Hi, Asher,
Bart and Doug,
What dpi printer setting do you use to optimally express the resolution in the 720 pixel per inch star target? It would be a reasonable assumption that one would go to 1440 dpi, so that the print would be more accurate?
Well, the first thing is that I have no way to deal with dots.
In any case, 1440 dpi is not an image resolution setting. It is description of a dot pitch used by the printer engine and its head as it uses several dots to make a pixel.
My printer (Epson SP R1900), in all combinations of settings (except Draft mode with toilet paper as the medium) reports to Qimage Ultimate that it wants to have social intercourse on the basis of 720 pixels/inch. I assume ti uses dots at the pitch of 1440 dots/inch in one direction (and I think 2880 in the other) as its "paintbrush" to paint pixels, 720 of 'em per inch.
Another clarification: Nyquist and printer dpi?
It occurs to me that one might ask why the printer is able to get practically the specified dpi as the input pixels/inch resolution.
I will respond with a brief (!) tutorial on the concept of the representation of a continuous phenomenon by sampling, which is heavily involved any time we work with discrete pixels.
Shannon and Nyquist teach us that if the have the"value" of a continuous phenomenon (for example, the color of the image on the focal plane) at intervals whose rate is Fs (the sampling frequency), and if all frequency components that make up the variation of that the phenomenon have frequencies less than Fs/2 (which we call Fn, the Nyquist frequency), then that set of values
completely describes the
entirety of the variation of the phenomenon (that is, the color at
every point - not just at every pixel location - on the image on the focal plane.
Woof!
This is at the center of the story, and although it may be hard to grasp, or accept, it is in fact so.
Now from that set of values (in our case,. the color representation at every pixel location), how to we fulfill Shannon and Nyquist's promise - how to we get back a complete image, corresponding to the image on the focal plane?
Well, imagine this conceptually (think for the moment in terms of reconstruction on a display, not on a printed page).
Suppose that on the display screen, at every pixel location (think in terms of the center of the pixel) we made a very tiny dot of light whose color was that indicated by the color code for that pixel.
Then we put in front of that an optical low-pass filter (spatial), whose cutoff frequency (spatial) is the Nyquist frequency.
What comes out of that filter (what we see) is an exact reproduction of the variation of color at every point on the original image on the focal plane.
Woof!
This may at first be hard to grasp, or accept, but believe me - it is so.
But how could we make such a spatial low-pass filter?. Well it turns out if, at each pixel location, the spot of light created from the pixel color code was not of infinitesimal size, but rather was spread out following a certain profile. That has the same as having dots of infinitesimal size behind a low-pass filter.
Another woof!
Now it turns out for this story to work precisely, that" profile" of intensity for each pixel spot would have to go negative at certain ranges of distance from the center, which of course cannot happen, so we could only hope to approximate this scenario. But what is important is not if we can do it "exactly" but rather for you to understand the underlying principle.
Now lets move to a printer. In the "first blackboard model", the printer would deposit a dot of ink of infinitesimal diameter for each pixel. Well, that has to be a dot whose color is that indicated by the color code for the pixel, so already we have a complication, having to make that "tiny dot" out of maybe six or eight tiny dots.
Now, having done this, we must "view" this array of tiny dots, one for each pixel, at the center of the pixel's real estate, through an optical low-pass filter (spatial), whose cutoff frequency is Fn.Then what we would see is an exact reproduction (larger of course) of the entire image on the focal plane (at
every point, not just at the pixel centers.
This is the wonder of what Shannon and Nyquist talk about.
But of course we don't want our viewer to lave to look at the pint through an optical low-pass filter. And in fact, having such would so spread out the color of each of those infinitesimal dots that the image seen by the viewer would be very feeble.
So instead we do something that parallels what I described above for the display. For each pixel color value, we deposit ink whose profile of intensity is such that it creates the low-pass filter function. It will have to spread over more than the area of that pixel to come close to performing the ideal concept.
So at each exact pixel on the print (to make an easy story), we will have ink that represents the color of that pixel, plus the color of each adjacent pixel scaled by the value of its profile at its distance from the point we are considering, and so forth.
And of course this is what all the computing power inside our printer does, among other things.
Now to circle back to your original pondering:
Suppose that we have a set of data (the image data sent to the printer) that gives the color of the image at every pixel location, suppose at a pitch of 720 per inch.
Then that should be able to "completely describe" the color all over the image insofar as the variations do not contain frequencies at or above the Nyquist frequency.
So in this case, that data would be able to, for example, convey a regular pattern with a frequency of almost 360 cycles/inch. We can think in terms of that having a line frequency of almost 720 lines/inc, or in term of pixels, of almost 720 pixels/inch.
And the printer should be able to paint that,. following the discussion above, painting the color encoded for each pixel as a pattern, centered on that pixel center, with a certain profile.
Now why might we not get very nearly 720 pixels/inch of resultion? Well, because various real things spoil my blackboard story. The printer may not be able to create that ideal profile (and in fact the really ideal profile is impossible to do physically, as it would require negative amounts of ink here and there).
Best regards,
Doug