Approximating nxnx(m-1)+1 Overall Intensity Levels

[s3a]

Homework Statement

PROBLEM STATEMENT:
Show that with an n × n pixel grid, where each pixel can take on m intensity levels, we can approximate n × n ×(m – 1) + 1 overall intensity levels.

TEXTBOOK'S SOLUTION TO THE PROBLEM:
Since the n ×n pixels can be set to a non-zero intensity value one after another to produce n ×n overall intensity levels, and there are m MINUS 1 non-zero intensity levels for the individual pixels, we can approximate a total of n × n × (m – 1) non-zero overall intensity levels. Finally, we need to add one more overall intensity level that corresponds to zero intensity (all pixels off).

Homework Equations

According to the solution of the textbook: Approximate amount of overall intensity levels = n × n × (m – 1) + 1

The Attempt at a Solution

(Having researched, I found that "a pixel intensity value describes how bright that pixel is and/or what color it should be.")

Why is an "overall intensity level" just one of the n × n pixels as a non-zero intensity, with the rest as zero? Shouldn't there be a huge amount of combinations, like, for example, pixel 0 and pixel 1 having non-zero values or pixel n – 2, n – 1 and n each having non-zero pixels, etc.? In other words, shouldn't the answer be something like "n × n choose m" = (n×n)Cm?

Any help in getting me to fully understand the problem and its solution would be GREATLY appreciated!

P.S.
Why is the answer that the textbook gives an approximation?

 

[.Scott]

We are assuming that every pixel has the same selection of intensity levels. Otherwise, the number of possible intensities is much higher.

The next assumption is whether or not 2 pixels with intensity 1 are the same brightness as 1 pixel with intensity 2. In other words, whether the intensity steps are equally spaced.
If so, then the formula would be as specified because...
* The minimum (all pixels off) is 0.
* The maximum (all pixels on) is n × n × (m-1).
* All integer values from 0 to n × n × (m-1) are possible by simply incrementing one of the pixel values from the previous overall intensity setup.

If that second assumption is not true, then the number can be different. For example. Let's say that the pixel intensities can be 1, (n × n), ... (n × n)^m:
Then we would be able to determine how many of each pixel intensity we had. The only information we could not determine is which pixel had which intensity.

 

[s3a]

Thanks a lot for your response. :)

I believe that I now get it, but before I jump from joy, I'd like to confirm some stuff with the the following questions.:
1. Is this statement correct (for the context of the main problem of this thread)?: "Each pixel of the set of pixels that represents an image has a pixel intensity value which describes how bright that pixel is and/or what color it should be."

2. Overall intensity level = sum of the intensity of each pixel (such that it doesn't matter where each pixel is located, and hence probably why you said that "the only information we could not determine is which pixel had which intensity.")?

 

[s3a]

Also:
3. Why is the answer that the textbook gives [n × n × (m – 1) + 1] an approximation?