuzzy colour naming,” Color Res. Appl. 31, 48-56(2006).
[CrossRef]

Bikdash, M.

M. Bikdash, “A highly interpretable form of sugeno inference systems,” IEEE Trans. Fuzzy Syst. 7, 686-696 (1999).
[CrossRef]

Boi, J. M.

G. Menegaz, A. Le Troter, J. Sequeira, and J. M. Boi, “A discrete model for color naming,” EURASIP J. Adv. Signal Process. 2007, 1-10 (2007).

CIE,

CIE, “Colorimetry technical report15,” 3rd ed. (CIE Central Bureau, 2004).

Darken, C.

J. Moody and C. Darken, “Fast learning in networks of locally-tuned processing units,” Neural Comput. 1, 281-294 (1989).
[CrossRef]

Douglas, R.

R. Douglas, “Precision of in vivo colorimetric assessments of teeth,” J. Prosthet. Dent. 77, 464-470 (1997).
[CrossRef]

Driankov, D.

D. Driankov, H. Hellendoorn, and M. Reinfrank, An Introduction to Fuzzy Control (Springer-Verlag, 1996).

Gebhardt, J.

R. Kruse, J. Gebhardt, and F. Klawonn, Foundations of Fuzzy Systems (Wiley, 1994).

Golub, G.

G. Golub and C. Loan, Matrix Computations (Johns Hopkins U. Press, 1961).

González, J.

H. Pomares, I. Rojas, J. González, and A. Prieto, “Structure identification in complete rule-based fuzzy systems,” IEEE Trans. Fuzzy Syst. 10, 349-359 (2002).
[CrossRef]

Guillén, A.

L. J. Herrera, H. Pomares, I. Rojas, A. Guillén, G. Rubio, and J. Urquiza, “Global and local modelling in radial basis functions networks,” Lect. Notes Comput. Sci. 5517, 49-56 (2009).
[CrossRef]

Gur, E.

Hellendoorn, H.

D. Driankov, H. Hellendoorn, and M. Reinfrank, An Introduction to Fuzzy Control (Springer-Verlag, 1996).

Herrera, L.

L. Herrera, H. Pomares, I. Rojas, O. Valenzuela, and A. Prieto, “Tase, a Taylor series based fuzzy system model that combines interpretability and accuracy,” Fuzzy Sets Syst. 153, 403-427 (2005).

Herrera, L. J.

L. J. Herrera, H. Pomares, I. Rojas, A. Guillén, G. Rubio, and J. Urquiza, “Global and local modelling in radial basis functions networks,” Lect. Notes Comput. Sci. 5517, 49-56 (2009).
[CrossRef]

Ishikawa-Nagai, S.

S. Ishikawa-Nagai, “Prediction of optical efficacy of vital tooth bleaching using regression analysis,” Color Res. Appl. 29, 390-394 (2004).
[CrossRef]

Johansen, T.

T. Johansen and R. Babuska, “Multiobjective identification of Takagi-Sugeno fuzzy models,” IEEE Trans. Fuzzy Syst. 11, 847-860 (2003).
[CrossRef]

Joiner, A.

A. Joiner, “The bleaching of teeth: a review of the literature,” J. Dent. Res. 34, 412-419 (2006).
[CrossRef]

A. Joiner, “Tooth color: a review of the literature,” J. Dent. Res. 32, 3-12 (2004).
[CrossRef]

Kakehashi, Y.

K. Shimada, Y. Kakehashi, H. Matsumura, and N. Tanoue, “In vivo quantitative evaluation of tooth color with hand-held colorimeter and custom template,” J. Prosthet. Dent. 91, 389-391 (2004).
[CrossRef]

Kang, J.

J. Kang, L. Min, Q. Luan, X. Li, and J. Liu, “Novel modified fuzzy c-means algorithm with applications,” Digit. Signal Process. 19, 309-319 (2009).
[CrossRef]

Klawonn, F.

R. Kruse, J. Gebhardt, and F. Klawonn, Foundations of Fuzzy Systems (Wiley, 1994).

Kocher, T.

J. Strackeljan, D. Behr, and T. Kocher, “Fuzzy-pattern recognition for automatic detection of different teeth substances,” Fuzzy Sets Syst. 85, 275-286 (1997).
[CrossRef]

Kruse, R.

R. Kruse, J. Gebhardt, and F. Klawonn, Foundations of Fuzzy Systems (Wiley, 1994).

Le Troter, A.

G. Menegaz, A. Le Troter, J. Sequeira, and J. M. Boi, “A discrete model for color naming,” EURASIP J. Adv. Signal Process. 2007, 1-10 (2007).

Li, Q.

Q. Li, H. Yu, and Y. Wang, “In vivo spectroradiometric evaluation of colour matching errors among five shade guides,” J. Oral Rehabil. 36, 65-70 (2009).

Li, X.

J. Kang, L. Min, Q. Luan, X. Li, and J. Liu, “Novel modified fuzzy c-means algorithm with applications,” Digit. Signal Process. 19, 309-319 (2009).
[CrossRef]

Liu, J.

J. Kang, L. Min, Q. Luan, X. Li, and J. Liu, “Novel modified fuzzy c-means algorithm with applications,” Digit. Signal Process. 19, 309-319 (2009).
[CrossRef]

Loan, C.

G. Golub and C. Loan, Matrix Computations (Johns Hopkins U. Press, 1961).

Luan, Q.

J. Kang, L. Min, Q. Luan, X. Li, and J. Liu, “Novel modified fuzzy c-means algorithm with applications,” Digit. Signal Process. 19, 309-319 (2009).
[CrossRef]

Matsumura, H.

K. Shimada, Y. Kakehashi, H. Matsumura, and N. Tanoue, “In vivo quantitative evaluation of tooth color with hand-held colorimeter and custom template,” J. Prosthet. Dent. 91, 389-391 (2004).
[CrossRef]

Mendlovic, D.

Menegaz, G.

G. Menegaz, A. Le Troter, J. Sequeira, and J. M. Boi, “A discrete model for color naming,” EURASIP J. Adv. Signal Process. 2007, 1-10 (2007).

Min, L.

J. Kang, L. Min, Q. Luan, X. Li, and J. Liu, “Novel modified fuzzy c-means algorithm with applications,” Digit. Signal Process. 19, 309-319 (2009).
[CrossRef]

Moody, J.

J. Moody and C. Darken, “Fast learning in networks of locally-tuned processing units,” Neural Comput. 1, 281-294 (1989).
[CrossRef]

Pomares, H.

L. J. Herrera, H. Pomares, I. Rojas, A. Guillén, G. Rubio, and J. Urquiza, “Global and local modelling in radial basis functions networks,” Lect. Notes Comput. Sci. 5517, 49-56 (2009).
[CrossRef]

L. Herrera, H. Pomares, I. Rojas, O. Valenzuela, and A. Prieto, “Tase, a Taylor series based fuzzy system model that combines interpretability and accuracy,” Fuzzy Sets Syst. 153, 403-427 (2005).

H. Pomares, I. Rojas, J. González, and A. Prieto, “Structure identification in complete rule-based fuzzy systems,” IEEE Trans. Fuzzy Syst. 10, 349-359 (2002).
[CrossRef]

Preston, J.

J. Preston, “Current status of shade selection and color matching,” Quintessence Int. 16, 47-58 (1985).

Prieto, A.

L. Herrera, H. Pomares, I. Rojas, O. Valenzuela, and A. Prieto, “Tase, a Taylor series based fuzzy system model that combines interpretability and accuracy,” Fuzzy Sets Syst. 153, 403-427 (2005).

H. Pomares, I. Rojas, J. González, and A. Prieto, “Structure identification in complete rule-based fuzzy systems,” IEEE Trans. Fuzzy Syst. 10, 349-359 (2002).
[CrossRef]

Reinfrank, M.

D. Driankov, H. Hellendoorn, and M. Reinfrank, An Introduction to Fuzzy Control (Springer-Verlag, 1996).

Rojas, I.

L. J. Herrera, H. Pomares, I. Rojas, A. Guillén, G. Rubio, and J. Urquiza, “Global and local modelling in radial basis functions networks,” Lect. Notes Comput. Sci. 5517, 49-56 (2009).
[CrossRef]

L. Herrera, H. Pomares, I. Rojas, O. Valenzuela, and A. Prieto, “Tase, a Taylor series based fuzzy system model that combines interpretability and accuracy,” Fuzzy Sets Syst. 153, 403-427 (2005).

H. Pomares, I. Rojas, J. González, and A. Prieto, “Structure identification in complete rule-based fuzzy systems,” IEEE Trans. Fuzzy Syst. 10, 349-359 (2002).
[CrossRef]

Rubio, G.

L. J. Herrera, H. Pomares, I. Rojas, A. Guillén, G. Rubio, and J. Urquiza, “Global and local modelling in radial basis functions networks,” Lect. Notes Comput. Sci. 5517, 49-56 (2009).
[CrossRef]

Sequeira, J.

G. Menegaz, A. Le Troter, J. Sequeira, and J. M. Boi, “A discrete model for color naming,” EURASIP J. Adv. Signal Process. 2007, 1-10 (2007).

Shimada, K.

K. Shimada, Y. Kakehashi, H. Matsumura, and N. Tanoue, “In vivo quantitative evaluation of tooth color with hand-held colorimeter and custom template,” J. Prosthet. Dent. 91, 389-391 (2004).
[CrossRef]

Strackeljan, J.

J. Strackeljan, D. Behr, and T. Kocher, “Fuzzy-pattern recognition for automatic detection of different teeth substances,” Fuzzy Sets Syst. 85, 275-286 (1997).
[CrossRef]

Sugeno, M.

T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modelling and control,” IEEE Trans. Syst. Man Cybern. 15, 116-132 (1985).

Takagi, T.

T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modelling and control,” IEEE Trans. Syst. Man Cybern. 15, 116-132 (1985).

Tanoue, N.

K. Shimada, Y. Kakehashi, H. Matsumura, and N. Tanoue, “In vivo quantitative evaluation of tooth color with hand-held colorimeter and custom template,” J. Prosthet. Dent. 91, 389-391 (2004).
[CrossRef]

Therapeutics, Council on Dental

Council on Dental Therapeutics, “Guidelines for acceptance of peroxide containing oral hygiene products,” J. Am. Dent. Assoc. 125, 1140-1142 (1994).

Urquiza, J.

L. J. Herrera, H. Pomares, I. Rojas, A. Guillén, G. Rubio, and J. Urquiza, “Global and local modelling in radial basis functions networks,” Lect. Notes Comput. Sci. 5517, 49-56 (2009).
[CrossRef]

Valenzuela, O.

L. Herrera, H. Pomares, I. Rojas, O. Valenzuela, and A. Prieto, “Tase, a Taylor series based fuzzy system model that combines interpretability and accuracy,” Fuzzy Sets Syst. 153, 403-427 (2005).

Vanrell, M.

R. Benavente, M. Vanrell, and R. Baldrich, “A data set for fuzzy colour naming,” Color Res. Appl. 31, 48-56(2006).
[CrossRef]

Wang, L.

L. Wang, Adaptive Fuzzy Systems and Control. Design and Stability Analysis (Prentice Hall, 1994).

Wang, Y.

Q. Li, H. Yu, and Y. Wang, “In vivo spectroradiometric evaluation of colour matching errors among five shade guides,” J. Oral Rehabil. 36, 65-70 (2009).

Yu, H.

Q. Li, H. Yu, and Y. Wang, “In vivo spectroradiometric evaluation of colour matching errors among five shade guides,” J. Oral Rehabil. 36, 65-70 (2009).

Zadeh, L.

L. Zadeh, “Fuzzy sets,” Inform. Contr. 8, 338-353 (1965).
[CrossRef]

Zalevsky, Z.

Appl. Opt. (1)

Color Res. Appl. (2)

R. Benavente, M. Vanrell, and R. Baldrich, “A data set for fuzzy colour naming,” Color Res. Appl. 31, 48-56(2006).
[CrossRef]

S. Ishikawa-Nagai, “Prediction of optical efficacy of vital tooth bleaching using regression analysis,” Color Res. Appl. 29, 390-394 (2004).
[CrossRef]

Digit. Signal Process. (1)

J. Kang, L. Min, Q. Luan, X. Li, and J. Liu, “Novel modified fuzzy c-means algorithm with applications,” Digit. Signal Process. 19, 309-319 (2009).
[CrossRef]

EURASIP J. Adv. Signal Process. (1)

G. Menegaz, A. Le Troter, J. Sequeira, and J. M. Boi, “A discrete model for color naming,” EURASIP J. Adv. Signal Process. 2007, 1-10 (2007).

Fuzzy Sets Syst. (2)

J. Strackeljan, D. Behr, and T. Kocher, “Fuzzy-pattern recognition for automatic detection of different teeth substances,” Fuzzy Sets Syst. 85, 275-286 (1997).
[CrossRef]

L. Herrera, H. Pomares, I. Rojas, O. Valenzuela, and A. Prieto, “Tase, a Taylor series based fuzzy system model that combines interpretability and accuracy,” Fuzzy Sets Syst. 153, 403-427 (2005).

IEEE Trans. Fuzzy Syst. (3)

H. Pomares, I. Rojas, J. González, and A. Prieto, “Structure identification in complete rule-based fuzzy systems,” IEEE Trans. Fuzzy Syst. 10, 349-359 (2002).
[CrossRef]

T. Johansen and R. Babuska, “Multiobjective identification of Takagi-Sugeno fuzzy models,” IEEE Trans. Fuzzy Syst. 11, 847-860 (2003).
[CrossRef]

M. Bikdash, “A highly interpretable form of sugeno inference systems,” IEEE Trans. Fuzzy Syst. 7, 686-696 (1999).
[CrossRef]

IEEE Trans. Syst. Man Cybern. (1)

T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modelling and control,” IEEE Trans. Syst. Man Cybern. 15, 116-132 (1985).

Inform. Contr. (1)

L. Zadeh, “Fuzzy sets,” Inform. Contr. 8, 338-353 (1965).
[CrossRef]

J. Am. Dent. Assoc. (1)

Council on Dental Therapeutics, “Guidelines for acceptance of peroxide containing oral hygiene products,” J. Am. Dent. Assoc. 125, 1140-1142 (1994).

J. Dent. Res. (2)

A. Joiner, “The bleaching of teeth: a review of the literature,” J. Dent. Res. 34, 412-419 (2006).
[CrossRef]

A. Joiner, “Tooth color: a review of the literature,” J. Dent. Res. 32, 3-12 (2004).
[CrossRef]

J. Oral Rehabil. (1)

Q. Li, H. Yu, and Y. Wang, “In vivo spectroradiometric evaluation of colour matching errors among five shade guides,” J. Oral Rehabil. 36, 65-70 (2009).

J. Prosthet. Dent. (2)

R. Douglas, “Precision of in vivo colorimetric assessments of teeth,” J. Prosthet. Dent. 77, 464-470 (1997).
[CrossRef]

K. Shimada, Y. Kakehashi, H. Matsumura, and N. Tanoue, “In vivo quantitative evaluation of tooth color with hand-held colorimeter and custom template,” J. Prosthet. Dent. 91, 389-391 (2004).
[CrossRef]

Lect. Notes Comput. Sci. (1)

L. J. Herrera, H. Pomares, I. Rojas, A. Guillén, G. Rubio, and J. Urquiza, “Global and local modelling in radial basis functions networks,” Lect. Notes Comput. Sci. 5517, 49-56 (2009).
[CrossRef]

Neural Comput. (1)

J. Moody and C. Darken, “Fast learning in networks of locally-tuned processing units,” Neural Comput. 1, 281-294 (1989).
[CrossRef]

Quintessence Int. (1)

J. Preston, “Current status of shade selection and color matching,” Quintessence Int. 16, 47-58 (1985).

Other (5)

CIE, “Colorimetry technical report15,” 3rd ed. (CIE Central Bureau, 2004).

L. Wang, Adaptive Fuzzy Systems and Control. Design and Stability Analysis (Prentice Hall, 1994).

G. Golub and C. Loan, Matrix Computations (Johns Hopkins U. Press, 1961).

R. Kruse, J. Gebhardt, and F. Klawonn, Foundations of Fuzzy Systems (Wiley, 1994).

D. Driankov, H. Hellendoorn, and M. Reinfrank, An Introduction to Fuzzy Control (Springer-Verlag, 1996).

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Figures (2)

Fig. 1
Fig. 1

Data and Vita Classical shade distribution in CIELAB space in the planes a * b * and b * L * . For plane b * L * , a fictitious line joins the Vita shades from C4 to B1 according to Table 2.

Fig. 2
Fig. 2

(a) Original μ 1 and μ 2 MFs for the one-dimensional example. (b) Normalized final MF activations using the modified calculation μ 1 ( x ) μ 1 ( x ) + μ 2 ( x ) and μ 2 ( x ) μ 1 ( x ) + μ 2 ( x ) .

Tables (3)

Tables Icon

Table 1 Prebleaching and Postbleaching Ranges for Coordinates L * , a * , and b *

Tables Icon

Table 2 CIELAB Coordinates for Vita Classical Shades

Tables Icon

Table 3 Postbleaching Vita Shades Expected for Each Prebleaching Vita Shade

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

IF     L i *     is         μ L * Vita- S AND     a i *     is     μ a * Vita- S AND     b i *     is     μ b * Vita- S THEN     L f * = L ˜ f * S AND     a f * = a ˜ f * S AND     b f * = b ˜ f * S ,
μ L * Vita- S ( L i * ) = e ( L i * L * ( S ) ) 2 2 σ S 2 , μ a * Vita- S ( a i * ) = e ( a i * a * ( S ) ) 2 2 σ S 2 , μ b * Vita- S ( b i * ) = e ( b i * b * ( S ) ) 2 2 σ S 2 ,
y ˜ ( x ) = k = 1 K μ k ( x ) y ˜ k k = 1 K μ k ( x ) ,
J = m D ( y ˜ ( x m ) y m ) 2 # D ,
y ˜ ( c k ) = k / μ k ( c k ) > 0 μ k ( c k ) y ˜ k k / μ k ( c k ) > 0 μ k ( c k ) .
μ k ( x ) = μ k ( x ) j = 1 ; j k K ( 1 μ j ( x ) ) .
y ˜ ( x ) = k = 1 K μ k ( x ) y ˜ k k = 1 K μ k ( x ) = k = 1 K ( μ k ( x ) j = 1 ; j k K ( 1 μ j ( x ) ) ) y ˜ k k = 1 K ( μ k ( x ) j = 1 ; j k K ( 1 μ j ( x ) ) ) .
μ k ( c k ) l = 1 K μ l ( c k ) = 1 ,     μ j ( c k ) l = 1 K μ l ( c k ) = 0 , j = 1 K , j k y ˜ ( c k ) = y ˜ k .
confid.value k = m D y ˜ m y m · μ k * ( x m ) ,
μ 1 ( x ) = μ 1 ( x ) ( 1 μ 2 ( x ) ) , μ 1 ( x ) l = 1 K μ l ( x ) = μ 1 ( x ) μ 1 ( x ) + μ 2 ( x ) ,
μ 2 ( x ) = μ 2 ( x ) ( 1 μ 1 ( x ) ) , μ 2 ( x ) l = 1 K μ l ( x ) = μ 2 ( x ) μ 1 ( x ) + μ 2 ( x ) .
μ 1 * ( c 1 ) = 1 ,       μ 2 * ( c 1 ) = 0 y ˜ ( c 1 ) = y ˜ 1 , μ 2 * ( c 2 ) = 1 ,     μ 1 * ( c 2 ) = 0 y ˜ ( c 2 ) = y ˜ 2 ,
IF     L i * is     B 2 ^ L * AND     a i * is     B 2 ^ a * AND     b i * is     B 2 ^ b * THEN L f * = 68.89 AND     a f * = 4.68 AND     b f * = 9.49 IF     L i * is     A 2 ^ L * AND     a i * is     A 2 ^ a * AND     b i * is     A 2 ^ b * THEN L f * = 60.39 AND     a f * = 4.49 AND     b f * = 8.15 IF     L i * is     C 2 ^ L * AND     a i * is     C 2 ^ a * AND b i * is     C 2 ^ b * THEN L f * = 53.29 AND     a f * = 7.12 AND     b f * = 11.86 IF     L i * is D 4 ^ L * AND     a i * is     D 4 ^ a * AND     b i * is     D 4 ^ b * THEN L f * = 57.87 AND     a f * = 4.08 AND     b f * = 10.90 IF     L i * is     D 3 ^ L * AND     a i * is     D 3 ^ a * AND     b i * is     D 3 ^ b * THEN L f * = 62.82 AND     a f * = 2.80 AND     b f * = 3.51 IF     L i * is     A 3 ^ L * AND     a i * is     A 3 ^ a * AND     b i * is     A 3 ^ b * THEN L f * = 64.24 AND     a f * = 6.14 AND b f * = 11.74 , .
IF     L i * is     B 3 ^ L * AND     a i * is     B 3 ^ a * AND     b i * is     B 3 ^ b * THEN L f * = 49.29 AND     a f * = 6.89 AND     b f * = 13.45 IF     L i * is     A 3.5 ^ L * AND     a i * is     A 3.5 ^ a * AND     b i * is     A 3.5 ^ b * THEN L f * = 55.65 AND     a f * = 6.84 AND     b f * = 11.72 IF     L i * is     B 4 ^ L * AND     a i * is     B 4 ^ a * AND     b i * is     B 4 ^ b * THEN L f * = 54.05 AND     a f * = 7.08 AND     b f * = 14.50 IF     L i * is     C 3 ^ L * AND     a i * is     C 3 ^ a * AND     b i * is     C 3 ^ b * THEN L f * = 53.46 AND     a f * = 5.16 AND     b f * = 10.72 IF     L i * is     A 4 ^ L * AND     a i * is     A 4 ^ a * AND     b i * is     A 4 ^ b * THEN L f * = 45.38 AND     a f * = 6.90 AND     b f * = 12.50 IF     L i * is     C 4 ^ L * AND     a i * is     C 4 ^ a * AND     b i * is     C 4 ^ b * THEN L f * = 47.32 AND     a f * = 5.66 AND     b f * = 11.78. ..

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