Atmospheric propagation effects on apparent luminance

In a day picture, particularly one with diffuse lighting (dull sky), the use of light propagation laws (absorption and diffusion) sometimes allows, from mensuration or estimation of apparent luminance based on pixel levels, estimation of a possible distance range between an object observed on the picture and the camera. Some main equations on which this approach is based will be presented later.

Atmospheric diffusion effects on sharpness

Depending on weather conditions, atmospheric diffusion effects on apparent sharpness of contours may be brought out and compared between various reference objects and the analyzed object, which leads to derive a range of possible distances between that object and the camera.

Flash range

On a picture taken by night, if an object appears lit by the flash, its distance from the camera cannot be larger than the flash range.


Radiometric parameters

Luminance of an object

Luminance associated with a photographed object is homogeneous with an emitted power per surface unit and per solid angle unit, in the direction of the lens. It is measured in lm/sr/m2 (lumens per steradian per square meter), while correspondence between lumens and watts depends, for each wavelength, on the luminous efficiency of the radiation.

This observed luminance may be due to the object’s own luminous emission, to transmission (transparent or translucent object) or to reflection of light coming from somewhere else, in particular from the sun.

In the case of a non-luminous object, one may assess its albedo: this is the fraction of received luminous flux reflected by the object, the value of which extends from 0, for a theoretical black body, to 1, for a white body. Evaluation of albedo will sometimes, through comparisons, provide indications on the material which makes up or covers the object under study.

The luminous flux F (expressed in lumens) – emitted, transmitted or reflected by the object – is simultaneously modified in two ways by the surrounding atmosphere:

- Atmospheric propagation, between object and camera, leads to attenuation due to atmospheric absorption by air molecules, along a Bouguer line:

F = F0 10-αx

where α is the extinction coefficient and x the thickness of the crossed atmospheric layer.

α value depends on weather conditions and on the wavelength, whereas αx represents the optical density of the considered atmospheric layer.

For light sources located beyond the atmosphere (such as astronomical objects and satellites), the atmospheric total thickness is crossed, the value of which only depends on the zenital distance of the source. The source intensity then varies according to that zenital distance, following a « Bouguer line ».

- In the case of day photos, the proper luminance L of an object located in the low atmosphere, at a distance x from the lens, is attenuated by atmospheric absorption as previously indicated, thus adding a contribution of atmospheric diffusion of daylight.

If LH represents sky luminance on the horizon (x -> ∞), apparent luminance L’ of the object is given by the relation:

L’ = 10-αx L + (1 - 10-αx) LH

...where the first term represents the extinction of light coming from the object, and the second one the contribution of atmospheric diffusion.

If it is a black object (or very dark), the formula simplifies as follows:

L’ = (1-10-αx) LH

In the particular case of an object of albedo R, under a uniformly dull sky, we have:

[align=center]L’ = [1 – (1 – R/2) 10-αx ] LH[/align]

Directly measurable data that quantify light received by a given pixel in the digital image are its gray level (along the black to white axis) and its respective luminance levels in the 3 primary colors axes (red, green, blue). Those values characterize the apparent luminance of corresponding points of the scene. In silver photography, as well as in digital in the case of the RAW format, one may sometimes establish a correspondence formula – more or less empirical – between luminance and gray level, through luminance calculations. Unfortunately this becomes practically impossible with JPEG format, because of all the optimization real-time processing performed inside the camera before storing the image (RGB demastering, delinearization with application of gamma factor, compression, accentuation, etc.).

For lack of means to estimate absolute luminance values, only relative calculations are possible, taking advantage reading across to monotonic level variation according to apparent luminance.

Nevertheless, these empirical interpolations or extrapolations are invaluable in many cases, for they allow definition of a range of possible distances of an object, by comparison with other elements of the scene that are located at known distances.

Sharpness of an object

Evaluation of the sharpness (or blurredness) of an object’s apparent shape may be essential in the frame of various distinct approaches.

- Movement blur: if, at shooting time, the object under analysis was moving and if this caused movement blur, it may be possible to quantify the object’s angular velocity at shooting time, using angular measurement of blurring and exposure time.

- Depth of field: if some blur may be related to the object being outside the depth of field, one may derive possible limits for the distance between object and camera (cf above).

- Atmospheric diffusion: independently from its consequences on apparent luminance of an object, atmospheric diffusion has degrading effects on sharpness of contours (MTF or Modulation Transfer Function), especially important if the object is remote. This feature is more or less visible, and thus measurable, according to weather conditions. In best cases (cloudy weather) it is possible to derive limits of possible distances by comparison with other objects in the scene, for which the distance from the camera at shooting time is known.


Color of an object

The object’s color may prove useful for analysis, as it provides (limited) information about the spectrum of light emitted, transmitted or reflected by that object. Besides, comparison with the dominant color of the scene may make it possible to bring out possible inconsistencies, that can be proof of a fake produced by image inlay.


Texture of an object

The object under analysis, if it covers a sufficient area in the photo, may display a texture, which depends on the material that constitutes or covers that object.

An image processing software will, in such a case, allow characterization of that texture, in view of comparison, if necessary, with a catalog of reference textures.

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Now, let's see practically how it works!

PRACTICAL ANALYSIS METHODOLOGY

This part sums up a standard sequence of actions to be taken in order to conduct analysis of an alleged UFO photo.

It will be illustrated, in the succeeding paragraphs, with the follow-up of a particular case in italics.

The file which we have chosen to illustrate – as far as possible – the different steps of photo analysis, concerns an incident which took place during the hot-air balloons competition Mondial Air Ballon 2007 which was organized, as every year, on the former military base of Chambley-les-Bussières (Meurthe-et-Moselle, France) in August 2007.

One of the participants to this meeting, who had shot 120 photos with his Nikon D200 digital camera, selected one of them, dated 5 August, on which appeared, in the upper left corner, among hot-air balloons, a quite singular unidentified object.

This case was reported by Mr. Christian Comtesse, who conducted a thorough on-site investigation , the final conclusions of which will be presented at the end.

Here is the photo – which we shall analyze – as it appeared, as well as a zoom on the unidentified object:






Data on the photographed scene

Depending on the circumstances and environment, one will try to identify elements appearing in the field – especially those which may be used for geometric or radiometric comparisons – and to assess their respective dimensions (size, distance from the camera, surface), in view of interpolating or extrapolating calculations, as well as characteristics in terms of luminance and color.

This may concern, for example, buildings, trees, relief, celestial bodies (sun, moon) or human beings.

The only visible elements of comparison in the scene photographed in Chambley are hot-air balloons. It was therefore important to make enquiries on the size of such objects.

Investigation on Internet taught us that a standard air-balloon has a volume in the order of 2500 m3, a height of 20 m and a diameter of 15 m.

Collection of technical data

Characteristics of the camera

All technical data of the camera as well as possible settings for the shot (focal length, focusing distance, exposure time) must be collected.

On the basis of a known camera model, it remains always possible to find out all characteristics, for if the operator himself does not have them to hand, the manufacturer can be consulted.

Besides, JPEG image files produced by digital cameras comprise, more and more systematically, metadata in the EXIF format, containing all required data on camera, date and settings. Access to original metadata, however, is only possible if the image file has not suffered any alteration.

Coordinates of a point A in pixels are given in the form A {ia,ja}, with:

ia : column number, from left to right on the screen, comprised between 0 and H-1
ja : row number, from top to bottom on the screen, comprised between 0 and V-1
H : total number of columns (thus of horizontal pixels)
V : total number of rows (thus of vertical pixels)

Coordinates of the center of the photo P being P {(H-1)/2,(V-1)/2}, distances in pixels between P and two points A and B are:



Those distances in pixels may then be converted into physical distances on the photosensitive medium, simply by applying the rule of three, if real physical dimensions of that medium are available.

Dimensions of the photosensitive array may be directly obtained with the technical characteristics of the camera, or inferred from the focal length conversion factor (if it is provided), either with technical characteristics, or with EXIF metadata associated with the photo (depending on manufacturers).

One should keep in mind that this conversion factor is to be applied to the diagonal of the rectangular (or square) surface of the sensor. Generally, it corresponds to the ratio of the diagonal of a 24x36 mm standard silver format – i.e. 43,3 mm – to the diagonal of the CCD array of the digital camera.

From the physical dimensions on the sensitive medium and from the focal length, one may derive the calculation of angular dimensions. Then, possible estimates of absolute dimensions may be obtained (cf above).

Regarding the estimation of distance between the object under study and the camera, a few approaches (cf above) require radiometric measures or sharpness estimates. One should then address respectively the two following sections.

The very relevant angular size mensuration on the photo from Chambley concerns the largest apparent dimension of the unidentified object (we shall refer to its « length ») and the hot-air balloons (we shall concentrate on the horizontal diameters of two of them).

The software permits direct calculation of such angular sizes in a few clicks, taking automatically advantage of available EXIF metadata. For the sake of illustration however, we shall go through the calculations step by step.

First, we interactively collect the coordinates of the 6 pixels designated in the image below (image size: 3872 columns, 2592 rows) :

Extremities of the unidentified object : A {550,371} B {582,368}

Side extremities of balloon 1 : C {591,1753} D {841,1753}

Side extremities of balloon 2 : E {1864,1288} F {2689,1288}

The calculation of angular size of the unidentified object is performed as follows, referring to above-mentioned formulas (where P is the center of the photo):

PA = √ { (550 – 1935,5)sqr + (371 – 1295,5)sqr } = 1666 pixels

PB = √ { (582 – 1935,5)sqr + (368 – 1295,5)sqr } = 1641 pixels

AB = √ { (550 – 582)sqr + (371 – 368)sqr } = 32 pixels

The focal length being equal to 80 mm (EXIF data), and the focal length conversion factor being equal to 1,5 (technical data of the camera), we are back to the case of a 24x36 mm camera with a focal length f = 120 mm (other EXIF piece of data called « Equivalent focal length 35mm camera »).

In that frame (24x36 mm), the effective dimension of one pixel equals:

36/3871 ≈ 24/2591 ≈ 0,00928 mm

thus:

a = PA = 15,46 mm

b = PB = 15,23 mm

d = AB = 0,30 mm

Applying the generalized Pythagoras’ theorem, we may calculate the angular size of the unidentified object:

[align=center]δ object = arccos [(2x120sqr+15,46sqr+15,23sqr–0,30sqr) / 2√{(120sqr+15,46sqr)(120sqr+15,23sqr)}][/align]

δ object = 0,14 °

In the same way, we calculate the angular size (horizontal diameter) of both reference balloons:

δ balloon1 = 1,10 °

δ balloon2 = 3,65 °

Luminance estimation

Directly measurable data which quantify light collected by a given pixel of a digital photo are gray levels (in black and white) or respective brightness levels in the 3 primary colors (red, green, blue). Those values represent the apparent luminance of corresponding points in the scene, with certain limitations.

It is only possible, in this domain, to produce estimates and to sort objects according to increasing or decreasing values of their apparent luminance. Nevertheless, this may be enough to perform a rough interpolation and to estimate a range of possible distances between the object and the lens.

One will use, for this type of mensuration, an image viewing or processing software providing levels of a selected pixel on the screen through a mouse, or statistics on level values in a given image area graphically selected on the screen, or a plot of level variation along a designated vector (a radiometric cross-section).

Here are examples to illustrate these types of mensuration, using our software:



To the cursor position (red cross) correspond the coordinates (position) of the selected pixel, as well as its RGB levels (red, green, blue) and gray level (average of the 3).



To the vector drawn on the right of the image (red arrow) correspond the plot of the radiometric cross-section showing variations of the levels (representing apparent luminance), displayed in the window.

Our photo sample was shot against the sunlight and we may consider that the darkest parts of the objects of the scene were submitted to variations of their apparent luminance mostly due to atmospheric diffusion. Consequently, we shall concentrate on the dark part of the unidentified object, as well as that of both reference balloons.

In a quite empirical approach, we shall content ourselves with noting down the darkest pixel value in each of these three areas, using the tool dedicated to the analysis of the radiometry of pixels within in a closed surface (here a red circle)..


Dark level object = 24

The same for both reference balloons’ baskets.

Balloon 1 :



Balloon 2:


Dark level balloon1 = 30
Dark level balloon2 = 12

Assuming – which is highly probable – that the object and both reference baskets are really dark, we may conclude that the distance of the object from the camera was somewhere between that of balloon 1 and that of balloon 2. In fact, those distances may be estimated, if we assume that both balloons have a standard diameter Ф = 15 m:

Distance balloon1 = (Ф/2) / tan (δ ballon1 / 2) i.e.: Distance balloon1 = 391 m

Distance balloon2 = (Ф/2) / tan (δ balloon2 / 2) i.e.: Distance balloon2 = 118 m

Through linear interpolation on the darkest pixel values (an empirical approach), we obtain an estimate of the distance to the unidentified object:

Distance object = 300 m

From which we may derive an estimate of its actual length:

Length object = 2 x 300 tan (0,14°/2) i.e.:

Length object = 0,73 m

Taking into account uncertainties and calculation approximations, we can conservatively conclude that the length of the object – if it was actually dark – was somewhere between 50 cm and 1 m.