Notes 20110217 CIS 6320 Image Processing

From SnOwy - Ed's Wiki Notebook

Arrived about half an hour late.

Contents 1 Start by defining a Set and Binary Operators on that Set 2 Vector Spaces 3 Specifics about the Functions 4 More on Vector Spaces 5 Read on these ideas 6 Rigourous, Precise 7 General

Start by defining a Set and Binary Operators on that Set

• let S be a nonempty set
• a binary operation on S is a total function from S² to S
• let * be a binary operation on S
• for any (s, t) in S²: *(s, t) is denoted by s * t
• * is commutative iff: ∀(s,t) ∈ s * t = t * s
• * is associative iff: ∀(s,t,u) ∈ S³, (s * t) * u = s * (t * u) -- NOT s * t * u
• let n be an element of S: n is a neutral element for * iff: ∀s∈s, s * n = n * s = s
• PROP: if there is a neutral element for *, then it is unique
• let s and t be two elements of S: Assume n is the neutral element for *, t is an inverse of s under * iff: s * t = t * s = n
• PROP: if there is an inverse of s under * then it is unique
  • this proposition is homework

Vector Spaces

• let V be a nonempty set.
• let + be a binary operation on v. (vector addition)
• let . be a total function from R×V to V (scalar multiplication)
• (V, +, .) is a real vector space iff:
• + is commutative
• + is associative
• there is a neutral element for + (denotred by 0, called the zero vector)
• there is an inverse under + for any element of V
• note: +(v, v) → v
• note: *(r, v) → v

Specifics about the Functions

• ∀(a, b)∈R², ∀v∈V, (a.b).v = a.(b.v)
• ∀(a, b)∈R², ∀v∈V, (a+b).v = (a.v)+(b.v)
• ∀a∈R, ∀(v,w)∈V², a.(v+w) = (a.v) + (a.w)
• ∀v∈V, 1.v=v
• note -- we may drop the brackets if we all agree that . precedes +
• an element of V is a vector
• an element of R is a scalar
• PROP: ...
  • ∀v∈V, 0.v = 0
  • ∀a∈R, a.0 = 0
  • ∀a∈R, ∀v&isinV, a.v = 0 → (a = 0 ∨ v = 0)
  • the inverse of any vector v is (-1).v (denoted by -v)
  • Homework: try to prove these four propositions

More on Vector Spaces

• let n be a positive integer and let (v_i)_i∈1..n ∈ Vⁿ
• (v_i)_i∈1..n is linearly dependent iff ∃ (a_i)_i∈1..n ∈(Rⁿ)^* Σ_ia_i.v_i = 0 ( where X^* excludes the zero element (or the zero vector) )
• (v_i)_i∈1..n is linearly independent iff ∀(a_i)_i∈1..n ∈Rⁿ, Σ_ia_i.v_i = 0 → ∀i∈1..n, a_i = 0
• (v_i)_i∈1..n generates V iff: ∀v∈V, ∃(a_i)_i∈1..n∈Rⁿ, v = Σ_ia_iv_i
• (v_i)_i∈1..n is a basis of V iff it generates V and is linearly independent
• this is the real vector space -- we can create a complex vector space by changing the field from R to C in all of the above assertions
• PROP: if we have a basis, there is only one way to specify a vector with respect to that basis -- this tuple is called the coordinates
• a basis is linearly independent ...
  • ∀(α,β,δ,γ)∈R⁴,αk_0,0+βk_0,1+γk_1,0+δk_1,1=0→α=β=γ=δ=0
  • ∀w∈W,∃(α,β,γ,δ)∈R⁴,w=αk_0,0+βk_0,1+γk_1,0+δk_1,1

Read on these ideas

• Hilbert space
• topological space

Rigourous, Precise

• we must challenge ourselves to be as mathematically precise as possible

General

• for this particular presentation ...
• don't be too technical
• must make it clear whether a particular transform is a good tool for a application
• need to do comparison and contrast against other techniques
• prefer figures to formulas
• example ...
• show and comment on a diagram is better than a bunch of formulas
• won't spend too much time on a slide
• at least half of the audience won't have enough time to understand half of the equations
• everyone in the audience should understand the principles illustrated in a diagram
• if we have an option between a function as math and a function as a plot, choose the plot
• mathematical formulas are not important in themselves
• principles, concepts useful
• show examples, results, illustrations first
• motivates audience -- we want to know more
• for instance, show the dithering demonstration before the dithering workflow
• (especially since we're unsure if everyone knows what dithering is)
• is the term dithering (and other technical terms) familiar to the audience?
• show the example first if it's not obvious
• control your presentation's length
• time control was poor
• best amount for [25, 30] is close to but not over 30 minutes
• check the watch on starting
• other comments
• don't continuously point at the slide
• must be more efficient or less confusing
• possibly add more slides, animations or use the white board
• don't apologize during the talk, don't self comment
• solid reference section
• numerous, recent papers
• honest answers are good
• talk two is a 50 minutes presentation with formulas -- the audience must understand everything
• should probably run more like a class