CAP representations (The right(?) way for generic MR analysis of - - PowerPoint PPT Presentation

CAP representations

(The right(?) way for generic MR analysis

• f Internet data )

Amos Ron University of Wisconsin∗ – Madison WISP: UCSD, November 2004

∗: Breaking news, 06/11/04: Wisconsin routed Minnesota 38:14,

• n its way to the national title.

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Outline

• Possible goals behind “generic analysis on Internet signals”
• Why is that a non-trivial task?
• Predictability and pyramidal algorithms
• Performance of pyramidal representation
• CAMP and my favorite pyramidal representation
• What parameters to extract from the representation?

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A mathematical view of Internet signals

• Main features in the signal:

– burst types – rate of their appearances

• This is non-trivial

(why? After all, nothing is easier than 1D signals...) – the amount of data may be overwhelming – there is no clear way to judge success

• It is also a cultural problem. really?

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maybe it is time to show some images?

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d4 d3 d2 d1

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Pyramidal algorithms I: MR representation

h : Z Z → I R is a symmetric, normalized, filter: h(k) = h(−k),

k∈Z Z h(k) = 1. ↓, ↑ are downsampling & upsampling:

y↓(k) = y(2k), k ∈ Z Z y↑(k) = 2y(k/2), k even, 0,

• therwise.

(yj)∞

j=−∞ ⊂ CZ Z s.t:

yj = Cyj+1 := (h ∗ yj)↓, ∀j. C is Compression or Coarsification yj+1 is then predicted from yj by yj+1 ≈ Pyj := h ∗ (yj↑). P is Prediction or subdivision

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Pyramidal algorithms II: the detail coefficients

• Define the detail coefficients:

dj := (I − PC) yj = yj − P yj−1.

• Replace yj by the pair (yj−1, dj).
• Continue iteratively.

ym ✲ C ym−1 ✲ C ym−2 . . . . . . y1 ✲ C y0 ❄ I − PC dm ❄ I − PC dm−1 ❄ I − PC dm−2 ❄ I − PC d1

• Reconstruction. Recovering ym from y0, d1, d2, . . . , dm is trivial:

y1 = d1 +Py0, y2 = d2 +Py1 and so on.

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Wavelet pyramids, Mallat, 1987

Decompose the detail map I − PC: I − PC = RD D : y → (h1 ∗ y)↓ =: w1,j−1, R : y → h1 ∗ y↑ with h1 a real, symmetric, highpass:

k∈Z Z h1(k) = 0.

ym ✲ C ym−1 ✲ C ym−2 . . . . . . y1 ✲ C y0 ❩❩❩❩❩ ⑦ D ❩❩❩❩❩ ⑦ D w1,m−1 w1,m−2 ❩❩❩❩❩ ⑦ D w1,0 Note that we can recover ym from y0, w1,0, w1,1, . . . , w1,m−1 since y1 = Rw1,0 +Py0, y2 = Rw1,1 +Py1 and so on.

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Performance

• Ability to predict. The best prediction are based on local

averaging, and on nothing else =spline predictors

• Time blurring: good prediction requires long averaging filter.

That blurs spontaneous events.

• Internet data exhibit different behaviour at “small” scales than
• ther scales. Hence: non-stationary representation
• Standard wavelet systems are mediocre for Internet data: they

blur time, and create artifacts, in order to gain unnecessary properties (orthonormality).

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Poor prediction

5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40

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My favorite representation

well, before we conducted any numerical tests Step I: Build an MR representation based on h1 = 1 4(1 2 1) Step II: Define the detail coefficients by: dj(k) =     

−yj(k+1)+2yj(k)−yj(k−1) 4

, k even,

yj(k−3)−9yj(k−1)+16yj(k)−9yj(k+1)−yj(k+3) 16

, k odd. The “performance grade” here is 2 in the strict sense. (To compare, Haar’s grade is 1 in the non-strict sense, and 0 in the strict sense.) This is an example of a new class of high-performance representations called CAMP

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what to analyse? what to extract?

for p ≥ 1, the p-norm is ap = (

• k

|ak|p)1/p the best thing to analyse is the “compressibility” of the detail coefficients: choose a number N, then (1) replace the N “most important” detail coefficients by 0, to

• btain a signal eN.

(2) reconstruct using eN to obtain YN. ‘ (3) define ep(N) := YNp. (4) find the a parameter α such that ep(N) ≈ CN −α.

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α(p) = the predictability parameter in the p-norm

“most important”=? (1) Non-linear: choose the largest ones (2) Linear: go from coarse scale to fine scale. Output: this way we have two functions p → α(p). Goal: learn how to judge properties of your signal based on these two functions it might be that the detail coefficients behave rather differently at different scale (small scale vs. large scale).

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CAP representations

Choose:

• two refinable functions φc, φr with refinement filters hc, hr.
• A third (Auxiliary-Alignment) lowpass filter ha.

Decompose: Fix f : I R → C. For all k, j ∈ Z Z, define yj(k) := 2j/2 f, (φc)j,k. The CAP operators are: C : y → (hc ∗ y)↓, (Coarsification-Compression), A : y → Ay := ha ∗ y, (Alignment), P : y → Py := hr ∗ (y↑), (Prediction-subdivision). Then Cyj+1 = yj, ∀j.

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The detail coefficients are: dj:= (A − PAC)yj = Ayj − PAyj−1. This is the CAP representation with (dj) the CAP coefficients. ym ✲ C ym−1 ✲ C ym−2 . . . . . . y1 ✲ C y0 ❄ A − PAC dm ❄ A − PAC dm−1 ❄ A − PAC dm−2 ❄ A − PAC d1 ym is recovered from y0, d1, d2, . . . , dm since Ay1 = d1+PAy0, Ay2 = d2+PAy1, . . . , Aym = dm+PAym−1 and deconvolving A from Aym.

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Summary Do they W F CAP implemented by fast pyramid algorithms ?

• provides good function space characterizations ?
• avoid mother wavelets ?
• very short filters, with no artifacts ?
• have simple constructions ?
• avoid redundant representations ?
• Wavelet are non-redundant. Caplets are only slightly redundant in

high dimensions. Their redundancy is non-essential.

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CAMP representations: Compression-Alignment-Modified Prediction

With CAP in hand, one can modify the process s.t.:

• The filters are shorter
• The performance (:= function space characterization) is the

same Example: Assume h is interpolatory. Define the details as: dj := yj − h ∗ yj,

• n 2Z

Zd, yj − h ∗ (yj↓↑),

• therwise.

Let φ be the refinable function of h. If φ ∈ Cs+ǫ

c

, then the above detail characterize Ls

p. 19

Example (2D): h =     1/8 1/8 1/8 1/4 1/8 1/8 1/8     . There are four (hidden) filters, for computing dj:   −1/8 −1/8 −1/8 +3/4 −1/8 −1/8 −1/8   ,   −1/2 +1 −1/2     −1/2 +1 −1/2   ,   −1/2 +1 −1/2   Those are 7, 3, 3, 3-tap. There are four (hidden) “CAMPlets”, whose average area of support is about 2. The performance is on par with tensor 3/5, whose filters are 25, 15, 15-tap. Each supported in 3 × 3 square.

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yj

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dV

j

dD

j

d*

j

dH

j

Figure 1: First level ˜ d CAMP coefficients, organized by cosets.

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Wisconsin

From left, 1st row: Julia Velikina, Youngmi Hur, Yeon Kim, Narfi Stefansson. 2nd row: Thomas Hangelbroek, Sangnam Nam, Jeff Kline, Steven Parker.

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Julia Velikina: undersampled MRI data

Schepp−Logan phantom Conventional recon. from 90 projections, acceptable quality Conventional recon. from 23 projections, unacceptable quality TV−based recon. from 23 projections

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Jeff Kline: new data representation in NMR

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Steven Parker: redundant representation of acoustic signals

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 coefs scaled to V0 8,0 8,1 7,1 8,4 8,5 8,6 8,7 8,8 8,9 8,10 8,11 7,6 7,7 7,8 7,9 7,10 7,11 6,6 8,28 8,29 8,30 8,31 6,8 6,9 7,20 7,21 6,11 5,6 5,7 2,1 1,1 Adpative framelet−based representation of a vibraphone recording 131072

• riginal signal (time)

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Narfi Stefansson: sparse framelet representations

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6/10 61440 coefficients cubic spline 34608 coefficients quartic spline 34452 coefficients box15,box17,box18 35085 coefficients

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FrameNet: on-line interactive framelet and wavelet analysis

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