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Project Ideas: Color Trees, Revisited
In the reading on trees, we encountered an interesting way to use trees: We can represent an image with a color tree, a tree whose leaves are all colors. We can then render the tree systematically. In that first rendering algorithm, when we encountered a pair, we split the image (or subimage) horizontally and then rendered the left half of the tree in the left half of the image, and the right half of the tree in the right half of the image.
It can be more interesting (and more valuable) to render a color tree by alternately decomposing the image horizontally and vertically. The following alternate definition does just that. (You may find it valuable to study it for a moment.)
;;; Procedures:
;;; image-render-color-tree!
;;; Parameters:
;;; image, an image
;;; ctree, a tree of colors
;;; left, a real number
;;; top, a real number
;;; width, a real number
;;; height, a real number
;;; Purpose:
;;; Render the tree into the portion of the image bounded at
;;; the left by left, at the top by top, and with the specified
;;; width and height.
;;; Produces:
;;; [Nothing; Called for the side effect.]
;;; Preconditions:
;;; image is a valid image
;;; ctree is a valid color tree
;;; 0 <= left < (image-width image)
;;; 0 <= top < (image-height image)
;;; width < 0
;;; height < 0
;;; 0 <= (+ left width) < (image-width image)
;;; 0 <= (+ top height) < (image-height image)
;;; Postconditions:
;;; The tree has now been rendered.
(define image-render-color-tree!
(lambda (image ctree left top width height)
(let kernel ([ctree ctree]
[hsplit? #t]
[left (exact->inexact left)]
[top (exact->inexact top)]
[width (exact->inexact width)]
[height (exact->inexact height)])
(cond
; If it's too small, just stop.
[(or (< width 1) (< height 1))]
; If it's a pair, and we're to split horizontally, do so
[(and (pair? ctree) hsplit?)
(kernel (car ctree) (not hsplit?)
left top
(/ width 2) height)
(kernel (cdr ctree) (not hsplit?)
(+ left (/ width 2)) top
(/ width 2) height)]
; If it's a pair, and we're to split vertically, do so
[(pair? ctree) ; NOT hsplit?
(kernel (car ctree) (not hsplit?)
left top
width (/ height 2))
(kernel (cdr ctree) (not hsplit?)
left (+ top (/ height 2))
width (/ height 2))]
; Otherwise, it's just a single color, so render in the
; given space.
[else
(image-select-rectangle! image REPLACE
left top (round width) (round height))
(context-set-fgcolor! ctree)
(image-fill-selection! image)
(context-update-displays!)
(image-select-nothing! image)]))))
How can color trees be useful for the project? Well, if you’re willing to put up with fairly “unpredictable” images, you can write a procedure that systematically builds a color tree from an integer. But how do we build an interesting color tree?
One strategy is to look at some characteristic of the number (e.g., the remainder after dividing by ten) and use that number to decide how one might split the number (e.g., if the remainder is 1 or 2, we build a color tree whose left subtree is built from some fraction of the number and whose right subtree is built from some other fraction of the number. If the remainder is 3 or 4, we build a color tree whose left subtree is a different fraction, and so on and so forth.
Here’s an implementation of that technique.
;;; Procedure:
;;; number->color-tree
;;; Parameters:
;;; n, an exact integer
;;; Purpose:
;;; Create an "interesting" color tree whose content depends only
;;; on n. Ideally, different values of n would give different
;;; color trees.
;;; Produces:
;;; ctree, a color tree
;;; Preconditions:
;;; n >= 0
;;; Postconditions:
;;; If x ≠ y, then (number->color-tree x) is unlikely to be the same as
;;; (number->color-tree y)
(define number->color-tree
; These are the colors used to build the tree
(let* ([colors (vector (irgb 255 0 0) (irgb 204 0 0)
(irgb 153 0 0) (irgb 255 0 102)
(irgb 204 0 102) (irgb 153 0 102)
(irgb 255 0 204) (irgb 204 0 204)
(irgb 153 0 204))]
[num-colors (vector-length colors)])
(lambda (n)
(let ([action (remainder n 10)])
(cond
; For small numbers, we simply grab them from the color tree.
[(< n num-colors)
(vector-ref colors n)]
[(< action 2)
(cons (number->color-tree (inexact->exact (round (* 0.4 n))))
(number->color-tree (inexact->exact (round (* 0.6 n)))))]
[(< action 4)
(cons (number->color-tree (inexact->exact (round (* 0.6 n))))
(number->color-tree (inexact->exact (round (* 0.4 n)))))]
[(< action 6)
(cons (number->color-tree (inexact->exact (round (* 0.25 n))))
(number->color-tree (inexact->exact (round (* 0.75 n)))))]
[(< action 8)
(cons (number->color-tree (inexact->exact (round (* 0.75 n))))
(number->color-tree (inexact->exact (round (* 0.25 n)))))]
[(< action 9)
(cons (vector-ref colors (remainder n num-colors))
(number->color-tree (inexact->exact (round (* 0.5 n)))))]
[else
(cons (number->color-tree (inexact->exact (round (* 0.5 n))))
(vector-ref colors (remainder n num-colors)))])))))
Here’s one way to incorporate that procedure into a series of images.
(define series-1
(lambda (n width height)
(let* ([ctree (number->color-tree n)]
[canvas (image-new width height)])
(image-show canvas)
(image-render-color-tree! canvas ctree 0 0 width height)
canvas)))
You’ll have a chance to explore these ideas a bit more in the lab.