Understanding the exposure-time effect on speckle contrast measurements for laser displays

Abstract

To evaluate the influence of exposure time on speckle noise for laser displays, speckle contrast measurement method was developed observable at a human eye response time using a high-sensitivity camera which has a signal multiplying function. The nonlinearity of camera light sensitivity was calibrated to measure accurate speckle contrasts, and the measuring lower limit noise of speckle contrast was improved by applying spatial-frequency low pass filter to the captured images. Three commercially available laser displays were measured over a wide range of exposure times from tens of milliseconds to several seconds without adjusting the brightness of laser displays. The speckle contrast of raster-scanned mobile projector without any speckle-reduction device was nearly constant over various exposure times. On the contrary to this, in full-frame projection type laser displays equipped with a temporally-averaging speckle-reduction device, some of their speckle contrasts close to the lower limits noise were slightly increased at the shorter exposure time due to the noise. As a result, the exposure-time effect of speckle contrast could not be observed in our measurements, although it is more reasonable to think that the speckle contrasts of laser displays, which are equipped with the temporally-averaging speckle-reduction device, are dependent on the exposure time. This discrepancy may be attributed to the underestimation of temporal averaging factor. We expected that this method is useful for evaluating various laser displays and clarify the relationship between the speckle noise and the exposure time for a further verification of speckle reduction.

Appendix

The cross-correlation was calculated by the following method. In [14], a zero-mean normalized cross-correlation is expressed by

$$R_{\text{ZNCC}} = \frac{{\mathop \sum \nolimits_{j = 0}^{N - 1} \mathop \sum \nolimits_{i = 0}^{M - 1} ((I\left( {i,j} \right) - \overline{I} )(T\left( {i.j} \right) - \overline{T} ))}}{{\sqrt {\mathop \sum \nolimits_{j = 0}^{N - 1} \mathop \sum \nolimits_{i = 0}^{M - 1} \left( {I\left( {i,j} \right) - \overline{I} } \right)^{2} \times \mathop \sum \nolimits_{j = 0}^{N - 1} \mathop \sum \nolimits_{i = 0}^{M - 1} \left( {T\left( {i,j} \right) - \overline{T} } \right)^{2} } }} ,$$

(A1)

where I (i, j) is a digital number of the original image, T (i, j) is a digital number of the template image with the size of M × N pixels, and the position of (i, j) is the address of template image in which the value of i is from 0 to M − 1 and the value of j is from 0 to N − 1. The values of $\overline{I}$ and $\overline{T}$ are expressed by

$$\overline{I} = \frac{{\mathop \sum \nolimits_{j = 0}^{N - 1} \mathop \sum \nolimits_{i = 0}^{M - 1} I(i, j)}}{MN} ,$$

(A2)

$$\overline{T} = \frac{{\mathop \sum \nolimits_{j = 0}^{N - 1} \mathop \sum \nolimits_{i = 0}^{M - 1} T(i, j)}}{MN} .$$

(A3)

In this work, the maximum value of R _ZNCC with the template image size of 128 × 128 was calculated by shifting every one pixel from the perfectly matched position between the original image and the template image to ± 128 pixels, and we plotted the maximum value as the cross-correlation R _CC in Figs. 9, 10 and 11. As a result, the value of R _CC was the same as the R _ZNCC at the perfectly matched position.

We calculated the speckle average grain size defined as an FWHM of the normalized auto-covariance function [15], which is expressed by

$$\begin{aligned} C_{I} \left( {x, y} \right) = \frac{{{\text{FT}}^{ - 1} \{ \left| {{\text{FT}}\left[ {I(x, y)} \right]} \right|^{2} \} - \left\langle {I(x, y)} \right\rangle^{2} }}{{\left\langle {I(x, y)^{2} } \right\rangle - \left\langle {I(x, y)} \right\rangle^{2} }} , \end{aligned}$$

(A4)

where I (x, y) is a digital number of the captured image at the position of (x, y), and FT and FT⁻¹ are Fourier and inverse Fourier Transform, respectively. $\langle \rangle$ is a spatial average. We plotted the FWHM of Eq. (A4) as S in Figs. 9, 10 and 11.