USE of the LOGITECH QUICKCAM PRO 9000

for photomicrography

 

History of a near-failure, or a semi-success


Walter Dioni                                                                               Cancún, México



INTRODUCTION

This article is NOT written for those with good quality photomicrography devices, already installed and working properly. The audience I address are those with a limited budget, and very little knowledge of the subject, who want to try, like me, to install an inexpensive (that is to say, within their budget) but reliable option.


All approaches I found to this problem always explain, for the described camera, an already working installation, and usually a successful one. Few people talk about the reasons underlying the options used, and it is difficult to find an account of why the other considered options were discarded... and which failures were suffered. And this is, from my point of view, what we really try to find, those who have a problem and a guide to aid us to prevent or minimize such failures.

This is a chronicle of the adventure in which I embarked, wanting to improve my photomicrography capacities. I knew little about the theme, to begin with, and I still have much knowledge to acquire. But I hope the data that I bring together here is not too wrong, and can provide guidance to some other beginner who also aims to investigate the capabilities of a webcam.

I think the article could really seem excessive. And somewhat naive. Anyone of average intelligence can imagine in advance the result of some of the tests I do.
But I feel that it is this visual and eyewitness testimony, which really interests me to share. 

This presentation is organized into 3 parts. The first one explains my problem, and my preliminary researches to decide which particular camera to test.

The second shows test samples, with the camera for photomicrography and the results (favourable or not) that I have achieved when working in brightfield.

The third looks at the results I got (favorable or not) trying special techniques such as oblique illumination, dark field illumination, episcopy, Rheinberg illumination, polarized light, and several contrast techniques which give more or less acceptable results when using my DC3 camera.

Unfortunately I can not provide data on the use of this camera with phase contrast, because I do not have the right equipment.

Maybe that with my obsession for this article, I shall exceed the limit of patience of my readers. But I'll never get to know this, if I do not share it.

 

My Current Equipment

 

I have a National Optical microscope with MOTIC camera and planachromatic objectives. The DC3 camera built into my microscope has a sensor that actually produces images of only 0.08 Mpx (320 x 240px). An internal resizing program (probably with bicubic algorithm, or better) and delivers acceptable images of 0.3 Mpx (640 x 480px) therefore at a size four times bigger than the original. 

It is included in the head of the microscope and cannot be removed, nor modified.

001 - Motic microscope

With the microscope I was given a superb management program (Motic Image Plus 2000 v.1.3), which allows the taking of still images, and videos from 320x240px to 30 fps (frames per second), in a very simple exercise. It also allows image capture in rapid succession manually (I can usually capture up to 5-6 frames per minute), or automatically. The camera is mounted in a way that captures the best rectangle inscribed in the field visible through the eyepieces.

002 - Motic Capture Screen

An overview of the screen from the Motic 2000 v 1.3 program with the capture screen displayed. A click on the button "Capture still image" displays a frame with an image of 640x480, behind the screen capture (showing image of a histological section of a tadpole 4 mm long, captured though 10x objective).

The height of the computer screen is 25.5 cm with a width of 41 cm. The display capture screen (320 x 240 px) is 98 x 68 mm. The Capture Viewer has as its main function the electronic adjustment of the image, because the capture itself can be simply done while viewing the subject directly through the binoculars. The camera is parfocalizable very easily.

The microscope is equipped with 4x, 10x, 40x and 100xOI planachromatic objectives and a "pre-centered" Abbe condenser 1.25 N.A. Lighting is provided with a 12V 20W lamp that generally is somewhat insufficient for the 100x objective in oil immersion (varies according to the subjects).  Although according to the manufacturers it has a preset Köhler illumination, the existence of a thin ground glass diffuser filter between the lamp and the front lens of the illuminator allows only "critical lighting", or "poor man’s Köhler" as the best possible light style.

Eight years ago this camera was justified, because although it was advertised as a "research microscope" it was in fact designed as a microscope for an advanced student of a science faculty, and at that time the VGA format was very common on computer monitors. The reasoning that led to the design of this microscope was apparently impeccable: the integrated camera will avoid all the problems of adapting a consumer camera, the sensor is fully protected, and located in the right place to record the largest rectangle inscribed in the field of view, the camera provided the ability to monitor the microscopic picture live on the computer screen.

This was easily parfocalizable, and, providing the microscope with an additional output for TV, and management software more than acceptable (that even now remains of high quality), facilitated the observation and taking pictures in quick succession, or even short videos while viewing directly by the eyepieces.... It wasn’t taken into account (I think that few people currently did, and I of course did not) the fast and explosive development of digital photography and the race towards offering more and more megapixels (which is not finished yet) and that soon made the VGA format obsolete. Although these models are still selling, most of the MOTIC microscopes of this kind are currently trinoculars. MOTIC now manufactures independent cameras: from the 1.3 Mpx Moticam 1000 (carefully reviewed by D. Walker at MICSCAPE), to 5 Mpx professional cameras.

http://www.microscopy-uk.org.uk/mag/artmar05/dwmoticam.html

More megapixels (despite the table included below, and all assertions to the contrary made repeatedly) represent better resolved images at digital level. Anyone can verify that this is extremely important for a good interpretation of the subject.  I believe that these two images will make the point clear enough:

4x Objective - two micrometer images respectively taken with the DC-3 (left) and a SX100is Canon of 8 Mpx. The DC·3 is the complete image. Both pictures trimmed from the originals

It is also true that every time that I submitted to others judgement, pictures displayed at the capture size, or others electronically enlarged; the largest (within reasonable limits of course) always got a better appraisal. (See below for Logitech images at different sizes).

 

To improve the capture size I decided to try adapting an external camera, supported on one tube of the binocular head.


According to the next table, a 1.3 Mpx camera is sufficient for the whole range of my objectives.

TABLE OF THE MINIMUM PIXELS AND REQUIRED NA (planachromatic objectives)

          Objective       N.A.              Recommended image size
               4x             0.10               1364 x 909 - 1.24 Mpx

             10x             0.25               1364 x 909 - 1.24 Mpx
             40x             0.65                 886 x 591 - 0.52 Mpx
          100 x             1.25                 665 x 436 - 0.29 MPx

 So I thought that adding a camera which produces 2 Mpx images would be enough.

Note: The calculation of this type of table depends on many factors (the NA and type of the used objectives, achromatic, planachromatic, planapochromatic), the number of photodiodes (2 or 3 e.g.) assigned to register each resolved point of the subject, sensor type (monochrome, colour and sensor type) the value of the relay lens reduction, if it is used, and also whether the photos are to be for screen use, published on the Web, or if they are to be printed, in which case there is of course an important interaction between the quality of the digital photo, the quality of the printer, and the size to which we intend to print the image.) But the important thing is that all calculations agree to allocate a higher demand in the quality of the sensor, and the number of photo diodes required, to low power objectives, with a ratio of 3 or 4 to 1 between the lower to bigger magnifications.

 Of course I have the option between the consumer cameras and the webcams. In the APPENDIX  1   I review very briefly my thoughts about the use of the consumer cameras

 

 SEARCHING FOR A CAMERA

The ease of capture and image quality, depend of course on the chosen camera. From my particular point of view I was seeking a camera that has these 13 characteristics.

a. Small size and light weight to allow an easy adaptation to a binocular head tube .*
b. Sensor of good quality .*
c. Image Capture 1.3 Mpx at least, according to the table above (2, 3 or Better 4 Mpx, to be preferred)
d. High capture speed, in order to photograph moving subjects
e. Capturing blank backgrounds for correcting background dirt *
f. Monitoring of the screen image in real time *
g. Adjust lighting, color, contrast and white balance *
h. Large video frame (monitor) screen for easy composition and focus
i. Electronic capture to eliminate vibrations .*
j. Manual sequential captures, as fast as possible
k. Video capture as fast as possible
l. Ability to apply darkfield, Rheinberg, oblique illumination and polarized light techniques
m. Ability to capture stacks to use in CombineZ * or similar software

 

Cameras similar to these would meet these requirements:

        Special cameras for photomicrography

        Motic MC 2000             2.0 Mpx                      U$ 799
        Motic MC 1000
                    1.3 Mpx                      U$ 654
        (The DC 3 integrated camera is only sold in the DC3-163 digital microscope system)
        Swift
                                      1.3 Mpx                      U$ 499
        MicroscopeNet (eBay) 1.3 Mpx
                             U$ 189
                                               0.3 Mpx                      U$ 100
        Bresser
                                0.3 Mpx                      U$ 130
                                               1.3 Mpx                     U$ 324


        Webcams, better suited for photomicrography


        Phillips SCP1300NC
                                 1.3 Mpx           U$ 156
        Philips SFP6500/00
                                   2.0 Mpx           U$ 128
        Logitech Quick Cam Pro 9000
                   2.0 Mpx           U$ 100 cost.2009
        Logitech C-950
                                           2.0 Mpx           U$ 100 cost 2009
        Logitech Quick Cam Pro 9000
                  2.0 Mpx           U$ 90 (my cost)

 

Note: Apart from the price of my Logitech, which is the price at which I acquired it in Dec. 2008, the rest are prices as for December 2009, obtained from online offers.

For all these cameras I would have to add shipping costs to Mexico that are usually around U$ 28-30 if imported from the United States, and insurance, because of the many risks of the transport. I do not know the price from Europe. Usually, in Mexico, customs also collects taxes on the importation of technical instruments.

 Until recently Philips cameras had CCD sensors. Both Philip and Logitech cameras, according to the current description by their manufacturers, now have CMOS sensors, although some unscrupulous sellers advertised them as the CCD’s. Until recently the CCD was considered unbeatable in terms of image quality. But over a couple of years some publications insist that CMOS have achieved the same or better quality, and that its manufacturing method allows them to have a much lower price.

 See Appendix 2 about the frequently required post-processing of images   APPENDIX 2

 

Why from the list I selected the Logitech QuickCam Pro 9000


1
Because some leading amateur microscopists use it and even post very acceptable videos on YouTube.
Many posts by users of this camera can be read in the Messages of the MICROSCOPE GROUP

2 Because it meets 9 and with certain conditions 10 of the 13 requirements for the best camera. The conditions not met, or fulfilled only in a limited way, are:

        a.Capture of images at least at 1.3 Mpx according to the above table. According to their current operating conditions (see below) it only captures images of 0.8 Mpx. (roughly 60% of desired)
        b. High speed capture, to photograph moving subjects. As will be seen below, 2.0 Mpx images (0.8 real) require 1 second to be recorded.
        c.
        Ability to apply darkfield and Rheinberg techniques. The tests so far suggest that it is possible to use oblique illumination and polarized light, but dark-field images are not very good (much "background noise").

3 Because it has a Zeiss Tessar lens with 5 elements.
4 Because it has a 2 Mpx sensor, advertised as providing high quality images
5 Because it has a program that has control of the lighting intensity, and which I thought could help to record better images at high magnification (with limited lighting)
6 Because it is totally manageable with the computer
7 Because the images monitor screen is large (20 x 23 cm on my monitor)
8 Because I have the additional programs (detailed above) necessary to process my photos and videos
9 Because it is very easy to disassemble and only weighs 30 g
10 Because it costs me $90 and was available in Cancun, while the European Philips are not available here, and are more expensive.
11 Because it cost me $0.50 to fit it to the microscope.

 In fact if I had been free to choose, I would have liked to buy a Logitech C-950 with exactly the same specifications, but with a more flat design and more easily adjustable to a homemade adapter.

 To convince me that the camera really qualifies for a trial I made some tests. The most simple are summarized inAPPENDIX 3

 

ADAPTATION TO THE MICROSCOPE


The first task to properly adjust this camera to a microscope, is to remove the brackets that allow its attachment to the computer screen, or support it over the desktop.

These are pictures of the entire camera.

005 - Logitech 1

006 - Logitech 2

In an article by Gary Honis (http://ghonis2.ho8.com/Pro9000a.html) on adapting the camera to a telescope, he illustrated step by step instructions not only to dismantle the camera, but to transform it into another to be attached directly to a telescope .... or even a microscope, of course.

The author also describes and illustrates very effectively with the camera itself the consequence of the inevitable removing of the
Infra Red (IR) included filter, and the need for replacement, as will be seen below.

This is the image of the camera as it was after I removed what I don’t need.

 007 - disassembled Logitech

The connection to the CPU uses a USB 2 port.
You only need to install the camera on a support that fixes it properly over the eyepiece. This is the one I use.

 008 - made in home adaptor

 This adapter was built entirely out of cardboard and plastic putty. It's sufficiently rigid, light weight, and fits well with the camera. If needed, slight vertical displacements can be adjusted with small wedges of thin cardboard, which is then permanently glued. (Or, with little additional effort, 3 screws can be installed, which can also help to “center” the camera, though I did not find it necessary). The camera is attached over the upper platform. For now I simply hold it with rubber bands. These fit well, but allow easy mini-movements to focus the objective on the exit pupil of the eyepiece. A recess into the top of the head allows the camera to settle so that ambient light does not leak to the objective. Whoever is more determined than I am, can glue it, using a suitable glue.

The preparation of the adapter needed the materials below and I believe that the description of the process is implicit. A ruler, a choice of cardboard tubes, cutting and gluing materials is almost all that is needed.

 008 - adaptor materials

HOW TO USE THE CAMERA

There are two methods that work with two well defined types of camera:

1 - Those with a fixed front lens, immovable, with which you can only use the system known as "afocal",  using the camera and microscope as they are, or at most changing the camera lens. What sort of pictures can be taken with this system?

  2 - Those with interchangeable lenses, so the front lens is unscrewed, leaving the sensor exposed, and with which you can use the system I call "astronomical" that only uses the microscope objective, and discards the eyepiece of the microscope and the objective of the camera.

 picture taking systems

As seen at left, the eye views through the front lens of the eyepiece  (e), the image ( i ) projected by the microscope objective in

the plane of the eyepiece stop (d) and the brain recognizes the magnified image (mi) - L, light path - t, tube of the microscope

The afocal system uses only part of the sensor.  The "astronomical" system uses only part of the field of view

 The Logitech can work straight away with the afocal system, and with a little additional work (remember the Ghonis article) as an astronomical device. But to eliminate the camera lens is a relatively delicate task, and therefore it is almost final, because there doesn’t exist an easy replacement method unlike the threaded objectives of higher quality digital cameras.

 “ASTRONOMICAL”  SYSTEM

The system I call "astronomical" because of its extensive use among amateur astronomers, is one that only uses the microscope objective, and discards the microscope eyepiece and the lens of the camera.

 

011 - an old camera

This was indeed the method we used for amateur microscopy in about  the 1950’s or 60’s, adding to the end of the tube of the microscope, without eyepiece, a bellows without lens, with a cone or a hollow pyramid with height proportional to the desired magnification, and using at the end a sensitive element (sensitized glass plates, silver bromide paper, or 35mm film). If the sensor is placed 25 cm from the exit pupil the magnification of the photo corresponded exactly to the nominal magnification of the microscope.
Attached picture: a professional Leitz instrument of the late nineteenth century (J.E.Barnard - 1911 - pict. 38a - pg. 114)

The amateurs of the twentieth century like myself often used cardboard cones, at the end of which was glued a wood and glass frame that we used to make prints on sensitized paper. Instead of the usual sandwich  "glass-negative-sensitive paper”, bare bromide paper was placed in the frame and we exposed it to the light projected by the objective of the microscope.  The image was later chemically reversed. Rigorously working in a darkroom of course.

As we speak of discarding the lens of the webcam we must make it clear that this is not a trivial action, and that we must be ready to counteract its effects. Read the APPENDIX IV

 

 Using the camera

In France the Philips webcams were used for years. (Until recently they were of 0.5 Mpx, and since a couple of years ago of 1.3 Mpx. Now they have 2.0 Mpx. Apparently sensors previously used were CCD, but the maker's specifications indicates that now they are using CMOS sensors). Webcams were used by adapting the method used by amateur astronomers: camera without lens, microscope without ocular

 The image generated by the microscope objective is then projected as a cone of light, which can be intercepted at any height after the exit from the tube of the microscope to get the real picture. In all cases tested, the diameter of the projected cone outside of the microscope tube is much bigger that the diagonal of the sensor. The sensor is exceeded in great excess by the image, and the Photographic Field is a small fraction (which can be only 5% to 7%) of the Field of View (FV)

 The following circular image represents the projected image, and the box in the middle of the field of view represents the area captured by the sensor. Image not to scale, the rectangle is here bigger than in reality.

 015- Field of view, Photografic field

Section of a blood vessel in a preparation of human duodenum. Fixation with formaldehyde in neutral buffer, embedded in paraffin, cut at 8 microns thick, HE staining.

Although through their eyepieces the microscopist would see the complete the picture above, the following would be an example that mimics the part of the field that the camera would record.

 016 - central clip

 This has given basis to the absurd claim of a web site that considers possible, using common and even low powered illuminated microscopes, by the only trick of removing the eyepiece of the microscope and the camera lens, to get images between 5000 and 8000 optical mags.

 Of course the resolution of the objective (in the optical sense of the term) cannot vary. If the objective 100x, NA 1.25 with oil immersion, resolves the image when illuminated with light of 550 nm to see as different 2 points distant only 0,02 micron, this is the maximum optical resolution achievable with any degree of projection distance. 

 The optical resolution depends on the microscope optical system (NA of the objective, Objective magnification, Eyepiece magnification) not the size of the image.

 This system is especially attractive to those wishing to study and photograph only tiny subjects, such as blood cells, diatoms and protozoa, for example. But it can be really annoying for those who want to photograph such things as a mosquito larvae. For even with a 4x normal scanning objective they must take several pictures to cover their entire body, and then set up a mosaic. If they tried to avoid mosaics, they should add to their equipment very low magnification objectives, as for example a 1.6x or 2.0x (expensive of course), and require very flat and uniform lighting of the large field of view covered, which is not a very common feature in budget microscopes.

People involved in higher invertebrates research, or histology, surely want to have greater field of view coverage. These people are more interested in

 THE  AFOCAL  SYSTEM

It is based on simply applying the camera to the eyepiece of the microscope, adjusting it properly to the exit pupil, and to be satisfied to record the total field of view at the magnification allowed by the sensor coverage.

This is an example of an image obtained under these conditions by the Logitech 9000.
The black rectangle is actually the recorded image of the full sensor. The central circle is the area of the sensor actually used. This effect is usually called "vignetting".

 

The same image shown earlier as the Logitech registers it in "afocal system". Reduced from the original 1600 x1200 picture

 As shown, and it is the normal situation with the webcam, and even with most consumer cameras with short focus objectives, the virtual image visible by the camera does not cover the sensor field. The picture obtained is the center circle. It uses only a portion of the photoreceptors of the sensor. Even when using a camera that normally produces images of 2 Mpx (the size of the entire sensor), some measurements and a quick calculation shows that the image uses barely 0.8 Mpx approximately.


Full sensor Rectangle: Image 1600 x 1200 px = 1.92 (2.0) Mpx

Full Circle (without zoom): 0.8 Mpx (only 40% of the sensor, but 10 times greater than that recorded by the DC3)

Maximum allowable cut with ratio 1:1.3: 800px x 600px = 0.48 Mpx = 24% (relation to DC3 = 0.48/0.08 = 6 times)

Cut made in scale 1:1.6 ratio: 860 x 513 = 0.44 MPx = 22% (5.5 fold increase)

Maximum square inscribed in the circle: 710 x 710 = 0.50 MPx = 25% (6.25 times higher)

Of course the ideal solution is this one.

 018 - max. rect.

The rectangle is the picture a dedicated photomicrographic camera takes from the Field of View

 Now the projection of the intermediate image has a diameter coincident with the diagonal of the camera sensor. The rectangular image is actually 2 Mpx obtained by covering the entire sensor. The special photomicrography cameras work that way. Webcams usually do not meet this requirement and instead stick to the geometry of the vignetted image shown earlier as the Logitech does.

SOLUTIONS USED - Faced with this reality microscopists have opted for some effective solutions.

 Installing a telephoto lens - Using the intermediate image size, equal to the size of the internal diaphragm of the eyepiece of the microscope which depends on the quality of the eyepiece (my eyepieces has a FN 18 mm) the focus of a new lens can be calculated.
To cover the sensor completely I need to replace the wide angle lens of the Logitech which have a focus of 3.7 mm, by one of 8 or 9 mm focal length, according to calculations that Charles Krebs kindly communicated to me. This means that I should provide it with a
telephoto lens.

There are on the Web some addresses of suppliers of suitable lenses. (Two recommended in the Forum MICROSCOPE are:
www.sunnex.com and www.optekusa.com)
The lenses themselves are not too expensive (depending on their quality of course). You can get up one for 20 to 40 dollars.

But the amateur microscopist must understand that these will not be Zeiss Tessars and they must also workout for themselves how to remove the original lens of the webcam. See how you can do it in

(http://ghonis2.ho8.com/Pro9000a.html)

and also how to adjust the new replacement lens to the camera. 

 The new lens does its job by reducing the image field of view so that it only covers all the photodiodes of the sensor, thus changing the geometry to fill the field.

 Some providers also offer to calculate and build the necessary adapter to turn your webcam into a camera similar to the so-called "Microculars”. (there are others, but see eg. Truetex)

 But that raises the price of the amendment, almost to the price of a commercial Microcular (sometimes more) and does, from my point of view, become a completely pointless venture.

Changing the microscope eyepiece - The second solution is to acquire and use in the microscope a SWF (super wide field) eyepiece with FN between 22 and 23 mm, and high relief point (eyepieces for short-sighted eye) in order to project onto the sensor a larger-diameter circle. These eyepieces are not easy to find, and are always very expensive. (A 15x Olympus lens with FN 22 was offered at $170 on eBay) and hardly provides a total solution.

 “I found that I needed a 15x ultra w.f. eyepiece to fill most of the field of view. The video frame is not completely filled, but it is not bad considering the wide angle lens of the webcam. Cavalue”

http://sfmicrosoc.org/newsletters/Issue%20VOL%202%20NO-2.pdf  (see page. 3)

 

 Both solutions can be used simultaneously, of course. As in the other cases, my concern is that with all these adjustments we are reaching the price level of a commercial camera, which could be acquired completely prepared. Except for the pleasure of making a homemade device, these adaptation solutions do not appear to have a favorable cost / benefit ratio.

Finally, if you don’t buy an adapter there is the problem of preparing one at home, to adapt the camera to the microscope. Solutions are also published on the web. Relatively few are economic ones. The efficient Cavalue solution, for example, not only provides a secure fixing of the camera in a working position, but permits at any time to retrieve the same for normal use. But the Lumicon adapter he uses, costs $50, over 50% of the price of my camera.

Others projects, based on PVC, or metal home designs, may have a favorable balance. But involve having tools that not everyone has. Perhaps my solution is the most economical, while being effective.

 A comparative calculation is below of the cost of adapting at home a Logitech, or purchasing a special camera for photomicrography. I limit it to the very cheap cameras like the Micro-oculars made in China.

Camera + teleobjective + highpoint eyepiece + Lumicon adapter
 90 dls         30 dls                       170 dls                         50 dls            total with eyepiece 340 dls

                                                                                                                      without eyepiece    170 dls 

 Add the custom and insurance fees if the material must be imported. Check prices for cameras with 2.0 and 1.3 Mpx in the table above. There are 1.3 Mpx cameras between 130 and 320 dls

 

 

A summary of my situation, after all this long analysis is:


As for many others I can’t purchase a special photomicrographic camera of a high-class brand. 

 My Canon A300 can by no means be successfully adapted, mainly due to its turn-on system which involves a difficult to operate, manually slide protection, for its lens.

But I did buy a Logitech for $90, to undergo adaptation.

After collecting the information summarized above, and doing some experiments with the camera, I decided to use it in afocal geometry,

for not to have to buy or import not matter what telephoto lenses, or eyepieces, or IR filters, or adapters of any kind.

That sentences me to have a Logitech with a wasted good sensor, using only 0.8 Mpx of the actual 2Mpx available. 

 

 The cost for me was $90. 50 and just needed to build the adaptor for the binocular tube. It was very affordable (not more than 50 dollar cents.)
To install the Logitech in these conditions was simple; it only weighs 30 g. and creates no tension on the head of the microscope.
Thus, my microscope is now equipped with the 0.08 Mpx
DC3, 0.3 Mpx digitally enhanced, that is still relatively useful, and which is managed since December with Motic Images Plus 2.0 software. It is usable with the 10x objective, but especially with the 40X and 100xOI objectives. And the Logitech, installed on one of the binocular head tubes, for all objectives, with 0.8 real Mpx. For occasional shots of absolutely immobile objects, I use the Canon A300, which I can manually hold over the eyepiece.


Go to the 2nd. of 3 parts (in March 2010 issue)

 In the second part I will explore the real target of this work: the Logitech 9000 web camera  as a photomicrographic camera, especially capturing still images in brightfield.

 In the third part we will see its behavior as a photomicrographic video camera and to capture snapshots of moving subjects, and the capabilities it has to record the results of special illuminations.

Acknowledgements
I wish to acknowledge the kind assistance of D. Walker and J-M Cavanihac, whose thoughtful suggestions allowed me to greatly improve the presentation of  this article.

 


Comments to the author, Walter  Dioni , are welcomed.

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