Fundamentals of Computer Science I: Media Computing (CS151.01 2008S)
Summary: We examine Scheme's list data type and an application of that type to representing images.
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've probably noticed that we have a number of ways to algorithmicaly describe images. That is, we can describe images in terms of a sequence of GIMP commands, in terms of a sequence of turtle commands, or as a drawing built from the unit drawings.
It's now time to consider another way to think about images. As you may recall, when we first discussed ways to describe images, we talked about images as collections of colored pixels. Clearly, for really big images, it may be unwieldy to describe each and every pixel on the screen, but if the image contains just a few swatches of color on a white background, this may be an appropriate representation. (In fact, most images are represented, or at least described, in terms of lots and lots of pixels, so the technique is not just appropriate for images with a little color.) For each pixel, we'll keep track of the column, row, and color. We call this combination a spot.
In the past, we've relied on other people to implement the data types we use. This time, we're going to try to implement it ourselves. That means we will need (a) a way to represent individual spots, (b) a way to represent collections of spots, and (c) an appropriate collection of procedures to manipulate individual spots and lists of spots.
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")
It's fairly clearly how we'll represent a collection of spots: We'll just make a list of all the spots in the collection. (In fact, most of the time Scheme programmers want to represent collections, they end up using lists, at least during initial development.)
How do we represent each spot? Well, a spot is a combination of a column, row, and color. We can represent that with another list. For example, here is the list that might correspond to “a green spot at position (10,11)”.
(10 11 "green")
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 "grey", and red by "red".
((10 1 "red") (10 2 "red") (10 3 "red") (11 2 "grey") (11 3 "grey") (11 4 "grey") (12 3 "black") (12 4 "black") (12 5 "black"))
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.
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. 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 1 2 3 4 5 6 7)
(1 2 3 4 5 6 7)
(list "red" 1 'red)
("red" 1 red)
(list 10 11 (* 10 11))
(10 11 110)
(list (list 10 1 "red") (list 10 2 "red") (list 10 3 "red") (list 11 2 "grey") (list 12 3 "grey"))
((10 1 "red") (10 2 "red") (10 3 "red") (11 2 "grey") (11 3 "grey"))
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.
list is a convenient way to
build lists, the
list operation is, itself,
composed of other, more primitive, operations. Behind the scenes,
for each element of the completed list, to hook that element onto
the previously created list. The
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,
cons constructs lists. (The habit of
naming procedures with the basic type they operate on, a dash, and
the key operation did not start until a few decades later.) The names
cdr were chosen for
very specific reasons that will not make sense for a few more weeks.
For now, just accept that you're learning a bit of computer-ese.
Scheme's name for the empty list is a pair of parentheses with nothing
(). Most implementations of Scheme suggest
permit you to refer to that list as
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.
You will find that we prefer to use a name for that list. If sample code does not work in your version of Scheme, try inserting the following definitions.
(define nil '()) (define null '())
Note that by using
nil, we can build
up a list of any length by starting with the empty list and
repeatedly prepending a valu.
(define singleton (cons "red" null))
(define doubleton (cons "green" singleton))
(define tripleton (cons "yellow" doubleton))
("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.
cons is a pure
function. That is,
does not change
lst. Rather, it
builds a new list whose first element is
val and whose remaining elements correspond to
the elements of
So far, so good. We now know how to build lists using two techniques:
list and using a combination of
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 procedure takes as a parameter a non-empty
list and returns the first element. The
procedure takes as a parameter a non-empty list and returns the
remaining elements (that is, all but the first element). In a sense,
cdr are the inverses
cons; if you think of a non-empty list as
having been assembled by a call to the
car gives you back the first argument
cdr gives you
back the second one.
(define colors (cons "red" (cons "black" nil)))
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
cdr in various ways.
cadr computes the car of the cdr
of a list (the second element),
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 "red") (list 10 2 "red") (list 10 3 "red") (list 11 2 "grey") (list 12 3 "grey")))
((10 1 "red") (10 2 "red") (10 3 "red") (11 2 "grey") (11 3 -2139062017))
(10 1 -16776861)
(10 2 "red")
(10 3 "red")
(11 2 "grey")
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
checks whether a value is a list (empty or nonempty).
(null? (list 1 2 3))
(list? (list 1 2 3))
It turns out that you can build any other list procedure with
nil?, and some other programming techniques.
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.
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 (list 1 2 3))
(length (list (list 1 2 3)))
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")
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 identity for
(append nil (list "blue" "yellow"))
(append (list "red" "green") nil)
(append nil nil)
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
cdr that many times) and
extracting the next element (by invoking
(list-ref sample 0) is the same as
(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."
We started this reading by exploring the another way to represent images, and ended up thinking about lists. Let's return to the original question. How do we use spots to represent images? We need a few more tools to deal with long lists of spots, but we can still think about individual spots.
To create a spot, we simply list the three components.
;;; Procedure: ;;; spot-new ;;; Parameters: ;;; col, an integer ;;; row, an integer ;;; color, a color (name, RGB, etc.) ;;; Purpose: ;;; Create a new spot. ;;; Produces: ;;; spot, a spot ;;; Preconditions: ;;; [No additional] ;;; Postconditions: ;;; (spot-col spot) = col ;;; (spot-row spot) = row ;;; (spot-color spot = color (define spot-new (lambda (col row color) (list col row color)))
Since it will be useful to extract the column, row, and color, we also write procedures that, given a spot, return the appropriate component of the spot.
;;; Procedure: ;;; spot-row ;;; Parameters: ;;; spot, a spot ;;; Purpose: ;;; Extract the row from a spot. ;;; Produces: ;;; row, an integer (define spot-row (lambda (spot) (cadr spot))) ;;; Procedure: ;;; spot-col ;;; Parameters: ;;; spot, a spot ;;; Purpose: ;;; Extract the col from a spot. ;;; Produces: ;;; col, an integer (define spot-col (lambda (spot) (car spot))) ;;; Procedure: ;;; spot-color ;;; Parameters: ;;; spot, a spot ;;; Purpose: ;;; Extract the color from a spot. ;;; Produces: ;;; color, an integer (define spot-color (lambda (spot) (caddr spot)))
Why don't we just use
Because we want to encapsulate our data. If we
decide tomorrow (or next week or next year) to change the way we implement
spots, we only need to change these four procedures, rather than every
single procedure that uses spots.
Of course, the spots we've just described aren't much use unless we can render them on an image. There are a number of techniques we can use. For now, let's just select a small rectangle around the spot and fill it.
;;; Procedure: ;;; image-render-spot! ;;; Parameters: ;;; image, an image ;;; spot, a spot ;;; Purpose: ;; Draw the spot on the image. ;;; Produces: ;;; [Nothing; Called for the side effect] (define image-render-spot! (lambda (image spot) (context-set-fgcolor! (spot-color spot)) (image-select-rectangle! image selection-replace (spot-col spot) (spot-row spot) 1 1) (image-fill! image) (image-select-nothing! image)))
We also might think about rendering the spot with some enlargement factor. For larger spots, we might use circles, rather than rectangles, to render. If we're scaling the size of the spot, we probably also want to scale the position. The following provides an initial approach, but you might find others useful, too.
;;; Procedure: ;;; image-scaled-render-spot! ;;; Parameters: ;;; image, an image ;;; spot, a spot ;;; factor, a number ;;; Purpose: ;; Draw the spot on the image, scaled by a factor of factor. ;;; Produces: ;;; [Nothing; Called for the side effect] ;;; Preconditions: ;;; factor >= 1 ;;; The position of the scaled spot is within the bounds of the image. ;;; Postconditions: ;;; The image now contains a rendering of the spot. (define image-scaled-render-spot! (lambda (image spot factor) (context-set-fgcolor! (spot-color spot)) (image-select-ellipse! image selection-replace (* factor (spot-col spot)) (* factor (spot-row spot)) factor factor) (image-fill! image) (image-select-nothing! image)))
How might you change the approach? As written, the circles start at the scaled position. You might want them centered at that position. You might also find that the circles work and look better if they overlap a bit, which means that we might use more than factor for width and height.
Now, let's steal an idea from the drawings-as-values model and think about the ways in which we might transform a single spot. It doesn't make a lot of sense to scale a spot (we'll just use the enlarged render procedure mentioned above). It does, however, make sense to shift a spot horizontally or vertically or to recolor it.
Let's start with the simplest shift to a spot, which we'll call “nudging” the spot. To nudge a spot right, we add one to the column. To nudge a spot left, we subtract one. To nudge a spot up, we subtract one from the row. And, to nudge a spot down, we add one to the row. Let's look at two examples.
(define spot-nudge-right (lambda (spot) (spot-new (+ (spot-col spot) 1) (spot-row spot) (spot-color spot)))) (define spot-nudge-up (lambda (spot) (spot-new (spot-col spot) (- (spot-row spot) 1) (spot-color spot))))
Copyright (c) 2007-8 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.
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