Jason Turuwhenua, "Corneal surface reconstruction algorithm using Zernike polynomial representation: improvements," J. Opt. Soc. Am. A 24, 1551-1561 (2007)

Recently Sicam et al. [J. Opt. Soc. Am. A 21, 1300 (2004)
] presented a new corneal reconstruction algorithm for estimating corneal sag by Zernike polynomials. An equivalent but simpler derivation of the model equations is presented. The algorithm is tested on a sphere, a conic, and a toric. These tests reveal significant height errors that accrue with distance from the corneal apex. Additional postprocessing steps are introduced to circumvent these errors. A consistent and significant reduction in height errors is observed across the test surfaces. Finally, Sicam et al. used the conic p-value p as a measure of algorithm efficacy. Further investigation shows that the finite Zernike representation affected the reported results. The p-value should therefore be used with caution as an efficacy measure.

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Estimates of R and p for the Sphere (Radial Order 10)^{
a
}

Pupil diameter (mm)

R (RC) (mm)

R$(\mathrm{RC}+\mathrm{M})$ (mm)

R (exact) (mm)

p (RC)

p$(\mathrm{RC}+\mathrm{M})$

p (exact)

7

7.0000

7.0000

7.0000

0.999

0.999

0.999

8

6.9998

6.9998

6.9999

0.996

0.997

0.998

9

6.9990

6.9991

6.9995

0.987

0.989

0.993

10

6.9958

6.9964

6.9978

0.957

0.964

0.976

11

6.9830

6.9856

6.9909

0.855

0.880

0.918

The Sicam (RC) columns estimate the original results presented in Ref. [8]. The differences between the Sicam (RC) and Sicam $(\mathrm{RC}+\mathrm{M})$ columns estimate the impact of deviations in corneal shape due to reconstruction method. The exact columns deviate from true $R=7\text{\hspace{0.17em}}\mathrm{mm}$ and $p=1$, as a result of the truncation of the finite Zernike representation.

Table 2

Estimates of R and p for the Conic (Radial Order 10)^{
a
}

Pupil diameter (mm)

R (RC) (mm)

R$(\mathrm{RC}+\mathrm{M})$ (mm)

R (exact) (mm)

p (RC)

p$(\mathrm{RC}+\mathrm{M})$

p (exact)

7

7.8700

7.8700

7.8700

0.820

0.820

0.820

8

7.8700

7.8700

7.8700

0.820

0.820

0.820

9

7.8699

7.8699

7.8700

0.819

0.819

0.819

10

7.8697

7.8697

7.8699

0.816

0.817

0.818

11

7.8690

7.8692

7.8695

0.810

0.812

0.815

The Sicam (RC) columns estimate the original results presented in Ref. [8]. The differences between the Sicam (RC) and Sicam $(\mathrm{RC}+\mathrm{M})$ columns estimate the impact of deviations in corneal shape due to reconstruction method. The exact columns deviate from true $R=7.87\text{\hspace{0.17em}}\mathrm{mm}$ and $p=0.82$, as a result of the truncation of the finite Zernite representation.

Table 3

Estimates of R and p for the Sphere (Radial Order 14)^{
a
}

Pupil diameter (mm)

R (RC) (mm)

R$(\mathrm{RC}+\mathrm{M})$ (mm)

R (exact) (mm)

p (RC)

p$(\mathrm{RC}+\mathrm{M})$

p (exact)

10

6.9998

6.9999

6.9999

0.997

0.997

0.998

11

6.9988

6.9990

6.9994

0.981

0.985

0.990

The table includes only the rows for $10\text{\hspace{0.17em}}\mathrm{mm}$ and $11\text{\hspace{0.17em}}\mathrm{mm}$ because errors were negligible otherwise. The results for the conic were not included because errors were completely negligible for all parameter values used.

Tables (3)

Table 1

Estimates of R and p for the Sphere (Radial Order 10)^{
a
}

Pupil diameter (mm)

R (RC) (mm)

R$(\mathrm{RC}+\mathrm{M})$ (mm)

R (exact) (mm)

p (RC)

p$(\mathrm{RC}+\mathrm{M})$

p (exact)

7

7.0000

7.0000

7.0000

0.999

0.999

0.999

8

6.9998

6.9998

6.9999

0.996

0.997

0.998

9

6.9990

6.9991

6.9995

0.987

0.989

0.993

10

6.9958

6.9964

6.9978

0.957

0.964

0.976

11

6.9830

6.9856

6.9909

0.855

0.880

0.918

The Sicam (RC) columns estimate the original results presented in Ref. [8]. The differences between the Sicam (RC) and Sicam $(\mathrm{RC}+\mathrm{M})$ columns estimate the impact of deviations in corneal shape due to reconstruction method. The exact columns deviate from true $R=7\text{\hspace{0.17em}}\mathrm{mm}$ and $p=1$, as a result of the truncation of the finite Zernike representation.

Table 2

Estimates of R and p for the Conic (Radial Order 10)^{
a
}

Pupil diameter (mm)

R (RC) (mm)

R$(\mathrm{RC}+\mathrm{M})$ (mm)

R (exact) (mm)

p (RC)

p$(\mathrm{RC}+\mathrm{M})$

p (exact)

7

7.8700

7.8700

7.8700

0.820

0.820

0.820

8

7.8700

7.8700

7.8700

0.820

0.820

0.820

9

7.8699

7.8699

7.8700

0.819

0.819

0.819

10

7.8697

7.8697

7.8699

0.816

0.817

0.818

11

7.8690

7.8692

7.8695

0.810

0.812

0.815

The Sicam (RC) columns estimate the original results presented in Ref. [8]. The differences between the Sicam (RC) and Sicam $(\mathrm{RC}+\mathrm{M})$ columns estimate the impact of deviations in corneal shape due to reconstruction method. The exact columns deviate from true $R=7.87\text{\hspace{0.17em}}\mathrm{mm}$ and $p=0.82$, as a result of the truncation of the finite Zernite representation.

Table 3

Estimates of R and p for the Sphere (Radial Order 14)^{
a
}

Pupil diameter (mm)

R (RC) (mm)

R$(\mathrm{RC}+\mathrm{M})$ (mm)

R (exact) (mm)

p (RC)

p$(\mathrm{RC}+\mathrm{M})$

p (exact)

10

6.9998

6.9999

6.9999

0.997

0.997

0.998

11

6.9988

6.9990

6.9994

0.981

0.985

0.990

The table includes only the rows for $10\text{\hspace{0.17em}}\mathrm{mm}$ and $11\text{\hspace{0.17em}}\mathrm{mm}$ because errors were negligible otherwise. The results for the conic were not included because errors were completely negligible for all parameter values used.