Abstract

A new correlation filter formulation (that we refer to as the minimax distance transform correlation filter (MDTCF) is presented that minimizes the average squared distance from the filtered desired (or true-) class training images to a filtered reference image while maximizing the mean squared distance of the filtered undesired (or false-) class training images to this filtered reference image. This approach increases the separation between the false-class correlation outputs and the true-class correlation outputs. Classification can be performed using the squared distance of a filtered test image to the chosen filtered reference image. We show that the previously introduced distance classifier correlation filter (DCCF) is similar to a special case of MDTCF. We also examine the differences between the DCCF and the MDTCF, and show that MDTCF can offer increased discrimination performance. Also, MDTCF performance is evaluated on two different face databases.

© 2002 Optical Society of America

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References

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  1. A. Mahalanobis, D. W. Carlson, B. V. K. Vijaya Kumar, “Evaluation of MACH and DCCF correlation filters for SAR ATR using MSTAR public database,” in Algorithms for Synthetic Aperture Radar Imagery V, E. G. Zelnio, ed., Proc. SPIE3370, 460–469 (1998).
  2. B. V. K. Vijaya Kumar, “Tutorial survey of composite filter designs for optical correlators,” Appl. Opt. 31, 4773–4801 (1992).
    [CrossRef]
  3. B. V. K. Vijaya Kumar, A. Mahalanobis, S. Song, S. R. F. Sims, J. F. Epperson, “Minimum squared error synthetic discriminant functions,” Opt. Eng. 31, 915–922 (1992).
    [CrossRef]
  4. A. Mahalanobis, B. V. K. Vijaya Kumar, S. R. F. Sims, “Distance classifier correlation filters for distortion tolerance, discrimination, and clutter rejection,” in Photonics for Processors, Neural Networks, and Memories, J. L. Horner, B. Javidi, S. T. Kowel, W. J. Miceli, eds., Proc. SPIE2026, 325–337 (1993).
  5. R. Duda, P. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973).
  6. A. Mahalanobis, B. V. K. Vijaya Kumar, “Important differences between distance classifier correlation filters and Fisher linear discriminant functions,” in Automatic Target Recognition VIII, F. A. Sadjadi, ed., Proc. SPIE3371, 263–274 (1998).
  7. http://amp.ece.cmu.edu —Advanced Multimedia Processing Lab at ECE CMU.
  8. A. Mahalanobis, B. V. K. Vijaya Kumar, S. R. F. Sims, J. F. Epperson, “Unconstrained correlation filters,” Appl. Opt. 33, 3751–3759 (1994).
    [CrossRef] [PubMed]
  9. http://www.uk.research.att.com/facedatabase.html —AT&T Laboratories, Cambridge, United Kingdom.

1994 (1)

1992 (2)

B. V. K. Vijaya Kumar, A. Mahalanobis, S. Song, S. R. F. Sims, J. F. Epperson, “Minimum squared error synthetic discriminant functions,” Opt. Eng. 31, 915–922 (1992).
[CrossRef]

B. V. K. Vijaya Kumar, “Tutorial survey of composite filter designs for optical correlators,” Appl. Opt. 31, 4773–4801 (1992).
[CrossRef]

Carlson, D. W.

A. Mahalanobis, D. W. Carlson, B. V. K. Vijaya Kumar, “Evaluation of MACH and DCCF correlation filters for SAR ATR using MSTAR public database,” in Algorithms for Synthetic Aperture Radar Imagery V, E. G. Zelnio, ed., Proc. SPIE3370, 460–469 (1998).

Duda, R.

R. Duda, P. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973).

Epperson, J. F.

A. Mahalanobis, B. V. K. Vijaya Kumar, S. R. F. Sims, J. F. Epperson, “Unconstrained correlation filters,” Appl. Opt. 33, 3751–3759 (1994).
[CrossRef] [PubMed]

B. V. K. Vijaya Kumar, A. Mahalanobis, S. Song, S. R. F. Sims, J. F. Epperson, “Minimum squared error synthetic discriminant functions,” Opt. Eng. 31, 915–922 (1992).
[CrossRef]

Hart, P.

R. Duda, P. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973).

Mahalanobis, A.

A. Mahalanobis, B. V. K. Vijaya Kumar, S. R. F. Sims, J. F. Epperson, “Unconstrained correlation filters,” Appl. Opt. 33, 3751–3759 (1994).
[CrossRef] [PubMed]

B. V. K. Vijaya Kumar, A. Mahalanobis, S. Song, S. R. F. Sims, J. F. Epperson, “Minimum squared error synthetic discriminant functions,” Opt. Eng. 31, 915–922 (1992).
[CrossRef]

A. Mahalanobis, B. V. K. Vijaya Kumar, “Important differences between distance classifier correlation filters and Fisher linear discriminant functions,” in Automatic Target Recognition VIII, F. A. Sadjadi, ed., Proc. SPIE3371, 263–274 (1998).

A. Mahalanobis, B. V. K. Vijaya Kumar, S. R. F. Sims, “Distance classifier correlation filters for distortion tolerance, discrimination, and clutter rejection,” in Photonics for Processors, Neural Networks, and Memories, J. L. Horner, B. Javidi, S. T. Kowel, W. J. Miceli, eds., Proc. SPIE2026, 325–337 (1993).

A. Mahalanobis, D. W. Carlson, B. V. K. Vijaya Kumar, “Evaluation of MACH and DCCF correlation filters for SAR ATR using MSTAR public database,” in Algorithms for Synthetic Aperture Radar Imagery V, E. G. Zelnio, ed., Proc. SPIE3370, 460–469 (1998).

Sims, S. R. F.

A. Mahalanobis, B. V. K. Vijaya Kumar, S. R. F. Sims, J. F. Epperson, “Unconstrained correlation filters,” Appl. Opt. 33, 3751–3759 (1994).
[CrossRef] [PubMed]

B. V. K. Vijaya Kumar, A. Mahalanobis, S. Song, S. R. F. Sims, J. F. Epperson, “Minimum squared error synthetic discriminant functions,” Opt. Eng. 31, 915–922 (1992).
[CrossRef]

A. Mahalanobis, B. V. K. Vijaya Kumar, S. R. F. Sims, “Distance classifier correlation filters for distortion tolerance, discrimination, and clutter rejection,” in Photonics for Processors, Neural Networks, and Memories, J. L. Horner, B. Javidi, S. T. Kowel, W. J. Miceli, eds., Proc. SPIE2026, 325–337 (1993).

Song, S.

B. V. K. Vijaya Kumar, A. Mahalanobis, S. Song, S. R. F. Sims, J. F. Epperson, “Minimum squared error synthetic discriminant functions,” Opt. Eng. 31, 915–922 (1992).
[CrossRef]

Vijaya Kumar, B. V. K.

A. Mahalanobis, B. V. K. Vijaya Kumar, S. R. F. Sims, J. F. Epperson, “Unconstrained correlation filters,” Appl. Opt. 33, 3751–3759 (1994).
[CrossRef] [PubMed]

B. V. K. Vijaya Kumar, A. Mahalanobis, S. Song, S. R. F. Sims, J. F. Epperson, “Minimum squared error synthetic discriminant functions,” Opt. Eng. 31, 915–922 (1992).
[CrossRef]

B. V. K. Vijaya Kumar, “Tutorial survey of composite filter designs for optical correlators,” Appl. Opt. 31, 4773–4801 (1992).
[CrossRef]

A. Mahalanobis, B. V. K. Vijaya Kumar, “Important differences between distance classifier correlation filters and Fisher linear discriminant functions,” in Automatic Target Recognition VIII, F. A. Sadjadi, ed., Proc. SPIE3371, 263–274 (1998).

A. Mahalanobis, D. W. Carlson, B. V. K. Vijaya Kumar, “Evaluation of MACH and DCCF correlation filters for SAR ATR using MSTAR public database,” in Algorithms for Synthetic Aperture Radar Imagery V, E. G. Zelnio, ed., Proc. SPIE3370, 460–469 (1998).

A. Mahalanobis, B. V. K. Vijaya Kumar, S. R. F. Sims, “Distance classifier correlation filters for distortion tolerance, discrimination, and clutter rejection,” in Photonics for Processors, Neural Networks, and Memories, J. L. Horner, B. Javidi, S. T. Kowel, W. J. Miceli, eds., Proc. SPIE2026, 325–337 (1993).

Appl. Opt. (2)

Opt. Eng. (1)

B. V. K. Vijaya Kumar, A. Mahalanobis, S. Song, S. R. F. Sims, J. F. Epperson, “Minimum squared error synthetic discriminant functions,” Opt. Eng. 31, 915–922 (1992).
[CrossRef]

Other (6)

A. Mahalanobis, B. V. K. Vijaya Kumar, S. R. F. Sims, “Distance classifier correlation filters for distortion tolerance, discrimination, and clutter rejection,” in Photonics for Processors, Neural Networks, and Memories, J. L. Horner, B. Javidi, S. T. Kowel, W. J. Miceli, eds., Proc. SPIE2026, 325–337 (1993).

R. Duda, P. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973).

A. Mahalanobis, B. V. K. Vijaya Kumar, “Important differences between distance classifier correlation filters and Fisher linear discriminant functions,” in Automatic Target Recognition VIII, F. A. Sadjadi, ed., Proc. SPIE3371, 263–274 (1998).

http://amp.ece.cmu.edu —Advanced Multimedia Processing Lab at ECE CMU.

A. Mahalanobis, D. W. Carlson, B. V. K. Vijaya Kumar, “Evaluation of MACH and DCCF correlation filters for SAR ATR using MSTAR public database,” in Algorithms for Synthetic Aperture Radar Imagery V, E. G. Zelnio, ed., Proc. SPIE3370, 460–469 (1998).

http://www.uk.research.att.com/facedatabase.html —AT&T Laboratories, Cambridge, United Kingdom.

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Figures (15)

Fig. 1
Fig. 1

Minimax distance transform visualization in the transformed feature space.

Fig. 2
Fig. 2

Sample images from the facial expression database used in our simulations. Each row contains different facial expressions from one individual with different rows representing different individuals.

Fig. 3
Fig. 3

Three images used as reference images for designing MDTCF; Goldhill (left-hand side), Mandrill (center), Saturn (right-hand side).

Fig. 4
Fig. 4

MSD values obtained from the MDTCF trained for Person 2 with the Goldhill reference image and three training images; solid curve shows the MSD values to the true class whereas the dotted curves show MSD values to various false classes.

Fig. 5
Fig. 5

MSD values obtained from the MDTCF trained for Person 2 with the Mandrill reference image and three training images; solid curve shows the MSD values to the true class whereas the dotted curves show MSD values to various false classes.

Fig. 6
Fig. 6

MSD values obtained from the MDTCF trained for Person 2 with the Saturn reference image and three training images; solid curve shows the MSD values to the true class whereas the dotted curves show MSD values to various false classes.

Fig. 7
Fig. 7

MSD values obtained from the DCCF trained for Person 3; solid curve shows the MSD values to the true class whereas the dotted curves show MSD values to various false classes.

Fig. 8
Fig. 8

MSD values obtained from the MDTCF trained for Person 3 with the true class mean as the reference image; solid curve shows the MSD values to the true class whereas the dotted curves show MSD values to various false classes.

Fig. 9
Fig. 9

MSD values obtained from the original MDTCF trained for Person 3 with the Goldhill image as the reference image; solid curve shows the MSD values to the true class whereas the dotted curves show MSD values to various false classes.

Fig. 10
Fig. 10

MSD values obtained from the modified MDTCF trained for Person 3 with the Goldhill image as the reference image; solid curve shows the MSD values to the true class whereas the dotted curves show MSD values to various false classes.

Fig. 11
Fig. 11

Sample images from the ORL database. Each row contains different images collected from an individual with different rows representing different individuals.

Fig. 12
Fig. 12

Plot showing EER results for both MDTCF and DCCF filters on each of 40 subjects in the ORL database.

Fig. 13
Fig. 13

Sample images showing the quality of a face image in the presence of different levels of AWGN (SNR dB).

Fig. 14
Fig. 14

Plot showing the average EER across all 40 subjects in ORL database in the presence of AWGN of different Signal to Noise Ratios (SNR dB) for both the MDTCF and DCCF filters.

Fig. 15
Fig. 15

Plot showing the improvement in performance with an MDTCF with some noise tolerance built in.

Tables (1)

Tables Icon

Table 1 Performance of the 13 Filters Synthesized for Each of the 3 Reference Images

Equations (37)

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cin=IFTFi*kHk,
EiΔ1Ln=1L|cdn-cin|2 =1Lk=1L|D*kHk-Fi*kHk|2,
E=1Ni=1NEi.
fi=Fi1, Fi2, Fi3,,FiLT, i=1, 2,,N,
h=H1, H2, H3,,HLT,
d=D1, D2, D3,,DLT.
Ti=diagTi1, Ti2, Ti3,,TiL.
Fi=diagFi1, Fi2, Fi3,,FiL,
D=diagd1, d2, d3,,dL.
ET=1NTi=1NT|D*h-Ti*h|2 =1NTi=1NTD*h-Ti*h+D*h-Ti*h=h+1NTi=1NTD-TiD-Ti*h=h+STh,
ST=1NTi=1NTD-TiD-Ti*.
mF=1NFi=1NFfi,
EF=|d+h-mF+h|2 =d+h-mF+h+d+h-mF+h =h+d-mFd-mF+h.
Jh=EFET=h+d-mFd-mF+hh+STh.
h=ST-1d-mF.
EF=|D*h-MF*h|2.
Jh=EFET=h+CFhh+STh
CF=1NFi=1NFD-MFD-MF*.
ST-1CFh=λh,
λ=h+CFhh+STh=Jh.
MSD=1L |X*h-D*h|2.
MSD=1LX*h-D*h+X*h-D*h, =1Lh+XX*h+h+DD*h-2h+XD*h,
p=h+XX*h and q=h+DD*h.
hf=HH*d,
MSD=1Lp+b-2xThf*.
Jh=h+mT-mFmT-mF+hh+Sh,
S=1NTi=1NTMT-TiMT-Ti++1NFi=1NFMF-FiMF-Fi+.
S=ST+SF,
EF=1NFi=1NF|Fi*h-MF*h|2=h+1NFi=1NFMF-FiMF-Fi+h=h+SFh.
Jh=EFET+EF=h+d-mFd-mF+hh+ST+SFh,
h=ST+SF-1d-mF.
JMDTCFh=h+d-mFd-mF+hh+SThh=ST-1d-mF, =d-mF+ST-1d-mFd-mF+ST-1d-mFd-mF+ST-1STST-1d-mF, =d-mF+ST-1d-mF.
ET=1NTi=1NT|D*h-Ti*h|2=1NTi=1NT|H*d-H*ti|2 =1NTi=1NTH*d-H*ti+H*d-H*ti.
dET=2NTi=1NTH*d-H*ti =H*2NTi=1NTd-ti=0.
doptimal=1NTi=1NTti.
Jnoise_tolh=EFET=h+d-mFd-mF+hh+αST+βIh,
hnoise_tol=αST+βI-1d-mF.

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