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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

In this study, a camera to infrared diode (IRED) distance estimation problem was analyzed. The main objective was to define an alternative to measures depth only using the information extracted from pixel grey levels of the IRED image to estimate the distance between the camera and the IRED. In this paper, the standard deviation of the pixel grey level in the region of interest containing the IRED image is proposed as an empirical parameter to define a model for estimating camera to emitter distance. This model includes the camera exposure time, IRED radiant intensity and the distance between the camera and the IRED. An expression for the standard deviation model related to these magnitudes was also derived and calibrated using different images taken under different conditions. From this analysis, we determined the optimum parameters to ensure the best accuracy provided by this alternative. Once the model calibration had been carried out, a differential method to estimate the distance between the camera and the IRED was defined and applied, considering that the camera was aligned with the IRED. The results indicate that this method represents a useful alternative for determining the depth information.

The geometrical camera model uses a mathematical correspondence between image plane coordinates and real word coordinates by modeling the projection of the real world onto the image plane [

This problem is known as depth estimation. A general approach to solving this problem is to introduce additional constraints into the mathematical system. These constraints can be obtained from other sensor devices (cameras, laser,

In addition, the geometrical model only uses the image coordinates as the principal source of information, and image gray level intensities are only used to ensure correspondence among images [

In references [

Images must be formed by the energy emitted by the IRED. The rejection of background illumination is obtained using an interference filter centered on 940 nm with 10 nm of bandwidth. This implies to using a 940 nm IRED.

The IRED must be biased with a constant bias current. This guarantees constant IRED radiant intensity.

The IRED and the camera must be aligned. This means that the radiant intensity as a function of the IRED orientation angle is also constant.

Under these conditions, the distance between the camera and the IRED can be obtained from relative accumulated energy [

Strategically, it would be advantageous to find another parameter and relate it to camera exposure time, IRED radiant intensity and camera to IRED distance. This process would increase the number of constraints extracted from images, thus improve the algorithm in future implementations.

By

The distance between the camera and the IRED is the main unknown, but in a future implementation the radiant intensity and the IRED orientation angle will need to be estimated or excluded from the final distance estimation alternative.

Regarding the IRED characteristic, the IRED radiant intensity is fixed by the bias current and varies with temperature, material aging, and other factors. Thus, radiant intensity will introduce drift into the distance measurement alternative.

To solve the ill-posed problem, at least one other parameter must be considered to define the final non-geometrical alternative for measuring the distance between the camera and an IRED.

In the representation shown in

The central peak that is shown in

This representation takes into account the gray level intensities of the image pixels, which were obtained by the energies that fall on sensor surface and were accumulated by the camera during the image capturing process.

Reference [

To estimate the relative accumulated energy, the camera inverse response function must be used. This function establishes a correspondence between pixel gray level intensity at the camera output and the energy accumulated by the camera during the exposure time [

In [

Comparing the parameters proposed in [

Therefore, similar to the procedure followed in [

In this case, the standard deviation of the image gray level intensities that were included in the region of interest and contained the IRED image is proposed as the empirical parameter to be extracted from the IRED image. Note that the standard deviation depends exclusively on the pixel gray level distribution, rather than on the pixel position that is used in projective models [

The standard deviation (Σ) provides a measure of the dispersion of image gray level intensities and can be understood as a measure of the power level of the alternating signal component acquired by the camera. Therefore, a relationship would exist between the standard deviation and the camera exposure time, IRED radiant intensity and distance, assuming that the IRED and the camera are aligned.

To use Σ in order to derive an alternative for measuring the distance between the IRED and the camera, a model for the standard deviation must be obtained. In other words, an expression for _{p}

To estimate the function _{p}

In all cases, a region of interest containing the IRED image is selected in the processed image. For example, _{i}

To characterize Σ obtained by

For each condition, 10 images were acquired. The final value for Σ in each condition was the median value over the 10 images acquired. The result of this behavior is shown in

The 10 images for each condition were used to perform a statistical model and also ensure the reduction of noise in the behavior of Σ in the model characterization process. Nevertheless, the consistency of the Σ parameter was measured before the behaviors were obtained.

As can be seen in

Under these conditions, Σ can be modeled by a linear function of camera exposure time. Mathematically it can be written as:
_{p}, δ_{1} and _{2} are the coefficients to model the linear relationship between

The non-linear behavior of Σ with the exposure time shown in

IRED radiant intensity can be controlled by the IRED bias current, as is shown in

To characterize the behavior of Σ with the IRED radiant intensity, images captured with different bias currents were used.

_{p}_{1} and _{2} are the coefficients used to model a lineal relationship between Σ and _{p}

To include the distance between the camera and the IRED in

To obtain the result shown in _{p}

However, distance behavior was considered quadratic, rather than linear as in references [

Finally, the behavior of Σ with the distance between the camera and the IRED were considered as a quadratic function. Therefore, in the _{i}

The behaviors measured and presented in Sections 2.1, 2.2 and 2.3 were integrated into a model that theoretically characterized the standard deviation of pixel gray level in the region of interest containing the IRED image.

Taking into account the

After parenthesis elimination, the standard deviation yields:
_{i}

To calibrate the model proposed in

The data used to obtain the values for the model coefficients are summarized in

For each distance, five IRED bias currents were considered. For each distance and IRED bias current, 16 images with different exposure times were used; therefore _{eqns}

From each of the equations, the error between the modeled and measured standard deviation can be defined. Thus:
_{modeled}_{measured}

The values for the model coefficients were calculated to minimize the error stated in

The coefficients k can be calculated by k = M^{+} × S, where M^{+} = (M^{t} × M)^{−1} × M is the pseudo-inverse matrix of M, which is formed with the calibration data summarized in _{i}

In

The relative error in the calibration process described, which is shown in

Once the coefficients had been obtained, the model proposed in

The differential method used two images captured with different exposure times. Consider two images (_{j}_{r}_{j}_{r}_{j}_{r}_{j}_{r}_{j}_{r}

From the

Then, the distance estimation can be obtained from the positive real root of

As can be seen from

Another question must be considered; for example, the

In reference [

Using the calibration data summarized in

Once model coefficient had been calculated in the calibration process, the model was written in the differential form to estimate the error in the calibration process. This process was used to evaluate the performance of the model and to detect the values of time differences where lowest error would be obtained for each bias current.

From _{r}

In addition, in

In the experimental tests, an SFH4231 IRED [

Because the IRED wavelength was 940 nm, the camera was fitted with an interference filter, centered at 940 nm with 10 nm of bandwidth, which was attached to the optic to reduce the influence of background illumination.

To exclude the influence of the orientation angle, the camera and the IRED were aligned by putting them in a line drawn in the floor. The alignment was obtained by rotating the IRED and the camera using goniometers, until a circular IRED image was obtained. To verify this alignment, several distances between camera and IRED were considered, and in all considered distances the images of the IRED were located in equals image's coordinates.

Once camera and IRED were put in the correct position and in the considered distance, they were raised from the floor using aluminum bars of squared-section and 1 m height to avoid the reflection on the floor.

Starting with an energetic study, the camera resolution in the radiometric model would not affect the performance. Note that the quantity of energy acquired by the camera will be the same in either a large or a small number of pixels. Evidently, if more pixels could be used, more accurate measurement could be obtained because an average could be used to reduce the spatially distributed noise. Alternatively, in the case of lower resolution cameras, the noise reduction could be achieved by temporal average, which implies to use more images for a single condition.

Currently, small resolutions do not constitute a strict problem from a practical point of view, because most camera sensors have higher than 640 × 480 pixels of resolutions. However, when a higher resolution could be used to estimate the Σ parameter, the result would be more noise-robust. Therefore, we recommend using square-ROI higher than 30 × 30 pixels to guarantee an average with more than 900 pixels. For the experiment performed to validate the distance estimation using the Σ parameter, a 60 × 60 pixels resolution was used.

To validate the standard deviation as an alternative method for estimating the distance between the camera and the IRED, a range of distances from 400 to 800 cm were considered. As shown in

The blue square marker represents the real distance and the colored circles represent the estimated distance. Circle color represents the relative error as a percentage of the distance estimation method. As can be seen from the color of the circles, most relative errors were lower than 5%.

The results shown in

A better result and greater efficiency in distance estimation could be obtained if optimum exposure times were used. As stated in Section 3.1 the optimum Δ

By using the optimum exposure time difference, the number of images used in the distance estimation process is reduced considerably. In this case, one optimum Δ

As can be seen in

In this paper, we have analyzed the estimation of distance between a camera and an infrared emitter diode. This proposal represents a useful alternative for recovering the depth information lost in projective models.

The alternative proposed in this paper follows the same idea that has been described in the references [

In addition, in this paper we have demonstrated the need to increase the number of constraints in order to reduce the number of degrees of freedom associated with the problem of estimating camera to emitter distance. The standard deviation alternative proposed here constitutes a helpful alternative.

The modeling process described in this paper was carried out in order to relate the standard deviation to the same magnitudes as those used in [

The standard deviation is a linear function of the camera exposure times and IRED radiant intensity.

The standard deviation is a quadratic function of the inverse-square distance between the camera and the IRED.

By using these conclusions, an expression for standard deviation was derived. The model for standard deviation had 12 coefficients, which were calculated in a calibration process.

The calibration process used images captured with different IRED radiant intensities values, camera exposure times and distances between the camera and the IRED.

By using a differential method, the distance between the camera and the IRED was obtained, using only the pixel gray-level information.

In addition, from data used in the calibration process and considering the differential method, an analysis of model fit was implemented in order to obtain the optimum exposure times to implement the measurement process. The maximum errors of distance estimations considering the optimum times were lower than 3%. Besides, in all experiments carried out to validate the distance estimation method proposed in this paper, the average relative errors were lower than a 1% in the range of distance from 440 to 800 cm.

The goal of this proposal was to define a useful alternative for extracting depth using only pixel gray level information. The main disadvantage of this proposal is that the relationship between the standard deviation and the IRED orientation angle was not considered in the modeling process.

This research was funded through the Spanish Ministry of Science and Technology sponsored project ESPIRA DPI2009-10143. The authors also want to thank the Spanish Agency of International Cooperation for Development (AECID) of the Ministry of Foreign Affairs and Cooperation (MAEC).

IRED image taken by the camera represented as a 3-D surface.

Behavior of Σ with camera exposure time (_{p}

Consistency of the Σ parameter extracted from the IRED images. The maximum dispersion over the 30 images was lower than a 0.5% of the mean value for each considered condition.

Behavior of Σ with

Relationship between IRED radiant intensity and the bias current.

Values of Σ obtained for a distance between the camera and the IRED of 520 cm, for different exposure times (6, 7, … , 16 ms) and for different IRED bias currents.

Values of Σ obtained for _{p}

The standard deviation as a function of ^{−2} From the behavior of Σ with

Result of the calibration process.

Relative error in the calibration process as a function of exposure time differences. In these figures different distances were considered: (a) 440 cm, (b) 560 cm, (c) 680 cm, (d) 720 cm and (e) 800 cm. From the performance of the calibration process it is possible to obtain the optimum exposure time differences to carry out the distance measurement process. Optimum Δ

Results of distance estimation process considering all available differences of exposure time for different bias currents: (a) for _{p}_{p}_{p}

Results of distance estimation process considering the optimum difference of exposure times (Δ_{p}_{p}_{p}

Data used in the calibration process.

Distance [cm] | 440, 560, 680 and 720 |

Exposure time [ms] | 2, 3, 4, … , 17 |

IRED's bias current [mA] | 7, 8, 9 and 10 |

Final distance estimation using the standard deviation of pixel gray-level intensities in an IRED image together with a differential method using the optimum exposure time difference.

| ||||||||
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_{p} |
_{p} |
_{p} |
_{p} |
|||||

440 | 439.3 | 0.2 | 437.3 | 0.6 | 439.3 | 0.2 | 439.6 | 0.1 |

480 | 478.8 | 0.3 | 474.2 | 1.2 | 471.5 | 1.8 | 468.5 | 2.4 |

520 | 520.1 | 0.0 | 514.5 | 1.1 | 511.7 | 1.6 | 510.2 | 1.9 |

560 | 561.1 | 0.2 | 554.4 | 1.0 | 556.1 | 0.7 | 553.0 | 1.2 |

600 | 601.3 | 0.2 | 598.2 | 0.3 | 598.6 | 0.2 | 597.3 | 0.4 |

640 | 639.1 | 0.1 | 635.9 | 0.6 | 638.7 | 0.2 | 639.0 | 0.2 |

680 | 682.2 | 0.3 | 680.5 | 0.1 | 684.7 | 0.7 | 686.0 | 0.9 |

720 | 718.7 | 0.2 | 721.6 | 0.2 | 724.4 | 0.6 | 728.8 | 1.2 |

760 | 755.4 | 0.6 | 759.3 | 0.1 | 772.2 | 1.6 | 773.6 | 1.8 |

800 | 790.7 | 1.2 | 800.8 | 0.1 | 813.1 | 1.6 | 820.7 | 2.6 |