## Theoretical estimation of moiré effect using spectral trajectories |

Optics Express, Vol. 21, Issue 2, pp. 1693-1712 (2013)

http://dx.doi.org/10.1364/OE.21.001693

Acrobat PDF (1260 KB)

### Abstract

Equations for spectral peaks and trajectories are found for *N* superposed one-dimensional gratings. The equations of trajectories are represented using the complex numbers. The number of geometric elements in the spectrum is found under various conditions and in the matrix form. The derivatives of trajectories are obtained. The orthogonal case is investigated in details, in particular, the regular structures (the square and the octagon) are found in the spectrum. The numerical simulation is in a good agreement with the theory. The proposed technique seems to be helpful in estimation of occurrence of moiré patterns in visual displays.

© 2013 OSA

## 1. Introduction

2. O. Bryngdahl, “Moiré: Formation and interpretation,” J. Opt. Soc. Am. **64**(10), 1287–1294 (1974). [CrossRef]

4. K. Patorski, K. Pokorski, and M. Trusiak, “Fourier domain interpretation of real and pseudo-moiré phenomena,” Opt. Express **19**(27), 26065–26078 (2011). [CrossRef] [PubMed]

5. S. Yokozeki, Y. Kusaka, and K. Patorski, “Geometric parameters of moiré fringes,” Appl. Opt. **15**(9), 2223–2227 (1976). [CrossRef] [PubMed]

6. I. Amidror, S. Chosson, and R. D. Hersch, “Moiré methods for the protection of documents and products,” J. Phys.: Conf. Series **77**, 012001 (2007). [CrossRef]

7. S.-G. Wang and Z. Fan, “Moiré-free color halftoning using 2x2 printer modeling,” Proc. SPIE **4300**, 397–403 (2000). [CrossRef]

8. Y. Kim, G. Park, J.-H. Jung, J. Kim, and B. Lee, “Color moiré pattern simulation and analysis in three-dimensional integral imaging for finding the moiré-reduced tilted angle of a lens array,” Appl. Opt. **48**(11), 2178–2187 (2009). [CrossRef] [PubMed]

9. V. Saveljev and S.-K. Kim, “Simulation and measurement of moiré patterns at finite distance,” Opt. Express **20**(3), 2163–2177 (2012). [CrossRef] [PubMed]

10. M. Dohnal, “Moiré in a scanned image,” Proc. SPIE **4016**, 166–170 (1999). [CrossRef]

9. V. Saveljev and S.-K. Kim, “Simulation and measurement of moiré patterns at finite distance,” Opt. Express **20**(3), 2163–2177 (2012). [CrossRef] [PubMed]

11. V. Saveljev, “Characteristics of moiré spectra in autostereoscopic three-dimensional displays,” J. Displ. Technol. **7**(5), 259–266 (2011). [CrossRef]

3. O. Bryngdahl, “Characteristics of superposed patterns in optics,” J. Opt. Soc. Am. **66**(2), 87–94 (1976). [CrossRef]

12. C. Huang, J. R. G. Townshend, X. Zhan, M. Hansen, R. DeFries, and R. Solhberg, “Developing the spectral trajectories of major land cover change processes,” Proc. SPIE **3502**, 155–162 (1998). [CrossRef]

13. P. S. Costa-Pereira and P. Maillard, “Estimating the age of cerrado regeneration using Landsat TM data,” Can. J. Rem. Sens. **36**(S2), S243–S256 (2010). [CrossRef]

11. V. Saveljev, “Characteristics of moiré spectra in autostereoscopic three-dimensional displays,” J. Displ. Technol. **7**(5), 259–266 (2011). [CrossRef]

*u*,

*v*). The ordered pairs of numbers can be treated as complex numbers (

*u*,

*v*) =

*u*+

*iv*and graphically drawn in the complex plane. This way one may employ the perfectly developed powerful and convenient mathematical tools to describe the moiré effect. Note that using the complex numbers and vector summation to describe moiré patterns in the Fourier space were primarily suggested earlier in [2

2. O. Bryngdahl, “Moiré: Formation and interpretation,” J. Opt. Soc. Am. **64**(10), 1287–1294 (1974). [CrossRef]

*N*gratings with some specific parameters. The topics include the number of elements (particularly in the compact matrix form as powers), the derivatives of the spectral trajectories of moiré waves. To illustrate the technique, we provide a number of analytical consequences from the basic equation together with graphical examples obtained in simulation and in experiment. In particular, we describe the regular structures in the spectrum, their visibility and appearance in the spatial domain, as well as relationships between them. The experiments are performed with non-sinusoidal gratings (square wave profile) for all four possible combinations of geometric characteristics of layers and are presented for two varying parameters.

*N*one-dimensional gratings is considered in Section 2, including the locations of the spectral peaks, the spectral trajectories with one parameter varying, and the number of elements. Section 3 covers the special case of 4 gratings arranged in 2 layers by orthogonal pairs which is important for practical applications; the theory and simulation for sinusoidal gratings is considered, including the trajectories and the number of elements; this includes the case of

*N*= 2. The regular geometric structures in the spectrum are described in Section 4. The experimental data are presented in Section 5. The terminative Section 6 finalizes the paper by discussion and conclusion.

## 2. General case of *N* gratings

### 2.1 Spectral peaks

*R*(

*z*) of the coordinates (

*u*,

*v*) or of the complex number

*z*=

*u*+

*iv*. Generally, such a function can be drawn as a surface, located “above” the complex plane, so as the height at each point

*z*is the value of the function

*R*(

*z*). However, this paper is focused on locations of the spectral peaks, especially in the neighborhood of the origin. This allows a simpler description of the two-dimensional moiré effect, which is capable to predict many essential features. This approximation gives the answer in terms of the wavevector, i.e., wavelength and direction of the moiré waves.

*p*-th peak of a comb can be written directly as the product of two numbers, integer and complex,where the integer

*p*is the index, which generally can take any integer value between –∞ and + ∞, while the complex

*k*is the wavevector corresponding to the period of a grating. For a limited spectrum with a finite number of peaks,

*p*= -

*q*, …, +

*q*is an integer number between -

*q*and +

*q*(i.e., max|

*p*| =

*q*) such that

*Q*= 2

*q*+ 1 is the total number of peaks in the spectrum of a one-dimensional grating.

*k*represents the fundamental spatial frequency; its polar coordinates are the modulus |

*k*| and the argument Arg

*k*, i.e., the fundamental wavenumber and the direction (orientation) angle, resp.

*Q*can be assigned to a finite number, if necessary, basing on the decay of Fourier coefficients and depending on the required accuracy.

*k*and the rotation of the grating about the origin by angle

*α*can be expressed as the product of the basic wavenumber Eq. (1) and the complex number

*e*basing on the Euler identity. Therefore, the locations of

^{iα}*Q*peaks of the transformed grating with a limited spectrum are given bywhere

*i*is the imaginary unit, and

*α*is the rotation angle.

*N*superposed gratings,orwhere

*N*is the number of gratings, and the values of

*k*,

_{n}*α*,

_{n}*p*,

_{n}*q*are attributed to the

_{n}*n*-th grating (

*n*= 1, …,

*N*) as follows: the two former values are the basic wavenumber and the rotation angle, while

*p*is an integer number between -

_{n}*q*and +

_{n}*q*. This equation explicitly indicates that the peaks of

_{n}*N*gratings are added to all combinations of other gratings. Equation (4) includes several vector sums (

*N*terms each) which involve the combinational peaks. An example is shown in Fig. 1, where two vector sums (of 25 in this set) are indicated by arrows originated at (0, 0), for illustration purpose. The other peaks can be similarly reconstructed as the vector sums of two components.

*p*

_{1}| = max |

*p*

_{2}| = 1,

*k*

_{1}=

*kσ*

_{1},

*k*

_{2}=

*k*,

*α*

_{1}= 0,

*α*

_{2}= π/2), so as

*e*

^{iπ}^{/2}=

*i*(derived from the Euler’s formula) was implicitly applied. The set of parameters in Eq. (6) is convenient for the rectangular grids, see Fig. 2. In this set, the parameters

*σ*

_{1}and

*σ*

_{3}are aspect ratios (the ratio of the vertical and horizontal periods of the rectangular cell of a grid), and

*ρ*is the relative size of grids (the ratio of the vertical periods of two grids).

### 2.2 Spectral trajectories

*k*or

*α*being a function of another variable

*t*; the variable

*t*is not necessarily the time, though.

*k*, the equation of trajectories looks like the follows,where

_{m}*p*= -

_{j}*q*, …,

_{j}*q*and

_{j}*k*

_{1}, …,

*k*

_{m}_{-1},

*k*

_{m}_{+1}, …

*k*are constants (which do not depend on

_{N}*t*). In this case, the

*m*-th term describe the lines, while all other terms describe the centers.

*m*-th depending term includes the integer number between -

*q*and +

_{m}*q*as a coefficient. In other words, the pairs of trajectories are locally symmetric with respect to their centers.

_{m}*k*(

*t*) can be found as follows,

*k*(

*t*) =

*A*+

_{k}t*B*, and recalling Eq. (2),where

_{k}*m*-th term of Eq. (8).

*α*(

*t*) should be used instead of

*k*(

*t*), and therefore

*α*(

*t*) =

*A*+

_{α}t*B*),

_{α}### 2.3 Number of elements

*N*terms in Eq. (3); the

*j*-th term contains

*Q*peaks. Then, the total number of the geometric elements in the spectrum can be expressed as follows,

_{j}*Q*

_{1}and

*Q*

_{2}spectral peaks each), the spectral peaks are arranged in the parallelogram (see Fig. 2) with

*Q*

_{1}and

*Q*

_{2}, peaks along each side. Therefore, there are

*Q*

_{1}∙

*Q*

_{2}peaks within this parallelogram.

*Q*=

_{j}*Q*, instead of Eq. (13), a simpler expression can be obtained,

_{0}*c*and

*l*characterize the type of geometric elements (centers and lines).

*Q*, where

_{m}*m*is the index of the parameter-dependent term. Particular cases can be analyzed in details as the orthogonal gratings in the next section.

## 3. Special case of 4 gratings in 2 layers by orthogonal pairs

### 3.1 Trajectories

*d*

_{1}and

*d*

_{2}(the geometric characteristics of the structure of the layers), the values of which are assigned as follows, 1 for a grating (one-dimensional structure) and 2 for a grid (two-dimensional). When these quantities are equal to each other, the common value

*d*=

*d*

_{1}=

*d*

_{2}can be used. The convenient set of parameters is described in the previous section, see Eq. (6). With using it, the trajectories of four kinds can be obtained from Eq. (7), for the running parameters

*α*,

*ρ*,

*σ*

_{3}, and

*σ*

_{1}, resp.

*d*

_{1}and

*d*

_{2}from (1, 1) to (2, 2) as follows,

*At*+

*B*), one can obtain

### 3.2 Number of elements

*p*

_{1},

*p*

_{2}and

*p*

_{3}between −1 and + 1; this means the triple amount of centers for varying

*σ*

_{1}and

*σ*

_{3}, as compared to varying

*α*and

*ρ*.

*ρ*and

*α*, the centers are always located along a horizontal line (for

*d*

_{1}= 1) or the sides of a square (for

*d*

_{1}= 2) by Eqs. (16) and (17). For varying

*σ*, the locations of centers are different: the horizontal/slanted line (

*d*= 1), the rotated square or parallelogram (

*d*

_{1}≠

*d*

_{2}), and 3 displaced squares/parallelograms (

*d*= 2). In the cases

*d*

_{1}= 2, the locations of centers include vertically displaced locations for

*d*

_{1}= 1 and corresponding

*d*

_{2}. Illustrations of that can be found in [18].

*d*=

_{c}*d*

_{1}+

*h*- 1, where and

*h*=

*d*

_{2}for varying

*σ*and

*h*= 1 for varying

*α*and

*ρ*.

*d*in (36) varies, depending on the combinations of the current varying parameter and numbers

_{c}*d*. To show the number of elements for all these combinations, the matrix form can be used as followswhere

_{j}*M*is the matrix for the total number of the geometric elements,

_{tot}*M*for centers and

_{c}*M*for lines. The matrices

_{l}*M*,

_{tot}*M*and

_{c}*M*are arranged as follows. Their columns correspond to the four combinations of

_{l}*d*

_{1}and

*d*

_{2}in the following order, (1, 1), (1, 2), (2, 1), (2, 2), while the rows correspond to the varying parameters arranged in two pairs, the first row for

*α*or

*ρ*, while the second row for

*σ*

_{1}or

*σ*

_{3}; this is based on the equal numbers for running

*α*or

*ρ*and for

*σ*

_{1}or

*σ*

_{3}The expanded expression Eq. (37) looks like follows,or

*M*and

_{tot}*M*are powers of 3; this exactly corresponds to Eq. (36). The matrix elements of

_{c}*M*are the products of powers of 2 and 3. One can also find the ratio of

_{l}*N*and

_{tot}*N*,where

_{c}*g*= 1 for varying

*σ*and

*g*=

*d*

_{2}for varying

*α*and

*ρ*, or to. Besides,

*g*< 3 (as it is in this paper). Alternatively, the ratios (40), (41) can be written in the matrix form with the division (/) treated as an entry-by-entry matrix operation,

## 4. Regular geometric structures in spectrum

*d*= 2. In this section, the only varying angle is implied, so as the functional notation “(

*t*)” in expressions like “

*α*(

*t*)” will be omitted here.

*u*and

_{j}*v*are the real and imaginary parts of the complex coordinates of three points.

_{j}*z*

_{1}and

*z*

_{2}means that their dot product equals zero,or, in terms of the real and imaginary parts,

### 4.1. Square

*p*

_{1}=

*p*

_{3}= 0,

*p*

_{2}= -

*p*

_{4}≠ 0 in Eqs. (48), (52), and

*p*

_{2}=

*p*

_{4}= 0,

*p*

_{1}= -

*p*

_{3}≠ 0 in Eqs. (50), (53). The remaining formulas Eqs. (48)-(55) are particular cases with non-zero integers as

*p*

_{1}=

*p*

_{2}= 1,

*p*

_{3}=

*p*

_{4}= −1 in Eq. (49),

*p*

_{1}=

*p*

_{4}= 1,

*p*

_{2}=

*p*

_{3}= −1 in Eq. (51),

*p*

_{3}=

*p*

_{4}= 1,

*p*

_{1}=

*p*

_{2}= −1 in Eq. (53), and

*p*

_{2}=

*p*

_{3}= 1,

*p*

_{1}=

*p*

_{4}= −1 in Eq. (55).

*σ*

_{1},

*σ*

_{3}, and

*ρ*. The similar statements about zero determinants can be proven for three other triplets. Therefore, the trajectories Eqs. (48)–(55) lie at the corners and at the midpoints of a quadrilateral with parallel sides.

*σ*

_{1}=

*σ*

_{3}, the dot product Eq. (59) equals zero, as required. It means a rectangle, not a parallelogram. The side of the rectangle is equal to the distance between two corners, say, Eqs. (49) and (51), as follows,

*a*

_{1}and

*a*

_{2}in Eqs. (60) and (61) are identical when

*σ*

_{1}=

*σ*

_{3}= 1. This condition means a square.

*v*

_{1}–

*v*

_{2})/(

*u*

_{1}-

*u*

_{2}), where

*u*and

_{j}*v*are defined after Eq. (45). From Eqs. (54) and (55) one can find that

_{j}*ρ*= 1, the side of the square from Eq. (60) isand when the angle

*α*reaches 90°, the side

*a*becomes 2 √2.

*ρ*= 1, Eq. (62) is simplified

### 4.2. Octagon

*σ*

_{1}=

*σ*

_{3}=

*ρ*= 1 is only considered.

*α*= π/4,

*d*=

_{eb}*d*=

_{bg}*θ*, where

*θ*is the central angle. The central angle can be determined analytically through the difference of two arctangents Eqs. (66) and (69) as follows

*α*= π/4 obtained above into Eq. (76), one can find the central angle π/4. Therefore, the interior angle of the polygon is equal to π - π/4 = 3π/4, as required for the regular octagon. Similar conditions can be proved for other pairs and triplets of vertices in Eq. (65)-(72). Thus, all angles are equal, as well as all sides are. Therefore, the trajectories Eqs. (65)-(72) at π/4 describe a regular octagon.

## 5. Experiments

*d*

_{1}and

*d*

_{2}with

*σ*= 1,

*ρ*between 0.71 and 1.4. The profile of gratings was rectangular (non-sinusoidal). The varying angle

*α*changed from 0° to approx. 70°.

*ρ*or one of

*d*was changed and the procedure repeated.

### 5.1. Varying angle α

*α*are shown in Figs. 9, 10, 11, and 12, where, the superposed spectra are only presented without intermediate stages of processing. The resulting examples of trajectories simulated by Eqs. (16)-(19) are presented in [18] and might be considered in parallel to Figs. 9–12.

### 5.2. Varying size ρ

*ρ*can be found. Such chunks of data can be assembled and treated as regular experiments with longer time interval between the successive measurements. As far as the time issues are not considered in this paper, this difference matters nothing, and all trajectories obtained by rearranging the existing data are equally valuable. The experimental trajectories obtained this way are shown in Fig. 13, each paired with corresponding sinusoidal simulation.

*ρ*, experimental at the left and simulated at the right side of Fig. 13, are evidentially similar.

### 5.3. Analysis of experimental data

*ρ*= 1 and

*ρ*= 2. This means the experimental observation of the moiré effect in non-sinusoidal gratings with

*q*≤ 2. The simulated sinusoidal trajectories (

*Q*≤ 5, up to 25 centers) for the identical periods (

*ρ*= 1) and for the period of grating twice shorter than grid (

*ρ*= 2) are shown in Fig. 14. This illustration represents the sketches of separate spectral components of the non-sinusoidal gratings.

*Q*x

*Q*centers are inscribed. There are 13 such centers for

*ρ*= 1 and 5 for

*ρ*= 2, as shown in Fig. 14(a) and Fig. 14(b), respectively.

*ρ*= 1, 2 are drawn over the experimental trajectories Fig. 11(b). This provides the direct comparison of several spectral components with experiment.

*ρ*and

*q*provide a suitable explanation of the experimentally observed spectral trajectories in non-sinusoidal gratings.

## 6. Discussion and conclusion

*N*overlapped one-dimensional gratings are found in the closed form. The practically important case of two layers with orthogonal gratings, typical for the autostereoscopic 3D display, is considered in details. All formulas of this case are confirmed by computer simulation. The simulation program is based on two-dimensional Fourier transformation and the visibility circle concept [1].

*σ*

_{1}=

*σ*

_{3}= 1, there always exists a square in the spectrum for any angle

*α*. In sub-section 4.2 it was proved that in the case of the case of

*σ*

_{1}=

*σ*

_{3}=

*ρ*= 1, the angle

*α*= π/4 provides the regular octagon in the spectrum. The corresponding structure in the spatial domain is a non-periodic tiling of the plane by almost regular octagons.

*ρ*may look less continuous than trajectories in Figs. 9–12. However, with an increased number of points per segment, the trajectories for varying

*ρ*should become similarly continuous, even if obtained by rearrangement. Trajectories for varying

*σ*can be built in a similar way, basing on the data for varying angle with different parameter

*σ*.

*n*along each coordinate virtually makes similar decay along any direction; practically it looks like a circle. Formally speaking, the number of the active centers is equal to the number of points of norm less than or equal to

*n*

^{2}in square lattice. The explanation of the experiment Fig. 11(b) additionally shows that practically, it is unnecessary to consider the infinite spectra. For practical purpose (like an estimation of the visual moiré effect in printed gratings), it is sufficient to consider the limited spectrum consisting of approx. 5 peaks per center (as in this paper).

3. O. Bryngdahl, “Characteristics of superposed patterns in optics,” J. Opt. Soc. Am. **66**(2), 87–94 (1976). [CrossRef]

## Acknowledgments

## References and links

1. | I. Amidror, |

2. | O. Bryngdahl, “Moiré: Formation and interpretation,” J. Opt. Soc. Am. |

3. | O. Bryngdahl, “Characteristics of superposed patterns in optics,” J. Opt. Soc. Am. |

4. | K. Patorski, K. Pokorski, and M. Trusiak, “Fourier domain interpretation of real and pseudo-moiré phenomena,” Opt. Express |

5. | S. Yokozeki, Y. Kusaka, and K. Patorski, “Geometric parameters of moiré fringes,” Appl. Opt. |

6. | I. Amidror, S. Chosson, and R. D. Hersch, “Moiré methods for the protection of documents and products,” J. Phys.: Conf. Series |

7. | S.-G. Wang and Z. Fan, “Moiré-free color halftoning using 2x2 printer modeling,” Proc. SPIE |

8. | Y. Kim, G. Park, J.-H. Jung, J. Kim, and B. Lee, “Color moiré pattern simulation and analysis in three-dimensional integral imaging for finding the moiré-reduced tilted angle of a lens array,” Appl. Opt. |

9. | V. Saveljev and S.-K. Kim, “Simulation and measurement of moiré patterns at finite distance,” Opt. Express |

10. | M. Dohnal, “Moiré in a scanned image,” Proc. SPIE |

11. | V. Saveljev, “Characteristics of moiré spectra in autostereoscopic three-dimensional displays,” J. Displ. Technol. |

12. | C. Huang, J. R. G. Townshend, X. Zhan, M. Hansen, R. DeFries, and R. Solhberg, “Developing the spectral trajectories of major land cover change processes,” Proc. SPIE |

13. | P. S. Costa-Pereira and P. Maillard, “Estimating the age of cerrado regeneration using Landsat TM data,” Can. J. Rem. Sens. |

14. | Y.-P. Lai and M.-H. Siu, “Hidden Spectral Peak trajectory model for Phone Classification,” Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing (2004), I-909 – 912. |

15. | S. W. Lee, F. K. Soong, and P. C. Ching, “Iterative trajectory regeneration algorithm for separating mixed speech sources,” Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing (2006), I-157 – 160. |

16. | J. S. Neto and J. L. A. de Carvalho, F.A. de O. Nascimento, A.F. da Rocha, and L.F. Junqueira Jr., “Trajectories of Spectral Clusters of HRV Related to Myocardial Ischemic Episodes,” Proc. XVIII Congresso Brasileiro de Engenharia Biomédica |

17. | M. Davy, B. Leprettre, C. Doncarli, and N. Martin, Tracking of spectral lines in an ARCAP time-frequency representation,” Proc. 9th Eusipco Conference (1998), 633 −636. |

18. | V. Saveljev and S. K. Kim, “Estimation of Moiré Patterns using Spectral Trajectories in the Complex Plane,” Computer Technol. Appl. |

**OCIS Codes**

(120.2040) Instrumentation, measurement, and metrology : Displays

(120.4120) Instrumentation, measurement, and metrology : Moire' techniques

(350.2770) Other areas of optics : Gratings

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: October 10, 2012

Revised Manuscript: January 4, 2013

Manuscript Accepted: January 7, 2013

Published: January 16, 2013

**Citation**

Vladimir Saveljev and Sung-Kyu Kim, "Theoretical estimation of moiré effect using spectral trajectories," Opt. Express **21**, 1693-1712 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-2-1693

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### References

- I. Amidror, The Theory of Moiré Phenomenon (Springer, 2009) vol. 1.
- O. Bryngdahl, “Moiré: Formation and interpretation,” J. Opt. Soc. Am.64(10), 1287–1294 (1974). [CrossRef]
- O. Bryngdahl, “Characteristics of superposed patterns in optics,” J. Opt. Soc. Am.66(2), 87–94 (1976). [CrossRef]
- K. Patorski, K. Pokorski, and M. Trusiak, “Fourier domain interpretation of real and pseudo-moiré phenomena,” Opt. Express19(27), 26065–26078 (2011). [CrossRef] [PubMed]
- S. Yokozeki, Y. Kusaka, and K. Patorski, “Geometric parameters of moiré fringes,” Appl. Opt.15(9), 2223–2227 (1976). [CrossRef] [PubMed]
- I. Amidror, S. Chosson, and R. D. Hersch, “Moiré methods for the protection of documents and products,” J. Phys.: Conf. Series77, 012001 (2007). [CrossRef]
- S.-G. Wang and Z. Fan, “Moiré-free color halftoning using 2x2 printer modeling,” Proc. SPIE4300, 397–403 (2000). [CrossRef]
- Y. Kim, G. Park, J.-H. Jung, J. Kim, and B. Lee, “Color moiré pattern simulation and analysis in three-dimensional integral imaging for finding the moiré-reduced tilted angle of a lens array,” Appl. Opt.48(11), 2178–2187 (2009). [CrossRef] [PubMed]
- V. Saveljev and S.-K. Kim, “Simulation and measurement of moiré patterns at finite distance,” Opt. Express20(3), 2163–2177 (2012). [CrossRef] [PubMed]
- M. Dohnal, “Moiré in a scanned image,” Proc. SPIE4016, 166–170 (1999). [CrossRef]
- V. Saveljev, “Characteristics of moiré spectra in autostereoscopic three-dimensional displays,” J. Displ. Technol.7(5), 259–266 (2011). [CrossRef]
- C. Huang, J. R. G. Townshend, X. Zhan, M. Hansen, R. DeFries, and R. Solhberg, “Developing the spectral trajectories of major land cover change processes,” Proc. SPIE3502, 155–162 (1998). [CrossRef]
- P. S. Costa-Pereira and P. Maillard, “Estimating the age of cerrado regeneration using Landsat TM data,” Can. J. Rem. Sens.36(S2), S243–S256 (2010). [CrossRef]
- Y.-P. Lai and M.-H. Siu, “Hidden Spectral Peak trajectory model for Phone Classification,” Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing (2004), I-909 – 912.
- S. W. Lee, F. K. Soong, and P. C. Ching, “Iterative trajectory regeneration algorithm for separating mixed speech sources,” Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing (2006), I-157 – 160.
- J. S. Neto and J. L. A. de Carvalho, F.A. de O. Nascimento, A.F. da Rocha, and L.F. Junqueira Jr., “Trajectories of Spectral Clusters of HRV Related to Myocardial Ischemic Episodes,” Proc. XVIII Congresso Brasileiro de Engenharia Biomédica 5 (2002), 365 −370.
- M. Davy, B. Leprettre, C. Doncarli, and N. Martin, Tracking of spectral lines in an ARCAP time-frequency representation,” Proc. 9th Eusipco Conference (1998), 633 −636.
- V. Saveljev and S. K. Kim, “Estimation of Moiré Patterns using Spectral Trajectories in the Complex Plane,” Computer Technol. Appl.3, 353–360 (2012).

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