Iterated Function Systems (IFS)

What is

A\cdot x

A\cdot x

Iterated Function Systems (IFS)

What is

y = A\cdot x

y = A\cdot x

Iterated Function Systems (IFS)

What is

y = A\cdot x

y = A\cdot x

Think of "A" as a "Transformation" on x

Iterated Function Systems (IFS)

y = A\cdot x

y = A\cdot x

Think of "A" as a "Transformation" on x

\begin{bmatrix} 1 \\ 0 \end{bmatrix}

\begin{bmatrix} 1 \\ 0 \end{bmatrix}

\begin{bmatrix} 0.5 & 3 \\ 0.5 & 4 \end{bmatrix}

\begin{bmatrix} 0.5 & 3 \\ 0.5 & 4 \end{bmatrix}

A

A

Iterated Function Systems (IFS)

y = A\cdot x

y = A\cdot x

Think of "A" as a "Transformation" on x

\begin{bmatrix} 1 \\ 0 \end{bmatrix}

\begin{bmatrix} 1 \\ 0 \end{bmatrix}

\begin{bmatrix} 0.5 & 3 \\ 0.5 & 4 \end{bmatrix}

\begin{bmatrix} 0.5 & 3 \\ 0.5 & 4 \end{bmatrix}

A

A

\begin{bmatrix} 0.5 \\ 0.5 \end{bmatrix}

\begin{bmatrix} 0.5 \\ 0.5 \end{bmatrix}

Iterated Function Systems (IFS)

y = A\cdot x

y = A\cdot x

Think of "A" as a "Transformation" on x

\begin{bmatrix} 1 \\ 0 \end{bmatrix}

\begin{bmatrix} 1 \\ 0 \end{bmatrix}

\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}

\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}

A

A

\begin{bmatrix} 0 \\ 1 \end{bmatrix}

\begin{bmatrix} 0 \\ 1 \end{bmatrix}

\begin{bmatrix} 1 \\ 0 \end{bmatrix}

\begin{bmatrix} 1 \\ 0 \end{bmatrix}

Iterated Function Systems (IFS)

y = A\cdot x

y = A\cdot x

\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}

\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}

A

A

\begin{bmatrix} 1 \\ 0 \end{bmatrix}

\begin{bmatrix} 1 \\ 0 \end{bmatrix}

\begin{bmatrix} cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta) \end{bmatrix}

\begin{bmatrix} cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta) \end{bmatrix}

\theta=\frac{\pi}{2}

\theta=\frac{\pi}{2}

"Rotation Matrix"

Iterated Function Systems (IFS)

y = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \cdot x + \begin{bmatrix} e \\ f \end{bmatrix}

y = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \cdot x + \begin{bmatrix} e \\ f \end{bmatrix}

\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}

\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}

A

A

\begin{bmatrix} 1 \\ 0 \end{bmatrix}

\begin{bmatrix} 1 \\ 0 \end{bmatrix}

\begin{bmatrix} cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta) \end{bmatrix}

\begin{bmatrix} cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta) \end{bmatrix}

\theta=\frac{\pi}{2}

\theta=\frac{\pi}{2}

"Rotation Matrix"

To Define such a generic linear transform, We will need 6 parameters: a,b,c,d,e,f

Iterated Function Systems (IFS)

Now, We shall consider a "Set of Such Transforms"

	a	d	e	f
1	0.5	0.5
2	0.5	0.5		0.5
3	0.5	0.5	0.5	0.5

Iterated Function Systems (IFS)

CC	AC	CA	AA
GC	TC	GA	TA
CG	AG	CT	AT
GG	TG	GT	TT

Introduction to Chaos Game Representation (CGR)

Problem at Hand

Iterated Function Systems (IFS)

Iterated Function Systems (IFS)

Iterated Function Systems (IFS)

Iterated Function Systems (IFS)

Iterated Function Systems (IFS)

Iterated Function Systems (IFS)

Iterated Function Systems (IFS)

Iterated Function Systems (IFS)

Iterated Function Systems (IFS)

Iterated Function Systems (IFS)

Iterated Function Systems (IFS)

Iterated Function Systems (IFS)

Do This Repetedly

Other Transformations

Chaos Game Representation

Chaos Game Representation

Chaos Game Representation

Chaos Game Representation

Chaos Game Representation

Chaos Game Representation

Chaos Game Representation

Introduction to Chaos Game Representation

Introduction to Chaos Game Representation

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