Since cameras integrate the incoming light during the interval of the exposure time, the image taken on a moving object may be blurred. The blurring effect depends on the speed of the object (the higher the speed the higher the blur) and the exposure time (the longer the exposure time the higher the blur). The latter effect can be utilized to measure the length of the exposure.
The method was applied in many forms to provide an estimate on the exposure time. Since rotating movements are easier to handle in measuring equipment than lateral movements, numerous methods used some form of rotating target; e.g., in the first published method a rotating disk with holes was applied [
26], or later a camera was rotated while taking a photo on a small fixed light source [
35]. For the sake of convenience, in simple measurement setups often conventional turntables were used to create controlled movement, as will be described in
Section 4.1. Later, instead of moving physical objects, electronic systems were utilized to simulate movement. In the era of cathode ray tubes (CRTs), the swiping electron beam on the display provided the moving target, which allowed higher precision and wider measurement range, as will be shown in
Section 4.2. Today’s measurement equipment utilizes
LED arrays, which will be discussed in
Section 4.3.
4.1. Moving Phisycal Target
In a convenient measurement setup, an image is taken of a small light object, which is rotated with known angular velocity. During the time of the exposure, the object will move, and so on the picture blurring will occur; instead of a point, an arc will be shown. For the sake of convenience, a turntable may be used for driving purposes [
36], and the measurement process may be automated [
37]. In
Figure 7, a measurement setup with a turntable is shown. The angular velocity
of the turntable is known, e.g.,
RPM (rotations per minute) or 45 RPM, and the measured angle of the blur is
. The length of the exposure
can be calculated as follows:
The measurement equipment and a photo taken with
s can be seen in
Figure 8.
The measurement uncertainty of (4) depends on the accuracy of angular velocity
and the accuracy of measurement
. Since normally the uncertainty of
is negligible compared to that of
, the uncertainty
can simply be approximated as follows:
and the corresponding relative uncertainty is
Since the maximum reading uncertainty of angle
on a photo is approx.
degrees, based on our experiments, and the uncertainty is independent of the actual value of
, the relative uncertainty (6) is practically inversely proportional with
. The maximum relative uncertainty of (6) are shown in
Figure 9 for a turntable with
RPM, where the measured angle (in degrees), as a function of exposure time is the following:
According to the results shown in
Figure 9, the uncertainty is around 1%, when the measured exposure time is higher than 1/4 s. For exposure times shorter than 1/40 s, the relative measurement uncertainty may be higher than 10%; thus, this measurement method is suitable only for long exposure times.
Measurements for cameras C2 and C3 were made by the turntable method, the results are shown in
Table 3, and the measured relative errors are also plotted on
Figure 9. In the case of C3, the error trend is similar to the theoretical measurement uncertainty, although the error is somewhat lower at shorter times, indicating a smaller reading uncertainty than 0.5 degrees. Since the error in this case is lower than the measurement uncertainty, it can be stated that the camera is more accurate than the measurement itself. In the case of C2, however, the error is significantly higher at longer exposure times: here, the camera has detectable deviation (in the range of 1–2%) from the nominal exposure values.
4.2. Moving Electron Beam: CRT Monitor
Instead of mechanical movement, a moving electron beam can be used to estimate the exposure time [
36]. A few decades ago, Cathode Ray Tubes (CRTs) were generally used in monitors and TV sets. This equipment provided an easily available and straightforward way to produce the necessary moving electron beams for the measurement. The principle is shown in
Figure 10. The CRT display creates the picture row-by-row, moving the electron beam from left to right in each row, then starting at the beginning of the next row. The picture is drawn on the screen frequently enough (60–120 times per second) so that the human eye does not see the blinking. When a photo is taken of the display, only those rows are shown in the pictures which were refreshed during the time of the exposure. (Note that other parts of the picture may also be visible, but not bright, due the phosphor persistence.) In
Figure 10, only a small slice of the full picture is visible. From the size of the visible part, the exposure time can be calculated.
A simple approach to estimate the exposure time is the following: let us calculate the number
of rows on the taken picture; for this, the display should contain a carefully designed pattern containing horizontal lines, e.g., in every tenth row. The time
necessary to draw one line can be calculated from the refresh rate and the number of lines of the monitor. A simple estimate of the exposure time can be the following:
This method, however, has bias, and may be significantly improved by investigating the process of exposure, as shown in
Figure 11a. Let us suppose that the monitor is refreshed from top to bottom, i.e., first the uppermost row is drawn, then the next, until the last row at the bottom of the screen. Also let us suppose that the camera is placed so that the curtains fall in the same direction as the rows follow each other on the image (since cameras create inverted images, this happens if the camera is oriented upside down).
As shown in
Figure 11a, the image is refreshed from top to bottom with speed
rows per second, where
. For sake of convenience, let us define the speed of the curtains as
rows per second (i.e., in one second the curtain would cover/uncover
rows of the monitor in the taken picture), the speed vector
pointing from top to bottom. Notice that in practice
.
At time instant , the monitor draws row A and the camera’s front curtain just opens before row A: it will be the first row shown in the picture. Since the curtain is faster than the beam, the front curtain will uncover the area below row A, followed by the slower beam. At time instant , the rear curtain reaches row A and covers it. At time instant , the rear curtain reaches the actually refreshed row B and covers it. It is the last row shown in the picture. The picture contains rows between A and B, their number is denoted by .
Notice that the number of lines refreshed during is always smaller than : the fact that the speed of the shutter is finite causes a bias in the measurement; the measured time according to (8) with is always longer than the real length of the exposure.
Notice that between time instants
and
the rear curtain covered the distance between rows
A and
B, thus
rows. The beam covered the same distance between time instants
and
Thus the following equation holds:
Let us consider the case when the camera is rolled 180 degrees (it is now in normal position), as shown in
Figure 11b. Now, the vertical speed of the beam points downwards, while the speed of the curtains points upwards. The process of exposure is the following: At time instant
, the front curtain uncovers row
C, which will be the first row shown in the picture. At time instant
the rear curtain reaches and covers the currently refreshed row
D. This will be the last row shown in the picture. Some time later, at
, the rear curtain reaches the position of row
C. In the taken picture rows between
C and
D are shown, their numbers being
.
Now let us notice that the rear curtain between time instants
and
covered the distance between rows
C and
D, altogether
rows. The same distance was covered by the beam between time instants
and
; thus, the following equation can be constructed:
Note that in this case there is a bias, too, if the naïve approach of (8) is used, but now the measured time is always smaller than .
From (9) and (10) the unbiased estimate of
can be expressed, as follows:
The uncertainty of the estimated exposure time can be calculated as follows. Using the partial derivatives
and
of (11), the variation of
, in the presence of measurement uncertainties
and
, can be approximated as follows:
If the reading uncertainties
and
are maximized by
, i.e.,
then the maximum estimation uncertainty
is the following:
and the maximum relative uncertainty is
Using approximate value
, the maximum relative uncertainty is estimated as follows:
Since
is constant, and the reading uncertainty
is also approximately constant (1–3 lines of uncertainty was experimented during the measurements), the relative estimation uncertainty (16) is inversely proportional with the exposure time.
Figure 12 shows the theoretical relative uncertainty, for
. Thus, reasonable measurements are possible between 1/125 and 1/4000; the uncertainty of the estimation may be below 1% above 1/500, but in the short exposure time region it can be as high as 10%.
An example measurement of C2 is shown in
Figure 13, with exposure time of 1/500 sec. The results were
and
, in normal and upside-down camera positions, respectively. The monitor draws 1 line in
(
), thus the naïve measurement results, according to (8) are
and 2.37
. The unbiased estimate of (11) is
, which is a good estimate of the nominal 1/500 s
value.
More measurement results are shown in
Table 4, the errors are also plotted on
Figure 12, for cameras C2 and C3. Since C3 has global shutter, for this camera
, so either approach gives the same unbiased estimate. The measurement results are quite close to the nominal values, for longer exposure time with error below 1%, while for shorter exposure times the error increased to 3%, possibly due to measurement inaccuracies, as was expected according to (16). C2 has rolling shutter, thus the naïve approach resulted in high errors. Notice that the error is always negative in normal, while positive in upside down position. The unbiased estimator shows good agreement between 1/125 s and 1/500 s, but there are significant differences for shorter shutter times (higher than the expected measurement uncertainties), thus the timing of the camera is probably not accurate in this range.
4.3. Running LED Array
The moving object can be replaced by an
LED array: in this setup, as shown in
Figure 14, one
LED is switched on at a time for time
. The
LEDs light up one after another, creating an effect as if one
LED was running circularly along the array. Such devices may use
LED stripes, as in [
38], where stripes of 100
LEDs were proposed, while the commercial product [
39] utilizes a 10 × 10 array of
LEDs.
Trivially, if a picture contains
bright
LEDs, then the exposure time
can be calculated as follows:
Notice that the array setup, shown in
Figure 14, results in the same problem that was discussed in the CRT case: the final speed of the rolling shutter will cause a bias. When the
LEDs are arranged in a single row, this effect is not present.
In practice the first and last
LEDs in the bright series may not be as bright as the other ones. Although the observed light intensity could be used to refine the estimate, it is safer to state that the reading uncertainty is not more than
in the count of
. The timing inaccuracy of the
LEDs can be neglected, thus the relative uncertainty of the measurement can be estimated as
Due to the limited resolution, the best result that can be obtained, according to (18), is 2/
, where
is the total number of
LEDs in the device. In a device containing 100
LEDs the relative uncertainty would be around 2%. The resolution, thus the accuracy, can be improved using multiple
LED timers, as shown in
Figure 15.
In the multi-timer device, the central
LED, illustrated as a wider
LED, has on-time
, while the on-time of the side
LEDs is
. Notice that
may be significantly higher than
. In the illustration
and
. If the central
LED is bright, on the left side there is
bright
LED, and on the right side there are
bright
LEDS, as shown in
Figure 15, then the exposure time is
resulting in
. Notice that the resolution is now determined by
, which may be a very small value, providing high resolution and high accuracy with a small number of
LEDs. The uncertainty is now estimated as
which in the example of
Figure 15 results in an error of 0.4%.
When the multi-timer device is used, for an unknown exposure time usually an iterative approach is necessary: first, the approximate exposure time is determined with , then is reduced and is set so that . Using the more and more accurate estimate , the values of and are updated using smaller and smaller values, until the required resolution is reached.
The utilization of a device, similar to
Figure 15, has disadvantages, too. Notice that the
LED pattern must fulfill the following requirements, R1 and R2, in order to contain meaningful measurement:
- R1:
The leftmost and rightmost LEDs must be dark (otherwise the numbers or would not be meaningful)
- R2:
The two side LEDs, next to the central LED must be bright (otherwise it would not be sure that the central LED was on for the full time of )
To capture such pattern, either the camera must be synchronized to the measurement device, or the user must be really lucky: the higher the ratio of the less probable that the image satisfies the requirements. If camera synchronization is not possible but the camera is able to record video, an alternative ‘quasi-synch’ method can be used, as follows:
The running
LED is not cycling continuously: the
LED runs along the line once and stops at the last
LED. The cycle will start again so that the repeat time of the cycle is
. If the camera’s framerate is
then
is tuned around
:
For
, the camera and the running
LEDs would be perfectly synchronized, thus pictures of the running
LED would be taken at exactly the same phase and all pictures of the video would contain the same image. Instead,
is used during the measurement, when each picture of the video is taken at a phase
time later than the previous one. Thus, the captured video stream scans the running
LED sequence, with offset changing by steps of
in each frame, eventually catching a desired time instant, similar to
Figure 15. After the video recording, a suitable frame, satisfying requirements R1 and R2, is selected and
and
are measured on the frame. Finally, (19) is used to calculate the exposure time estimate. Notice that the “quasi-synch” method in fact uses equivalent sampling [
40,
41], which will be discussed in
Section 5.
Figure 16 shows the multi-timer measurement equipment and a photo taken by C3 with settings
,
. From
Figure 13b, the values
and
can be read, thus, according to (19), the measured exposure time is
.
Measurement results of C3 can be seen in
Table 5. The nominal value, set by the user, is internally rounded and slightly modified by the camera, and the exact value can be queried. Thus, the column
Reported exposure time shows the exact timing reported by the camera. Column
shows the estimated exposure times, along with the maximum uncertainty. The uncertainty values were calculated as
; e.g.,
means that the side
LEDs on-time was
, causing maximum
reading uncertainty. As the results show, there is a systematic error of approximately
, which is especially visible at the lower time region: the camera has higher exposure time than it is actually reported by the camera’s software. Similar effects were observed concerning other camera types of the same manufacturer [
42]. The last column
Relative error(br) shows the relative measurement error, after the
bias was subtracted from the reported values. The accuracy of the measurement is really good: at lower speed the relative error is way below 1%, while around the few microseconds range the error increased to 7%. This measurement method allows exposure time measurement even with
accuracy.