e-book Fractals: A Programmers Approach

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Recursion is a critical programming concept, but it can be pretty abstract. But Art showed students fractals and prompted students to create their own. With a touch of Algebra and looking at a few lines of sample code, Mikaela jumped in to make her own Sierpinski Triangle. Whenever I think of recursion, now I think of nature and tree rings, Mr. Simon showed us this picture of a leaf that is a leaf inside of a leaf inside of a leaf type thing. Art encourages his students to look at similar projects on GitHub for inspiration and to ask other students for input.

He structures the assignments in terms of milestones, layering in a different element or feature of the project at each turn. We get the ship working first. We get the asteroids on the screen. We get the ability to remove asteroids and look for collisions. Then we add bullets, and you got a working game. People are going to create objects that explode differently and encounter collision detection differently. Mikaela has run out of computer science courses to take at Lowell, but she participates in an after-school program called Mission Bit and plans to continue with an engineering internship this summer.

And Mr. I have to compliment Mr. There is a whole lot of action at a distance.

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The language is full of global and implicit state. Who the fuck knows. In, say, Python, the equivalent. In PHP , these functions return false. This is bad! Here, PHP has actively created a subtle trap for me to fall into, and I have to be vigilant even with such mundane things as string operations and equality comparison. PHP is a minefield.

I have heard a great many stories about the PHP interpreter and its developers from a great many places. So I have to fit this in here, because it bears repeating: PHP is a community of amateurs. Oh, dear reader, you are of course a rare exception! Those who do grow a clue tend to drift away to other platforms, reducing the average competence of the whole. Unlike literally!

Fractal Programming Tutorial Overview

If you try, PHP will… er, throw an exception. See above about int. PHP is a high-level, dynamically-typed programming language. There is none. How do you sort backwards? In Python, you might do. Programs are nothing more than big machines that chew up data and spit out more data.


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A great many languages are designed around the kinds of data they manipulate, from awk to Prolog to C. Array functions often have confusing or inconsistent behavior because they have to operate on lists, hashes, or maybe a combination of the two.

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What will this code do? I leave it to the reader to figure out what happens if the arrays are mixed. Every time a page is hit, PHP recompiles the whole thing before executing it. Zend, the company behind PHP , has made this part of their business model. For quite a long time, PHP errors went to the client by default—I guess to help during development. However, since the conception of automatic fractal image compression [11], researches on improvement of the compression time are widespread [7].

The points considered in this comparison are image fidelity, encoding time and amount of compression on the image file. To modeling this problem let E be the space of gray scale images. Then, we choose a space of functions f : X G , where the set X is taken as the set of spatial coordinates of the image and G represents the set of intensity values of this image. Also, let d be a metric, such that E , d is a complete metric space.

A detailed description of basic aspect of this theory and implementations of automatic approaches can be found in [5]. The decoding process consists of iterating the mapping T from any initial image until the iterates converge to the atractor - an approximation of the original image [1]. PIFS basic idea is simplify the problem of compress the whole image by partitioning the image into blocks and then seek self-similarity between larger parts and smaller parts.

The nature of the PIFS is illustrades in figure 1. The encoding process consists of the construction of the operator T , which is a system of affine transformation, each one defined by matching the better couple of sets, formed from the original image [2]. Several researchers have taken up the challenge to improve the basic automated algorithm for compression [7, 8, 15]. Some of the main subjects addressed are compression ratio, time reduction and image fidelity [3, 12].

We show, in the folowing that, the time of compression can be improved without lost of the image fidelity considering the contractive factor and the local image complexity. Depending on the block complexity, different contractive factors have been used to form two groups on the image partitioned for compression [11].

Fractal dimensions FD have been used to obtain shape information and distinguish between smooth small FD and sharp large FD regions [6, 13, 14]. In this work we propose it to characterize local complexity in images. In other words, the partitioned image is preprocessed and each local FD is computed to generate separate pools depending on local image complexity. To derive the contractive transformation only the relation between elements on pools of same FD are considered improving compression time and quality of reconstruction.

Daniel Shiffman

Let t i : Dj Ri be a contraction defined by matching the better t i Dj for each Ri , which means that t i is chosen such that the distance d Ri , t i Dj is as small as possible figure 2. So by the Collage theorem [1, 2]:.

Lecture - 32 Fractals (Contd.)

This is the main idea of our work: the utilization of variable contractive factor, based on image complexity values. Let D be a collection of subsets of from which D j are chosen. The number of elements on D determines how much computation time the encoding takes on finding t i D j for each R i. For encoding, the mapping t i must be specified and the better domain squares D j must be chosen from a set of potential candidates. We use local fractal dimension represented by LFD to subdivide the blocks into classes of complexity see figure 5.

Some methods on LFD estimation do not give satisfactory results in all possible range for images from 2 to 3.

File Extensions and File Formats

For fast LFD evaluation a simple and efficient algorithm has been used [6]. On finding each t i , the domain sets D j must be chosen from a set of potential candidates, i. Although, real images present semi-fractal behavior, they are considered in this analysis ideal fractal models. Of course it must be considered when one is addressing to the problem of finding the real fractal dimension of the image, but here it is only a way for representing its local complexity [13]. Table 1 presents the results of this proposition on compression and reconstruction of 13 images.

The efficiency of this image compression approach is quantified by considering both image fidelity and time reduction.

Fractal Explorer - Fractal Programming Basics

In this table, for no hardware dependece and installation interference on the results, the parameter related to the time spending on encoding is relative to the average results obtained on the same 13 images using the classical approach [2] see table 2. Image fidelity is evaluated using 3 measures: the root mean square error, e rms , the signal to noise ratio, SNR and the peak signal to noise ratio, PSNR of the each image.

These are defined as:. For comparison the same set is compressed by two other methods: the Heavy Brute Force Algorithm or Exhaustive Approach [2, 11] table 2 and figure 7 and Restricted Area Search [12] table 3 and figure 8. On third column, for better verification the absolute value of these subtractions are amplify by a scale factor of 5.

On fourth column, an offset of the middle of the gray level is used to turn possible visualization of negative or positive values. Considering now the compression ratio. They need bits or bytes and 8 bpp bits per pixel before compression.