Building upon the idea of representation, we will discuss how images are represented in digital form. We'll work up to it, first starting with how color is represented (which is based on the physiology of the human eye), then looking at images as rectangular arrangements of spots of pure color. Finally, we'll calculate the file size of an image and discuss one way of compressing the file so that it is smaller and therefore faster to download. This compression is, in fact, a different representation of the information.
In the present day, modern browsers support 140 color names. This means that we can use color names such as black, aqua, or chocolate as values for the CSS properties that except a color value, such as color, background-color, etc. Many years ago, browsers could only support 17 color names, known as standard colors: aqua, black, blue, fuchsia, gray, green, lime, maroon, navy, olive, orange, purple, red, silver, teal, white, and yellow. However, later that list was expanded with 123 more colors. W3Schools maintains a complete list of the 140 recognized color names. While you can achieve a lot by using only these named colors, very often you want something more specific from the color spectrum. It turns out that we can use numerical codes to refer to colors, because inside the computer, colors are represented by numbers. How? For that, we need to understand additive colors and color vision.
black
aqua
chocolate
color
background-color
Human retinas happen to have rod-shaped cells that are sensitive to all light, and cone-shaped cells that come in three kinds: red-sensitive, green-sensitive, and blue-sensitive. Therefore, there are three (additive) primary colors: Red, Green and Blue or RGB. All visible colors are seen by exciting these three types of cells in various degrees. (For more information, consult these Wikipedia articles on additive color and color vision.)
Color monitors and TV sets use RGB to display all their colors, including yellow, chartreuse, you name it. So, every color is some amount of Red, some amount of Green, and some amount of Blue.
On computers, RGB color components are standardly defined on a scale from 0 to 255, which is 8 bits or 1 byte.
Play with the Color slider page to get a feel for this.
Here is a list of examples:
We can use this knowledge about colors being represented as a mix of red, blue, and green when specifying color values in CSS. There are three ways to do this:
color: rgb(64,224,208); /* three RGB numbers in the range 0-255 */ color: rgb(25%,88%,82%); /* three RGB percentages */ color: #40E0D0; /* three RGB numbers expressed as a hexadecimal triple */
The first two ways are self-explanatory, since they use decimal numbers and percentage values with which you are familiar. In the following, we will explain the meaning of the hexadecimal color codes such as #40E0D0. The # sign is used in this case to simply indicate that the sequence of digits and letters is in hexadecimal.
#40E0D0
#
People use decimal (base 10), computers use binary (base 2), but programmers often use hexadecimal (base 16) for convenience.
Binary numerals get long very fast. It is not easy to remember 24 binary digits, but you can more easily remember 6 hexadecimal digits. Each hexadecimal digit represents exactly four binary digits (bits). (This is because 2^{4}=16.)
One way to understand hexadecimal is by analogy with decimal, but we're all so familiar with decimal numerals that our reflexes get in the way. (In fact, humans throughout history have used many different numeral systems; decimal is not sacrosanct.) So, we first need to break down decimal notation so that you can see the analogy with hexadecimal. For now, we'll stick with two-digit numerals, but the same ideas extend to any larger numbers.
Decimal notation works by organizing things into groups of ten, then counting the groups and the leftovers: Suppose you had a bunch of sticks on the ground and you bundled them all into groups of 10 with some left over (fewer than 10). Now, use a symbol to denote the number of bundles and another symbol to denote the number of sticks left over. You've just invented two-digit numbers in base 10.
Hexadecimal: Do the same thing with bundles of 16, and you've invented two-digit numbers in base 16. For example, if you had thirty-five sticks , they could be bundled into two groups of sixteen and three left over, so the hexadecimal notation is 23. Careful! That numeral isn't the decimal number twenty-three! It's still thirty-five sticks, but we write it down in hexadecimal as 23.
To distinguish a decimal numeral from a hexadecimal numeral, we use subscripts. So, to say that thirty-five sticks is written 23 in hexadecimal, we can write:
35_{10} = 23_{16}
Both decimal and hexadecimal notations are based on place value. We say that 23_{16} means 35_{10} because it's a "2" in the sixteens place and "3" in the ones place, just like 35_{10} has a "3" in the tens place and a "5" in the ones place.
Let's take another example. Suppose we have 26_{10} sticks. That's one group of 16 and 10 left over. How do we write that number in hexadecimal? Is it 110_{16}? That is, a "1" in the sixteens place followed by a "10" in the ones place? No; that would be confusing, since it would look like a three-digit numeral. We need a symbol that means ten. We can't use "10," since that's not a single symbol. Instead, we use "A"; that is, A_{16}=10_{10}. Similarly, "B" means 11, "C" means 12, "D" means 13, "E" means 14, and "F" means 15. We don't need any more symbols, because we can't have 16 things left over, since that would make another group of 16. The following table summarizes these correspondences and what we've done so far.
To convert a big decimal number to hexadecimal, just divide. For example, 230_{10} divided by 16 is 14_{10} with a remainder of 6_{10}. Thus, the hexadecimal numeral is E6_{16}. To convert a hexadecimal number to decimal, just multiply: E6_{16}=E*16 + 6 = 14*16 + 6 = 230.
E6_{16}=E*16 + 6 = 14*16 + 6 = 230
Try the following conversions as an in-class exercise. You can use a calculator, you can ask your neighbors, anything you like.
You can check your work with the following form:
Now that we know both hexadecimal and binary, you can convert binary to hexadecimal (and vice versa). However, you would probably do so by converting the binary number to decimal and then the decimal number to hexadecimal. There's a better way, involving almost no arithmetic (or, rather, all the arithmetic is with one-digit numbers you can add in your head). Indeed, this technique is the reason that computer scientists like using hexadecimal. (Well, this and the fun of getting to spell words like ACE and DEADBEEF with hex digits.)
Let's start with an example. Suppose you need to convert the following from binary to hexadecimal:
01010100 = ??_{16}
What we're going to do is to take the bits in chunks of four bits, so to mark the chunks we'll insert a period in the middle of the number:
0101.0100 = ??_{16}
Now, we just convert each chunk directly into hex. The first chunk, 0101, is just the number 5. The second chunk, 0100, is just the number 4. Those are already in hex, so we are done:
0101.0100 = 54_{16}
(Try doing it via decimal, to check. The decimal value corresponding to both of these is 80+4=84.)
Let's do another one, this time with slightly larger values:
10101100 = ??_{16}
Again, take the bits in chunks of four bits:
1010.1100 = ??_{16}
Now, we just convert each chunk directly into hex. The first chunk, 1010, is 8+2 or 10_{10}, which is the digit A in hex. The second chunk, 1100, is 8+4 or 12_{10}, which is the digit C in hex. So we are now done:
1010.1100 = AC_{16}
(Again, check our work by doing it via decimal. The decimal value corresponding to both of these is 160+12=172.)
Why does this work? Suppose we needed to convert 172 from decimal to hex: our first step would be to divide the number by 16. In binary, moving the binary point to the left by one place is equivalent to dividing by two, so moving the binary point four places is equivalent to dividing by 16. So when we put a period in the middle of the 8-bit binary number, it is exactly the same as dividing by 16. We then have the quotient to the left of the binary point, and the remainder to the right of the binary point. Just convert each to hex, and we are done.
binary point
Notice that the only arithmetic we have to do is converting each chunk of four bits to the equivalent hex digit. The mental arithmetic involved is limited: we know that (1) we are adding one-digit numbers, (2) at most four of them, and (3) the sum will always be less than 16.
Watch the rest of Prof. Kurmas from Grand Valley State University on binary numbers and hex numbers. This is a version he edited for us. You watched the first 5 minutes for last time; watch the rest for today.
Even better, here's a video with Tom Lehrer singing New Math. It's about 4 minutes long; you'll enjoy it.
New Math
We already know that every color in a computer is a combination of some amount of each of the three primary colors: red, green and blue. The amounts are always given in the same order: red, green, blue. The amounts are numbers from 0 to 255_{10}, or, in hexadecimal, 00 to FF_{16}. Each primary is expressed as a two-digit numeral in hexadecimal, using a leading zero if necessary so that the numeral is always two digits. Three pairs of hexadecimal digits completely specifies a color. Finally, the notation for a color always starts with a pound sign (#). For example, a color like (35, 230, 10) would be written #23E60A.
Experiment with defining a color numerically. In the form below, enter a color value in the syntax #RRGGBB and press return/enter. The box will change its background color to display the entered color value.
Now that we know how to represent a color, we can represent images. You can think of an image as a rectangular 2D grid of spots of pure color, each represented as RRGGBB. A spot of pure color is called a pixel, short for picture element, the atom of a picture. Pixels are better seen if you blow up an image several times; here are some examples. Click on the picture to enlarge it.
Every image on the computer monitor is represented with pixels. The images on a web page are saved in files that, in addition to the image data, contain information on the size of the image, the set of colors used, the origin of the image, etc. Depending on how exactly this information is saved, we refer to them as image formats. GIF, JPEG, PNG, and BMP are some of the well-known image formats. We will talk more about image formats below. For now, we will focus on the number of pixels and the representation of each pixel, and consequently, the file size of the image.
We said above that the amount of each primary color is a number from 0 to 255_{10} or 00 to FF_{16}. It is no coincidence that this is exactly one byte (8 bits). A byte is a convenient chunk of computer memory, so one byte was devoted to representing the amount of a single primary color. Thus, it takes 3 bytes (24 bits) to represent a single spot of pure color.
With 256 values for each primary, we have 256 x 256 x 256 = 16,777,216 colors. Humans can distinguish over 10 million colors, so 24-bit color is sufficient to represent more colors than humans can distinguish. All modern monitors use this so-called 24-bit color. Some old monitors used 16-bit or 8-bit color, which were relatively impoverished, being only able to represent 65,536 colors (for a 16-bit monitor) or 256 colors (for an 8-bit monitor). Of course, a black-and-white monitor can only represent two colors, which could be called 1-bit color. An example is the Scottish terrier picture, above.
In an uncompressed file format, every pixel needs 24 bits (3 bytes) to be stored. Let's suppose you are going to take pictures of all your 30 class peers for a class website, using your iPhone4 camera. According to the phone specifications, its screen has 2592 x 1936 pixels, which amounts to about 5 million pixels, or 5MP (mega-pixels). Thus, if every pixel takes 3 bytes, and a photo with your camera has 5MP, to store the image you need 15MB (mega bytes). For all your peer photos, you will need 30 x 15MB = 450 MB.
Imagine now that you put all these photos online on your website, in one single page (using the attributes width and height to make them fit in one screen), and then you send the link to this page to your parents. They might have an average Internet connection (e.g. Verizon offers offers a 1-3 Mbs (mega bit per second) to non-FiOS subscribers).
The amount of time that it will take to load a page with all these pictures on your parent's computer can be calculated as below:
content size (450MB) x 8 bits/byte / 1Mbs = 3600 seconds or 1 hour.
If each of your photos have been only around 100KB instead (as we require in some of your homework assignments), then the amount of time to load all of them on the page would have been 24 seconds.
So, how do we get our images to be so small in size? There are two ways: resizing (decrease the number of pixels in the image by judicious cropping), and compressing (decreasing the necessary number of bits per pixel). We will discuss compression in the next section.
Short of making our images smaller (fewer pixels), what can we do to speed up the downloads? We can compress the files.
There are two classes of compression techniques:
We will look in detail at one kind of lossless compression, which is indexed color (GIF encoding), because it gives us a window into the kinds of ideas and techniques that matter in designing representations of information.
The idea behind indexed color is that if a particular color is used many times in an image, we can create a "shorthand" for it. In fact, if we limit the number of colors, each one can be assigned a shorthand. What will be confusing is that the colors are, of course, represented as numerals and so are the shorthands! For example, instead of saying (for the umpteenth time), color #D619E0, we'll just say, for example, color number 5. This will only work, however, if the shorthands really are shorter. They are, and we'll see exactly how much.
One way to think about indexed color is that we are creating a "paint-by-numbers" picture. We choose:
You can see the general scheme at work: we create a table of all the colors used in the picture. The shorthand for a color is simply its index in the table. We will limit the table so that the shorthands will be at most 8 bits. Since the shorthands are all replacing 24-bit color specifications, the shorthand is at most one-third the size. In the example above, the shorthand is 1/24th the size.
Let's continue with the example. What is the file size if the image uses 4 colors, say red, yellow, blue and lime? In that case, the table looks like this:
As you can see, the shorthand is now two bits instead of one. Therefore, the 150,000 pixels require 300,000 bits or 300,000/8=37,500 bytes or about 37.5kB. Obviously, this is about twice the size of the previous example, since each shorthand is now twice as big. Nevertheless, it's still much smaller than the 450 kB uncompressed file.
What about the size of the palette? That's now twice as big, too. Four entries at 3 bytes each adds 12 bytes to the file size, which is a negligible increase to the 37.5 kB.
What's the pattern here? The number of colors in the original image determines the size of the palette, which determines the number of bits in each shorthand, which then determines the size of the file as a whole. The shorthand for a color is simply the binary numeral for the row that the color is in the table. For example, the color red in the last example was in row zero (00 in binary) and the color lime was in row 3 (11 in binary).
You can see that the number of bits required for each pixel is the key quantity. This quantity is called bits per pixel or "bpp." It's also often called "bit depth" so that the file size of an image is just width x height x bit depth, almost as if it were a physically 3D box.
width x height x bit depth
Finally, we can state the rule:
The bit depth of an image must be large enough so that the number of rows in the table is enough for all the colors. If the bit depth is d, the number of rows in the table is 2^{d}.
Here's the exact relationship, along with the size of a 300x500 image:
Consider an image that is 80 x 100 (pixels).
In summary, you can reduce your image file size by using fewer colors. Of course, this may reduce the quality of your image. It's a tradeoff.
We've learned how indexed color works and how it affects file size. This is important not only for the theoretical understanding of why representations matter, but also for the practical usefulness of understanding how to reduce the sizes of your images. In this section, we'll review how to compute the approximate size of an indexed-color image. (Indexed color is one of the tricks used in GIF files, though GIF files use other tricks as well.) Why do we do this? Because it combines all the conceptual issues into one small calculation.
A key concept in the computation is the bit-depth of the image. Read on page 19 the definition of bit-depth. It's the number of bits necessary to represent the desired number of colors. Remember that the number of colors is 2^{d}, where d is the bit depth. It's an exponential relationship. Adding just one bit to the bit-depth doubles the number of colors you can have.
Recall that the indexed-color representation comes in two parts:
Thus, our computation breaks down into two parts.
width * height * bit_depth / 8
num_colors * 3
To find the rough size of an image, we first determine the bit-depth, then we compute the file size using the two formulas above. (This is the rough size because, remember, we are omitting some fixed overhead and further compression techniques.) You can combine them into one formula:
rough
(width * height * bit_depth) / 8 + (num_colors * 3)
Finally, because the file size will usually be large (thousands or millions of byte), we divide by 1000 or 1,000,000 to convert to kilobytes or megabytes, as appropriate.
We will continue to discuss file size calculations in lab and homework assignments.