Article about Fractal Image Compression

[David Makin]

I tend to think of the Universe/Existence as a limitation of an infinite mathematical construct.
Most seem to consider things the other way round 

Also copies can contain copies can contain copies - at least mathematically speaking - just zoom into the Harter-Heighway Dragon for example.

[David Makin]

Also - on the question of a set containing everything.

Would it be true to say "everything" can only truly be defined as two sets: a set of everything except nothing and an empty set ?
(I confess my formal education on such ended with Venn diagrams)

[rloldershaw]

Thanks Lycium for your thoughtful and interesting response to my comments on infinite sets.

Someday someone is going to explain exactly why the M-set generates the recurive copies-within-copies structure wherein the copies are self-similar versions of the whole M-set, rather than little pictures of Richard M. Nixon or a discrete hierarchy of completely different shapes. The mystery of self-similarity is a very fundamental one, I think, and its explanation will involve a big step forward in human understanding of mathematics and nature.

Anybody got any ideas?

Here's one off-the-wall idea. If the M-set were reformulated in a version wherein the hierarchy were not capped at the top, i.e., the M-set continues without bound in the "larger scale direction", as it does in the "smaller scale direction", then maybe the discrete self-similarity would be exact at all levels.  So the fact that the copies are are all slightly deformed, would be a result of artificially capping off the hierarchy. Unfortunately I do not have the math skills to check this out.

Rob

[lycium]

btw,

Infinite magnification is impossible.

am i the only calculus student that feels downright violated by that remark?

[bh]

Someday someone is going to explain exactly why the M-set generates the recurive copies-within-copies structure wherein the copies are self-similar versions of the whole M-set

Unfortunately I don't know much serious maths about the Mandelbrot set, but I think your problem is something really well understood. Here is what I can scratch as a beginning of explanation.

Take the largest minibrot at the left of the main part. You will find its center is at about c = -1.7548. In fact, this value is one of the four roots of the polynomial (c²+c)²+c. The roots are the values of c for which 0 makes a cycle of order 3 (question: where are the other three roots? It's not too difficult).

We call c₀ this root and set c = c₀ + d. Since 0 makes a cycle of order 3, we consider in place of f(z) = z²+c the function:

g(z) = f(f(f(z)))
       = ((z²+c₀+d)²+c₀+d)²+c₀+d

It is some trouble to expand, but only the general form is important. We can assume that d is small. We can also assume that z is small, because to draw the mandelbrot set we want to iterate the value 0, and for d=0 we have g(0)=0. We also make use of the fact that (c₀²+c₀)²+c₀ = 0. Keeping only the smallest powers of z and d (because the others are negligible) we get something like:

g(z) = a z² + d

for a constant a, which makes a nearly standard mandelbrot set.

A complete proof would certainly need to control exactly how those d and z keep being small. But it's a math theorem, not a philosophy theorem.

Obvious generalizations for higher order cycles: the eight roots of ((c²+c)²+c)²+c ...

Can someone point to a reference where this would be explained in length?

[bh]

Infinite magnification is impossible.

am i the only calculus student that feels downright violated by that remark?

Charles took my virginity some time ago, and this one doesn't really bother me.

May I beg everyone to clearly distinguish several ways that a set can "contain" another?

• Set B can be an element of set A (like the set {1,{3,4},{2,7,8},{0,{0}}} "contains" the set {2,7,8})
• Set B can be a subset of set A (like the set {1,{3,4},{2,7,8},{0,{0}}} "contains" the set {1,{2,7,8}}, and in this sense it is worth noting that every set contains itself)
• The image f(B) of set B by an injective function can be a subset of set A (like the Sierpinsky triangle "contains" multiple copies of itself via similarity transformations, or the set of even integers "contains" the set of integers via f(n)=2n)
  

and please only try to invoke Russell's paradox in the first case?

[David Makin]

Infinite magnification is impossible.

am i the only calculus student that feels downright violated by that remark?

Charles took my virginity some time ago, and this one doesn't really bother me.

May I beg everyone to clearly distinguish several ways that a set can "contain" another?

• Set B can be an element of set A (like the set {1,{3,4},{2,7,8},{0,{0}}} "contains" the set {2,7,8})
• Set B can be a subset of set A (like the set {1,{3,4},{2,7,8},{0,{0}}} "contains" the set {1,{2,7,8}}, and in this sense it is worth noting that every set contains itself)
• The image f(B) of set B by an injective function can be a subset of set A (like the Sierpinsky triangle "contains" multiple copies of itself via similarity transformations, or the set of even integers "contains" the set of integers via f(n)=2n)
  

and please only try to invoke Russell's paradox in the first case?

Many thanks for that - it certainly clarifies the situation for me (having next to no education on set theory).

[Charleswehner]

May I beg everyone to clearly distinguish several ways that a set can "contain" another?
..........
and please only try to invoke Russell's paradox in the first case?

Firstly, there is a very superficial individual who continues to fight a flame war against me. His commentaries are ill-considered, and uninformative. He is peeved because I would not accept his lectures. He seems determined to score "points" off me.

So when you are guided by someone who has an ulterior motive - other than truth itself - you get led astray.

In this case, for example, bh has been misled (1) into thinking I am as flippant as my detractor, and (2) into believing that set theory is small-talk.

Think again.

From first principles, Russell's Paradox considers only a set containing itself.

The first case cited by bh was of a set containing another set - which has nothing to do with the paradox.

I had hoped that there would be friendly discussion here, amongst people interested in Fractals. Personal attacks and ego-trips have no place. I was obliged to say before that life is not war. I repeat it. People like bh are being distracted by side-issues caused by the rancour of another.

Charles

[heneganj]

Can we get back on topic please?

[jehovajah]

The symbolic logic that is at the heart of this set inclusion difficulty is what most people including myself are not familiar with. This is simply because i am familiar with a more natural collection of relations which the symbolic logic of the past never set out to model. Nevertheless whatit has produced is a model of certain systematic relations among defined and ill defined entities. Russel in setting out his paradox or rule set it out in many complementary systems, but the most telling is in the formal definition of a logical statement: the predicate. Again the predicate is ill defined and custom and usage among philosophers of logic linguistics and symbolic logic fail to appreciate the iterative nature of some of the concepts that seem so hard and fixed. After iterating through many written expressions of a statement the defined parts resolve into entities not on paper or in the statement structure but in our perception faculty. The referent for predicate is a set of perceptions within me which are triggered by certain symbolic patterns or more directly certain language patterns which i distinguish auditorily. Once i can distinguish a predicate auditorily or visually manipulations of this entity do not necessarily maintain its value as a predicate within a statement structure that is more complex than a simple statement. Thus to hold that the recognised entity will have that value throughout a complex argument may be true for an algorithm but is not consistent with any known language system used for communication between individuals. The iteration process of understanding produces widely differing referents and iterations between various entitic states. These i will call paradoxes, but they are only expected outcomes from an iteration that has no convergence to a single value. They are not monsters and may not be very useful in designing axiom systems to exclude them. in fact they are showing the lie of the land we exist in and should be included in our models of experience not excluded. We can then  classify them by how many states they iterate between. Unlike Russel who had a classical view of Nature, I view the experiential continuum within which i exist as having infinite possibilities of iterative outcomes: That is to say everything is a fractal, that is a product of an iteration process.

So to the point of fractal compression: the fact that a partition scheme is the de facto solution is in line with the other properties of iteration especially boundarisation and regionalisation. To assume that any random shape can be made up from a self similar shape arranged at smaller scales does not allow for structures within the regions that themselves are bounded by sub iterations in the overall iteration schema.

[Collin237]

But as I see it, the purpose of fractal image compression is neither to theorize on the origin of the original shape, nor to predict the further detail that would be found by magnifying the original shape at any particular location. The purpose is to provide an un-intelligent, and thus un-biased, interpretation of the known, admittedly incomplete, data, in a way that our perception, which is optimized for viewing the high-resolution fractals of nature, can better understand it.

As such, it is useful only for applications in which an abstract representation is at least as good as the actual further details.

Of course, if you're interested only in pretty pictures, that's almost always the case.

[jehovajah]

I had hoped that Charles, who has had health problems would return to the forum.
However, the compression of image or data as charles points out has some pretty fundamenal definition issues. Within a computer paradigm a set can be nested within itself, and the issue is related to scale. Scale is related to what i might call sequence load, which is to say that a certain set of sequences have to be carried out at each nest level befor thenext level can be affected. This translates into time and space constraints in an objective space, ie the real world.

The simplest and natural solution is to do things in lower levels smaller and quicker. This is a natural programming ,structural consequence of any constraint on the action of running the sequence.

What this means is that copies of a the larger or higher level will play out at lower or smaller levels  with less and less influence on the higher level output depending on spatial availability.

Because we can now print in 3d ,the mandelbrot set looks different simply because more room is available to allow the substructure to be formed, and regions are not obscured , overlapped or over printed simply because there is no space on the printing media to represent the region.

The paradoxes, though interesting are always subjective. In "real life", things resolve one way or another. Russels paradox resolve into an "infinite" undecidability loop in he computing paradigm, and the solution is nested "levels" which of course translate in Gödel's terms to sets that enclose sets indefinitely, which is of course the structure of a fractal .

Barnsley's practical solution involves recognising and using these structurl boundaries and developing the coreect minimum sequence load at each level. Once that has been achieved for the given image, enlarging the image by scale does not effect the sharpness of the final iterated image, as the smaller sequence load at the lower levels has negligible to vanishing impact, but the spatial relationship of the enlarged pixel areas remain the exact same . Infilling in the larger canvas is then not by block colour but by iterated repetition of the structure for that region. We subjectively see then a variegated region that we would expect to see not an unexpected block.

Our process of perception undoubtedly uses many of the same techniques Barnesly has discovered to compute surface outputs for our own subjective experience. No one is saying that the extra detail is in fact what we would find if we looked through a microscope. What is being said is that the extra detail is consistent with expectations for that surface. Most of us would find a microscopic image of a surface totally contrary to expectations.

It is also True, that at the right scale many macro features are indistinguishable from microfeatures, thus expected surfaces are a food approximaion to reality, and a nested fractal paradigm provides that self similarity.

It seems likely to me that we may come to realise that we can not progress beyond a fractal self similarity in pursuing our quest for a deeper understanding of reality, to which i say "thank God!" Endless mindless questioning is very draining!

[jehovajah]

<a href="http://www.youtube.com/v/Lte3xpmH2_g&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/Lte3xpmH2_g&rel=1&fs=1&hd=1</a>

Hopefully an update.

[jehovajah]

Imagine for a moment a fractal compression scheme fast enough to compress the data in a keyframe, whilst also compressing changes to the keyframe during the cycle of compressing the key frame. Thus as soon as the compression is complete, or the function for searching for change becomes a search of the whole key frame a new keyframe is declared and compressed.

Thus video could be stored directly to a memory chip as a compressed data set, saving memory space. However the cost of frame rate must not be compromised. Presumably the advantage would be clearer images at post processing . However the multiplexing of sound is another technical issue..