Let’s continue on our discussion of image sharpness and resolution.  So far we’ve limited our discussion to sharpness as limited by the number of pixels and seen how that determines required pixels per inch when you are displaying your wonderful image on a computer screen.  Today let’s consider the issue of displaying your images on paper – that is printing them.

The situation here is very similar to that of displaying on a computer screen.  The limitation here is how fine can the human eye resolve?  This number is something like 200 pixels per inch.  For good measure I like to crank that up  to 300 pixels per inch.  Ok then, this means that if you’re going to print an 8” X 12” print  you need 2400 pixel by 3600 pixels; so 8.64 M pixels.  Boy, that was easy!

Let’s consider the maximum number of pixels required for different size prints.  The first column gives you the number of megapixels,  MPz, requirement assuming 200 pixels per inch while the second column gives you the required MPz assuming 300 pixels per inch.

The other way to look at this is that for a Canon T2i with its 5186 pixel by 3457 pixel (or 18 MPz) sensor array, if we assume 200 pixels per inch as a requirement, then the biggest picture you should print is 5186/200 x 3457/200 , then the biggest picture you should print is 5186/200 x 3457/200 or 26” x17”.  For a 300 pixel per inch requirement this becomes 17” X 12”.  So now you can understand while I won’t print from my Canon T2i any bigger than 18” X 12”.

It was the perfect combination: good friends, great weather, great food, andCape Cod.  It was a late summer weekend onMassachusetts’Cape Cod.  For those of you not intimate withMassachusettsgeography, it divides into five distinct photography zones: Boston, the North Shore, The Berkshires, Northwest Mass, and theSouth Shore andCape Cod.

Provincetown, MA photographer on the beach

For photographers the two words, “Cape Cod,” conjure up images of Photographer “Joel Meyerowitz” and his two magnificent works: “Cape Light” and “Red Heads.”  Meyerowitz has made an art form of the apparent but deceptive “snap shot.”  Meyerowitz is also now famous for his important chronicle, “Aftermath,”of the World Trade Center terrorist attack and the subsequent rescue efforts.  A century from now this will be recognized as a significant photochronicle.

The Little Store, Provincetown, MA

This Saturday we wandered up to Wellfleet and Provincetown.  The first stop was the PB Boulangerie Bistro  in Wellfleet.  According to Zagat: “This has to be the best restaurant on the Cape and definitely on par with the best in Boston.” I have no photographs from PB.  I was too busy eating the wonderful French sandwiches, soups, and pastries.  Screech to a halt and leave your camera in the car.  Although Chef Philippe Rispoli might allow you to take some pictures of him cooking in the wonderful open kitchen.   In the meanwhile check out their mouth-watering photogallery!

Pedicars, Provincetown, MA

My first photograph of the day, or more accurately photo-disaster was a picture of a woman photographing a seagull on the beach.  I always set my camera for the next likely shot.  And I highly recommend that you do so.  Whatever your shooting mode, anticipate the light.  But I had just gotten out of the car…  Actually there’s no excuse.  At the very least I should have set the camera on one of the auto modes.  So instead, I got an over-exposed.(should have set compensation by at least  a stop because of the bright sand) and I certainly shouldn’t have had the ISO set at 6400.  So I wound up with a pretty substandard silhouette.

Spank the Monkey, Provincetown, MA

Provincetown on a late summer’s day is a symphony of color; vivid, sometimes bordering on the garrish, but always fun.  You can start with the very bright red “Little Store,” and then move on to the pink “P-Town Pedicars,” and the piece de resistance of color is the wonderful Joey Mars murals that now adorn the “Spank the Monkey” Art Gallery.  Check this graphic pop artist out at his website.  It is well worth the visit.

Ocean View, Provincetown, MA

I also stopped to photograph some glass in a shop window that was particularly bathed in sunlight. These weren’t too successful, because despite the best efforts of my polarizing filter reflections from the window glass were overpowering.  I had to console myself with a picture of an ornamental frog to add to a photoessay that I have been doing for several years now.  There is a special appeal to frogs.  Maybe it’s the smile or the eyes.  Or maybe it’s childhood recollection of Kermit and “it’s not essay being green.”

Ornamenal Frog, Provincetown, MA

Of course, on Cape Cod you’re never far from the dunes and the ocean accentuated by sun and atmosphere.  Everyone in P-Town, including us, was having a good time.  We must have seen at least six weddings.  And every time a couple was wed, the street erupted in applause.  It was a most excellent day.

While we’re on the subject of image sensors and pixel limited resolution, it is worth considering the question of what size your images should be for computer reproduction – say to put on a website like this one.

If we again take the Canon T2i are typical, the 18 M pixel sensor chip consists of a 5184 X 3456 pixel array.  As discussed this defines some ultimate resolution for an image. For computer monitors a typical high resolution pixel density is 1024 X 768.  this is 0.786432 M pixels.  That said there is a trend to higher and higher pixel density monitors.For a fairly comprehensive and useful list of monitor pixel densities his site.  The point is that this is what your monitor is capable of displaying.  Anything else is wasted space or more accurately wasted memory.  And it can seriously slow down your website, because of download times.

In addition, you don’t usually display an image as full screen.  A good size, based on aesthetics and the need to have text with your pictures, is about 600 pixels X 400 pixels = 0.24 M pixels. Also, remember that bigger displays, like the ones at sports events and Times Square are meant to be viewed at greater distances.  So the issue with monitor display is usually number of pixels as percentage of screen not pixels per inch as it is with printing. So the happy news, is that where the purpose is displaying images on computers: websites, social media, and emails to aunt Tilly there’s no need send huge images.

One caveat to keep in mind is that Aunt Tilly might want to have the picture printed and there, as we shall discuss, the critical number is pixels per inch and the size of the picture.  I recently wanted to print a family picture someone had posted on Facebook.  It was great on the monitor but as a print – no way!

BTW – there is an interesting and well written website that I would recommend as further reading on the topic of “the myth of dots per inch.”  This site also explains several of life’s more profound mysteries like: why does Photoshop default to 72 pixels per inch, why is your computer screen called a desktop, and why does it have a virtual trash barrel – not to mention the all consuming question: “what’s a pica.”

It appears that in 1737 Pierre Fournier created the ciceros as a unit to measure printing type.  Six ciceros were almost and inch, actually 0.998 in. Then around1770, François-Ambroise Didot made the ciceros bigger; so that it evenly divided the French foot into 6 X 12 or 72 equal parts.  This is 0.1776 in.  Today, and as established in 1886, the American Point System defines a “pica” as 0.166 in.; so that 6 of these make up 0.996 inches.  We have, (ready for this?)12 points per pica and 6 picas per inch.  That is 72 points per inch.  Heard that number before 72 dots (or points) per inch?

When Apple introduced the Macintosh in 1984, they wanted something people could relate to.  Type was meant to follow the rules of print, as in 72 points per inch.  How many people can relate to that – I mean really? We worked on “desktops” and threw old files in the “trash.”

There’s a rule-of-thumb that for every equation that you use in writing you lose 90% of your audience.  So to the less than one percent of you that remain after my last blog, I’d like to point out that by determining the magnification of a camera lens, we are finally in a position to begin to address the very import questions: how sharp and image can we hope to achieve?

We showed in the previous post that the magnification, or actually demagnification of a lens is approximately the focal length divided by the distance from the camera to what you are photographing, the so called object distance.  This is great.

Pixel limited resolution, the smallest thing that you could see, for different lenses as a function of distance from the object

Now recognize that if you had some kind of perfect lens your resolution would be limited by the pixel spacing in your detector.  Remember that the pixels are like a graph paper on which your image is projected.  We can ask a very simple question.  How far apart need two white spots be to be distinguished in a photograph.  Well, there has to be a back spot in between otherwise they would merge into one.  For my Canon T2i this is 8.6 um (that’s .0086 mm). So an estimate of image resolution is to calculate the magnification as a function of distance and then divide the .0086 mm by the magnification to determine the equivalent separation in at the object.  I’m calling this number the pixel limited resolution.

I’ve done this for you in the figure, for a set of typical lens focal lengths.  It’s pretty impressive. Even at 100 m, more than the length of a football field, you would be able to resolve 3 mm.  Effectively we are looking at the finest detail that you could see if your camera’s resolution were pixel limited.  Is it pixel limited?  Well stay tuned!

Let’s continue our discussion of the technical basics of photography, building upon our discussions of the pinhole camera. As we’ve seen, pinhole cameras require tiny pinholes to create sharpness. As a result, they collect very little light. How do we solve this problem?

Well, I’m sure you know the answer. You use a lens to collect the light. It’s kind of like trying to drink rain. If you open your mouth to collect rain drops you catch very few. You need a funnel with a very large aperture. Lens are light funnels. The bigger the lens the more light we collect.

In photography this relates to the concept of f-number. The bigger the lens the smaller the f-number and the more light it can collect. Functionally, if you have a dim scene to photograph, you decrease the f-number. The bigger the lens the further down you can go. Significantly, what is important in optics is not the absolute size of the lens aperture, but the ratio of the focal length to the aperture, the f-number.

Figure 1 – Refraction causes a glass of wine to bend and partially focus light

In Figure 1, I use a glass of wine as a lens to focus my wolf picture. Because the glass is more curved in one direction than the other the image is smaller in the horizontal direction than the other. But importantly, we observe the bending of light as it moves from air to wine. This bending is referred to as light refraction.

Figure 2 – Light refraction obeys Snell’s law

Refraction is described by Snell’s law or Snell-Descartes’ law. Descartes (1596-1650) is best remembered for his famous statement: “I think therefore I am.” The Snell part refers to the Dutch Astronomer Willebord Snellius (1580-1626). Interestingly, it was first described and used by the Arab scientist Ibn Sahl in 984. Snell’s law describes what happens to light when it moves from air into a denser medium. Transparent media are described by their indices of refraction, let’s call them n1 for the air and n2 for the wine. Referring to Figure 2 we see that when the light ray in the first medium strikes the surface of a medium of higher index of refraction it is bent towards the normal (the dashed line perpendicular to the surface). When it comes back out it bends back away from the normal and is parallel to the original ray, except displaced. If you’ve ever tried to spear a fish from above the surface of a lake (hasn’t everyone?), you find that he’s not where he appears to be. You’ve got to compensate for refraction.

Figure 3 – Light from a distant point are parallel and focus by a lens a distance, f, called the focal distance from the lens

By shaping a lens appropriately you can design it so that all of the light from a distant object (physicists speak about an object at infinity, that’s the distant mountain that you are photographing.) to a point a distance f from the lens, called the focal point and focal length respectively. This is shown in Figure 3. Significantly, all the rays from the object are parallel and parallel rays go through the focal point. I say significantly because we can now consider the more general case where the object is somewhere other than at infinity.

Let’s continue our discussion of the technical basics of photography, building upon our discussions of the pinhole camera. As we’ve seen, pinhole cameras require tiny pinholes to create sharpness. As a result, they collect very little light. How do we solve this problem?

1/o+1/i=1/f.

Figure 4 – Lenses cause and image of an object in front of the lens to form a distance i, called the image distance, behind the lens.

So let’s consider the omnipresent arrow again at some distance that we call o in front of the lens. A ray of light coming from the point of the arrow parallel to the center line, referred to as the optical axis of the lens, must pass through the focal point. A ray that hits the center of the lens passes right through it. These two rays cross.The other significant point is that the point the ray that strikes the center of the lens passes straight through it. The two rays cross a distance I behind the lens and form an inverted (aka upside down) image. This is referred to as the image distance. The relationship between o, i, and f is the so call “thin lens equation.Note that in the case that I have drawn the arrow becomes magnified. This is sometimes the case in macrophotography. The more typical; case is when the image becomes demagnified. A tree is usually a lot smaller as a picture than in real life. In any event the magnification, M, is given by

M = (image size)/(object size) .

And geometry teaches us that

(Object size)/(object distance) = (image size)/(image distance).

Finally, and as a result,

M = i/o = 1/(1/f-1/o)o.=1/(o/f-1).

If we note that in general, except for close-up macrophotography, the object distance is much larger than the focal length, the o/f is much greater than 1 so

M ~ 1/(o/f)=f/o.

We will use this simple relationship in a subsequent blog, where we will start to explore the important question of what sharpness can be achieved in a photograph with a given lens.

Let’s summarize what we have discussed so far in these technical blogs from a practical point-of-view.. Cameras typical give us the ability to control three things:

• We move the lens so as to change the lens position relative to the detector surface and   thereby change what’s in focus

• We change the lens or the zoom that we are using so as to change the magnification

• We change the aperture so as to change the depth of focus.

The physical basis of these first two points has been given in this blog post. The physical basis of the last point was touched upon in our discussion of the pinhole camera and will be discussed in more detail in a subsequent post.

I’d like to begin to explore the technical aspects of photography today.  A lot can be learned from what is ultimately the simplest of cameras, namely the pinhole camera.  You can make a pinhole camera simply by taking a pin and punching a hole in the front of a cardboard box (see the accompanying figure A) and putting some sort of photosensitive material: film, paper, or imaging chip on the opposite side.

Suppose that we have some object that we wish to photograph.  In physics books this is usually a candle, or a tree, or a minimalist arrow.  Let’s go with an arrow.  The thing is that every point on the arrow emits light rays in all directions.  However, the pinhole only allows a single ray to enter the camera.  This is true for every point of the arrow.  As a result a perfect inverted image of the arrow forms on the camera’s image plane.  The image plane, defined by the back of the box, is a distance f (for focal length) from the pinhole.  Interestingly for a perfect pinhole camera f can be any value.  Also, regardless of how far the object is from the pinhole the object is, a sharp image of it forms at the image plane.  The camera has infinite depth of focus or field.

We typically define a parameter called the f-number of the camera that defines depth of focus.  It is the focal length f divided by the diameter of the pinhole d.

I recently went to see an Ansel Adams exhibit entitled At the Water’s Edge,” at the Peabody Essex museum in Salem, Massachusetts.  I have had several such encounters over the years, and encounters are truly what they are.  Years ago I went to an exhibit at the Palace of Fine Arts in San Francisco, then again in San Francisco about ten years later, and then there was a major retrospective at the Museum of Fine Arts in Boston.  What I find interesting is how my view of photography and these works,  in particular, has evolved.

Early on I was taken by the sharpness of detail, how you can see every blade of grass or every nuance in the bark of a tree.  Sharpness in a picture relates to what we call resolution.  Later, particularly at the MFA exhibit, I was taken, really overwhelmed, by the dynamic range in these pictures.  Dynamic range, as already suggested, is the number of shades of grey between black and white in your picture.  The power of Adams; pictures, their inherent luminescence, owes widely to careful placement of the dynamic range.   This is the zone system that Adams’ created.  He would painstakingly measure the brightest and dimmest elements in his picture and then develop, first the negative, and then the print so as to just place these levels within the gamut of grey levels that the photograph allows.  Ultimately, the choice of film developer, paper developer, exposure times, and development times – not to mention some very well chosen dodging and burning in, modified the linearity of response in the final print, the so called gamma.  Linearity means, plain and simple, if I double exposure does recorded density also double.  If it isn’t then gamma isn’t one.

In the next few weeks, I’d like to explore all of these photographic elements in detail.  The point, for now, is that achieving a “technically good” image requires matching.  You’ve got to match the resolution of your lens  to that of your detector, be it digital or film, and then to the resolution of your print, be it a computer screen or printer,  And then the printer resolution needs to be matched to the resolution of the paper as the eye sees it.  Similarly you’ve got to match the dynamic range in the scene to that of the detector and then again to computer screen, printer, and paper. Again, as always, the eye is the ultimate arbiter and task master.

In the world of analogue photography Adams’ zone system was very hard to implement and exploit for 35 mm roll film cameras. Simply because it was hard to keep every picture in the roll within the same set of exposure and development conditions, there was a lot of compromising to be done.  However, just as Eastman’s invention of roll film made it so easy to take pictures, that they for the most part became bland and mediocre, digital photography, particularly if you shoot in raw format, has brought technical proficiency easily within everyone’s reach.  You are left to concentrate on artistic vision.  And as I walked through At the Water’s Edge, I came to realize that when all was said and done, Ansel Adam’s greatness lay in his vision.

The form of photography we spoke about in the last blog is called autoradiography.  You might ask whether this is really a form of photography and whether it truly belongs in a blog about photography.  The thing is that light comes in many flavors.  Light is an electromagnetic wave, much like the waves at a beach.  The distance between one wave and the next is referred to as the wavelength.  Visible light has wavelengths between about .350 microns and .750 microns.  Remember that a micron is a thousandth of a millimeter. But the spectrum of electromagnetic radiation extends beyond visible light.  Increasing wavelength we have infrared light, microwaves, sub mm waves, and radio waves.  In the other direction we have ultraviolet light, X-rays, and gamma rays.  So really we are taking a pretty narrow view, if we confine ourselves to visible light simply because that is what the human eye can see.

So suppose we had a painting that contained a pigment that contained arsenite, ⁷⁶As.  ⁷⁶ As has a half life of just over a day.  This means that if you start off with a gram of ⁷⁶As, it will give off gamma rays so fast that a day later you will only have ½ a gram, two days later ¼ gram etc. On the other hand if you have a gram of copper ⁶⁴Cu, it decays twice as fast; so a 1/4 gram after one day and 1/8 gram after two days.  So suppose you have a green pigment in a painting and want to determine if it contains arsenite, an arsenic ore, or malachite, a  copper ore, you simple place a sheet of photographic film under the painting for an an hour or so develop it and repeat the process a day and then two days later.  After 24 hours the density of the film below a green region decreases in half if it is made of arsenic or to 1/4 if it is made of copper.  Then to a 1/4 for arsenic or a 1/16 for copper after 48 hours.

It’s as simple as that, except for one thing.  There is very little ⁷⁶As in arsenite.  It is almost completely ⁷⁵As.  There’s a similar story for copper, which likes to be either ⁶³Cu or ⁶⁵Cu.  Atoms consists largely of two types of particles: protons and neutrons.  All arsenic atoms have 33 protons.  The so called isotopes differ by the number of neutrons.  In ⁷⁵As there are 22 neutrons.  In ⁷⁶As there are 23.  Now here’s the problem, natural samples of arsenite have precious little ⁷⁶As, certainly not enough to expose a photographic film in an hour.

To turn ⁷⁵As into ⁷⁶As you’ve got to shoot neutrons into your painting.  To accomplish this you have to take your painting to your local nuclear reactor and transmutate the metals in your pigments..  Most art museums don’t have nuclear reactors in their basements.  Fortunately, many physics departments do.  These are the basics of neutron activated autoradiography.  Remember this next time you want to catch a Spanish forger.

* For those who want to dig into this a bit more deeply check out Neutron Activation of Paintings.

I can’t quite remember the details, but a while back one of my Facebook friends posted a video dealing with some previously unknown Amazon tribe, or some such, losing their culture, anonymity, and homes to the press of lumbering.  The video was very sad, but viscerally didn’t ring true.  So I made some comment like “sad, if true.”  One of this friend’s friends (what does that make them to me?) fired back angrily that it didn’t matter if it was true, it was still sad.  I’m sorry, but my mother taught me that above all, and despite whatever the political agenda, the truth matters.  In my adult life I have frequently been confronted with the dilemma of whom should I go with, this guy (fill in the name) or my mother.  Experience has taught me, always go with what your mother taught you.

My mother was Sally Wolf (1917-1988).  She was not an opera singer, but she was very wise.

This desire to deceive with image manipulation is not new – witness the Shroud of Turin.  The desire to believe suspends all credulity. But let’s consider the Spanish Forger, who was a late 19^th century forger of medieval illuminated manuscript pages.  Scientists used a form of photography where the paintings were overlaid with photographic film so as to measure the rate of radioactive decay of pigments in the pictures.  The greens were found to contain a pigment called “emerald green” that consists of copper arsenite, not known to be used in artists’ paints until 1814.  The story of green pigments and their evolution in the 19^th century is itself very interesting, so perhaps another time.  As for the Spanish Forger, he won in a sense.  His forgeries are collected today for his artistry and in 2009 the Victoria and Albert Museum purchased a large collection of them.