An Empirical Study of Continuous Connectivity Degree Sequence - - PowerPoint PPT Presentation

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An Empirical Study of Continuous Connectivity Degree Sequence - - PowerPoint PPT Presentation

Mar 28, 2024 203 likes •725 views

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SLIDE 1

An Empirical Study of Continuous Connectivity Degree Sequence Equivalents

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SLIDE 2
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SLIDE 3

Unfortunate Timing

Spoiler Alert

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SLIDE 4

Standard Connectomics

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SLIDE 5

Standard Connectomics

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SLIDE 6

Standard Connectomics

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SLIDE 7

Standard Connectomics

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SLIDE 8

Standard Connectomics

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SLIDE 9

Standard Connectomics

= ⎡ ⎢ ⎣

  • · · ·
  • · · ·
  • ⎤

⎥ ⎦

  • = |{(, ) : ∈ , ∈ }|
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SLIDE 10

Standard Connectomics

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SLIDE 11
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SLIDE 12

Some Known Problems

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SLIDE 13

Some Known Problems

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SLIDE 14

Some Known Problems

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SLIDE 15

Choice of Parcellation Problems + + + + + + + +

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SLIDE 16

Choice of Parcellation Problems + + + + + + + +

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SLIDE 17

So what can we do?

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SLIDE 18

Continuum Graphs

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SLIDE 19

Continuum Graphs

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SLIDE 20

Continuum Graphs

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SLIDE 21

Continuum Graphs

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SLIDE 22

Continuum Graphs

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SLIDE 23

Point Processes [] λ : → R+ [] ⊂ ( ) = (

  • λ())
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SLIDE 24

Point Process Connectomics = Ω × Ω Ω ∼ ∪

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SLIDE 25

Point Process Connectomics = Ω × Ω Ω ∼ ∪

  • λ

Ω × Ω

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SLIDE 26

Continuous Connectivity λ : Ω × Ω → R+ Choice:

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SLIDE 27

Continuous Connectivity λ : Ω × Ω → R+ Choice: λ

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SLIDE 28

Procedure

  • ∼

(, ) κ = (, )(, ) × ˆ λ

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SLIDE 29

Recovery of the Intensity Function

(, , σ) =

  • =

+ π {−σ( + )}

( · )

σ(, ) =

  • (, , σ)

,

  • κσ((, ), (, )) = σ(, )σ(, ) Ω × Ω
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SLIDE 30

Pretty Pictures

() =

  • λ(, )
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SLIDE 31

Degree Equivalents

Marginal connectivity ⊂ Ω () =

  • λ(, )

λ(, ) ()

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SLIDE 32

Degree Equivalents

λ : Ω × Ω → R+ () = Ω() =

  • Ω

λ(, ) = {}

  • : × → R+

() =

  • ∈

(, )

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SLIDE 33

Choosing σ

κ((, )|) =

  • (,)∈

σ(, )σ(, ) =

  • +

π + π

  • {−σ( + + + )}
  • ,

×

  • (,)∈
  • ( · )

( · )

  • σ, once
  • (||) σ

( + ||)

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SLIDE 34

Degree Equivalents

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SLIDE 35

Degree Equivalents

Consistency and Asymptotic Normality λ ˆ λ ˆ λ (λ, ˆ λ) = |λ − ˆ λ| consistent

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SLIDE 36

DTI vs CSD

  • ∼
  • ˆ

λ ˆ λ ˆ λ ˆ λ −

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SLIDE 37

Degree Equivalents

σ = . σ = . σ = . σ = .

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SLIDE 38

Other Applications

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SLIDE 39

Continuum to Discrete λ : Ω × Ω → R+

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SLIDE 40

Continuum to Discrete λ : Ω × Ω → R+ G : {}

=×{} = → R+

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SLIDE 41

Continuum to Discrete λ : Ω × Ω → R+ G : {}

=×{} = → R+

G(, ) =

  • ×

λ(, ) = E[ ]

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SLIDE 42

Comparison of Parcellation Sets

ℓ() =

  • (,)∈
  • ×
  • − λ(, )

ℓ() =

  • (,)∈

L() () = ℓ − = ℓ − ||

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SLIDE 43

Parcellation Choice

  • ℓ()

. × − 1.0257 × 10−5 . × − ℓ() 85256.0

  • ()

175068.1

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SLIDE 44

Some Known Problems

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SLIDE 45

Some Known Problems

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SLIDE 46

Some Known Problems

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SLIDE 47

References

80 34 249 59 50

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SLIDE 48

Thank you!

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SLIDE 49

Questions?

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SLIDE 50

Thank You!

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SLIDE 51

Questions?