Fundamentals of Computer Science I: Media Computing (CS151.02 2007F)

Representing Images as Lists of Spots

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Summary: We examine Scheme's list data type and an application of that type to representing images.


Introduction: Representing Data

As we've said a few times, computer scientists study the algorithms we write to manipulate information and the ways in which we represent information. Since an emphasis of this course is images, we have been emphasizing kinds of data related to images. You may not have noticed it, but we have developed a number of ways to represent both colors and images.

For the case of colors, we can represent them as names (e.g., "blood red"), as RGB colors (often computed with the, as in ( 102 0 0)), and as strings that list the three components (e.g., "102/0/0").

For the case of images, we can represent them as a grid of rgb color values that you can get or set. However, as we've discovered with our experiments with region.compute-pixels!, a function from position to color is sufficient to describe many images.

It's now time to consider another way to think about images. Particularly for images that have only a few pixels set, we can describe the image by indicating the positions and colors of those pixels. (We can also describe a small part of the image in this way. Such descriptions can then act as a form of stamp for the image.) We call call the combination of position and color a spot.

Scheme's Lists to the Rescue

Fortunately, Scheme, like Lisp before it, provides a simple and elegant way to represent collections of data, the list. In contrast to Scheme's unstructured data types, such as symbols and numbers, lists are structures that contain other values as elements. In particular, a list is an ordered collection of values. In Scheme, lists can be heterogeneous, in that they may contain different kinds of values.

As we will see throughout this semester, lists provide a simple, relatively convenient, and elegant mechanism for organizing collections of information.

How does Scheme show you that something is a list? It surrounds the list with parentheses and separates the items with spaces. For example, the following is the list of three color names that we like.

("red" "grey" "black")

For our problem of representing simple images, we can create a list of spots, where each spot consists of a column, row, and color. (How do we represent the combined column, row, and color? With another list.) For example, here is the list that might correspond to a white spot at position (10,11) (recall that -1 seems to be the integer that DrFu uses for white).

(10 11 -1)

Similarly, here is a list that might represent red spots at (10,1), (10,2), and (10,3); grey spots at (11,2), (11,3), and (11,4), and black spots at (12,3), (12,4), and (12,5). Note that black seems to be represented by 255, grey by -1465341697, and red by -16776961.

((10 1 -16776961) (10 2 -16776961) (10 3 -16776961)
 (11 2 -1465341697) (11 3 -2139062017) (11 4 -2139062017)
 (12 3 255) (12 4 255) (12 5 255))

Of course, to achieve the goal of usefully representing images as lists of spots, we need to know more than what those lists might look like when printed. We must also know basic operations we can use for processing those lists, and others. These include operations for creating lists, for extracting information from lists, for extending lists, and for dealing with each element in a list.

Creating Lists with list

The easiest way to create a list is to invoke a procedure named list. This procedure takes as many parameters as you care to give it and packs them together into a list.

many of them there may be, and packs them into a list. Just as the addition procedure + sums its arguments and returns the result, so the list procedure collects its arguments and returns the resulting list:

> (list "red" "grey" "black" "white")
("red" "grey" "black" "white")
> (list)
> (list 1 2 3 4 5 6 7)
(1 2 3 4 5 6 7)
> (list "red" ( 255 0 0) (rgb->string (cname->rgb "red")))
("red" -16776961 "255/0/0")
> (list 10 11 ( 255 255 255))
(10 11 -1)
> (list 11 2 color.grey)
(11 2 -1465341697)
> (list (list 10 1 
        (list 10 2 
        (list 10 3 
        (list 11 2 color.grey) 
        (list 12 3 color.grey))
((10 1 -16776961) 
 (10 2 -16776961) 
 (10 3 -16776961) 
 (11 2 -1465341697) 
 (11 3 -2139062017))

Nested Lists

You may recall that we said that a list can contain any types of Scheme values. As the last two examples above suggest, lists can even contain other lists as values. The technique of placing one list inside another is called nesting lists, and is useful in a wide variety of situations.

Most typically, we nest lists when the element lists are used for a different purpose than the enclosing lists. For example, in the examples above, the element lists each represent one spot (that is, column, row, and color) and the list of elements represents a collection of spot.

We are not limited to this simple nesting. We can nest lists within lists within lists within lists, as deep as we desire to go. Not all of our nested values need to be lists. We can, for example, make lists some of whose elements are lists, some of whose elements are strings, some of whose elements are numbers, and so on and so forth.

Constructing Lists with cons

Although list is a convenient way to build lists, the list operation is, itself, composed of other, more primitive, operations. Behind the scenes, list invokes cons once for each element of the completed list, to hook that element onto the previously created list. The cons procedure takes two parameters, a value and a list, and builds a new list by prepending the value onto the list. The prepended value is called the car of the resulting list. The previous list (that is, the list that has now had a new value added to the front) is called the cdr of the resulting list.

> (cons 5 (list 1 2 3))
(5 1 2 3)

Why is it called cons instead of list.prepend or something similar? Well, that's the name John McCarthy, the designer of Lisp, chose about forty years ago. It's short for construct, because cons constructs lists. (The habit of naming procedures with the basic type they operate on, a period, and the key operation did not start until a few decades later.) The names car and cdr were chosen for very specific reasons that will not make sense for a few more weeks.

Scheme's name for the empty list is a pair of parentheses with nothing between them: (). Most implementations of Scheme suggest permit you to refer to that list as nil or null. A few permit you to type it verbatim. All permit you to describe the empty list by putting a single quote before the pair of parentheses.

> '()
> nil
> null

You will find that we prefer to use a name for that list. If sample code does not work in your version of Lisp, try inserting the following definitions.

(define nil '())
(define null '())

Note that by using cons and nil, we can build up a list of any length by repeatedly prepending a value to the list.

> (define singleton (cons "red" null))
> singleton
> (define doubleton (cons "green" singleton))
> doubleton
("green" "red")
> (define tripleton (cons "yellow" doubleton))
> tripleton
("yellow" "green" "red")
> (cons "senior" (cons "third-year" (cons "second-year" (cons "freshling" null))))
("senior" "third-year" "second-year" "freshling")

You may note that lists built in this way seem a bit backwards. That is, the value we add last appears at the front, rather than at the back. However, that's simply the way cons works and, as the last example suggests, in many cases it is a quite sensible thing to do.

Taking Lists Apart

So far, so good. We now know how to build lists using two techniques: Using list and using a combination of cons and nil. Of course, creating lists is not very useful unless we can extract the individual elements of the list.

The two most basic operations one uses to recover elements from a list are car and cdr. The car procedure takes as a parameter a non-empty list and returns the first element. The cdr procedure takes as a parameter a non-empty list and returns the remaining elements (that is, all but the first element). In a sense, car and cdr are the inverses of cons; if you think of a non-empty list as having been assembled by a call to the cons procedure, car gives you back the first argument to cons and cdr gives you back the second one.

> (define colors (cons "red" (cons "black" nil)))
> colors
("red" "black")
> (car colors)
> (cdr colors)

We can, and often do, nest these calls. For example, to get the second element of a list, we first get the cdr of the list and then take the car of that result.

> (car (cdr colors))

To reduce the amount of typing necessary for the programmer, many implementations of Scheme provide procedures that combine car and cdr in various ways. For example, cadr computes the car of the cdr of a list (the second element), cddr computes the cdr of the cdr of a list (all but the first two elements), and caar computes the car of the car of a list.

> (define spots(list (list 10 1 
                       (list 10 2 
                       (list 10 3 
                       (list 11 2 color.grey) 
                       (list 12 3 color.grey)))
> spots
((10 1 -16776961) (10 2 -16776961) (10 3 -16776961) (11 2 -1465341697) (11 3 -2139062017))
> (car spots)
(10 1 -16776861)
> (cadr spots)
(10 2 -16776961)
> (caddr spots)
(10 3 -16776961)
> (cadddr spots)
(11 2 -1465341697)
> (caar spots)
> (cadar spots)
> (caddar spots)

List Predicates

Scheme provides two basic predicates for checking whether a value is a list. The null? predicate checks whether a value is the empty list. The list? predicate checks whether a value is a list (empty or nonempty).

> (null? nil)
> (list? nil)
> (null? (list 1 2 3))
> (list? (list 1 2 3))
> (null? 5)
> (list? 5)

Other Common List Procedures

It turns out that you can build any other list procedure with just nil, cons, car, cdr, and nil?. Nonetheless, there are enough common operations that most programmers want to do with lists that Scheme includes them as basic operations. (That means you don't have to define them yourself.) Here are four that programmers frequently use.


The length procedure takes one argument, which must be a list, and computes the number of elements in the list. (An element that happens to be itself a list nevertheless contributes 1 to the total that length computes, regardless of how many elements it happens to contain.)

> (length nil)
> (length (list 1 2 3))
> (length (list (list 1 2 3)))


The reverse procedure takes a list and returns a new list containing the same elements, but in the opposite order.

> (reverse (list "white" "grey" "black"))
("black" "grey" "white")


The append procedure takes any number of arguments, each of which is a list, and returns a new list formed by stringing together all of the elements of the argument lists, in order, to form one long list.

> (append (list "red" "green") (list "blue" "yellow"))
("red" "green" "blue" "yellow")

The empty list acts as the identify for append.

> (append nil (list "blue" "yellow"))
("blue" "yellow")
> (append (list "red" "green") nil)
("red" "green")
> (append nil nil)


The list-ref procedure takes two arguments, the first of which is a list and the second a non-negative integer less than the length of the list. It recovers an element from the list by skipping over the number of initial elements specified by the second argument (applying cdr that many times) and extracting the next element (by invoking car). So (list-ref sample 0) is the same as (car sample), (list-ref sample 1) is the same as (car (cdr sample)), and so on.

> (define rainbow (list "red" "orange" "yellow" "green" "blue" "indigo" "violet"))
> (list-ref rainbow 1)
> (list-ref rainbow 0)
> (list-ref rainbow 5)
> (list-ref rainbow 7)
Error:  --  list-ref "Too few elements in list." 




Disclaimer: I usually create these pages on the fly, which means that I rarely proofread them and they may contain bad grammar and incorrect details. It also means that I tend to update them regularly (see the history for more details). Feel free to contact me with any suggestions for changes.

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Samuel A. Rebelsky,

Copyright © 2007 Janet Davis, Matthew Kluber, and Samuel A. Rebelsky. (Selected materials copyright by John David Stone and Henry Walker and used by permission.) This material is based upon work partially supported by the National Science Foundation under Grant No. CCLI-0633090. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. This work is licensed under a Creative Commons Attribution-NonCommercial 2.5 License. To view a copy of this license, visit or send a letter to Creative Commons, 543 Howard Street, 5th Floor, San Francisco, California, 94105, USA.