Notes 20110217 CIS 6320 Image Processing
From SnOwy - Ed's Wiki Notebook
Arrived about half an hour late.
Contents |
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 (vi)i∈1..n ∈ Vn
- (vi)i∈1..n is linearly dependent iff ∃ (ai)i∈1..n ∈(Rn)* Σiai.vi = 0 ( where X* excludes the zero element (or the zero vector) )
- (vi)i∈1..n is linearly independent iff ∀(ai)i∈1..n ∈Rn, Σiai.vi = 0 → ∀i∈1..n, ai = 0
- (vi)i∈1..n generates V iff: ∀v∈V, ∃(ai)i∈1..n∈Rn, v = Σiaivi
- (vi)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,αk0,0+βk0,1+γk1,0+δk1,1=0→α=β=γ=δ=0
- ∀w∈W,∃(α,β,γ,δ)∈R4,w=αk0,0+βk0,1+γk1,0+δk1,1
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