TRAITEMENT des IMAGES MEDICALES · group element TAG VR length 128 bytes 44 49 43 4D data set data...

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TRAITEMENT des IMAGES MEDICALES MASTER PhyMed GMPH308 - Physique de l’imagerie médicale Fayçal Ben Bouallègue [email protected] http://scinti.edu.umontpellier.fr

Transcript of TRAITEMENT des IMAGES MEDICALES · group element TAG VR length 128 bytes 44 49 43 4D data set data...

Page 1: TRAITEMENT des IMAGES MEDICALES · group element TAG VR length 128 bytes 44 49 43 4D data set data set data set image EXPLICIT group element length IMPLICIT value VR value , = = ,

TRAITEMENT des

IMAGES MEDICALES

MASTER PhyMed

GMPH308 - Physique de l’imagerie médicale

Fayçal Ben Bouallè[email protected]://scinti.edu.umontpellier.fr

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𝑓 𝑥, 𝑦 : ℝ2 → Ω (ℝ,ℝ+, ℕ) 𝑓 𝑖, 𝑗 : … → …

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N

N

𝑓 𝑖, 𝑗 : 1: N × 1: N → …

𝑥, 𝑦 → 𝑖, 𝑗 ∶ 𝑥𝑖 = 𝑖 𝑑𝑦𝑖 = 𝑗 𝑑

𝒅 × 𝒅

𝒙

𝒚

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N

N

𝑓 𝑖, 𝑗 : 1: N × 1: N → Λ = {𝑞𝑚𝑖𝑛: 𝑒: 𝑞𝑚𝑎𝑥}

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N

N

𝑓 𝑖, 𝑗 : 1: N × 1: N → Λ = {𝑞𝑚𝑖𝑛: 𝑒: 𝑞𝑚𝑎𝑥}

3 octets

1 octet

½ octet

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N

N

𝑓 𝑖, 𝑗 : 1: N × 1: N → Λ = {𝑞𝑚𝑖𝑛: 𝑒: 𝑞𝑚𝑎𝑥}

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N

N

𝑓 𝑖, 𝑗 : 1: N × 1: N → Λ = {𝑞𝑚𝑖𝑛: 𝑒: 𝑞𝑚𝑎𝑥}

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N

N

𝑓 𝑖, 𝑗 : 1: N × 1: N → Λ = {𝑞𝑚𝑖𝑛: 𝑒: 𝑞𝑚𝑎𝑥}

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N

N

𝑓 𝑖, 𝑗 : 1: N × 1: N → Λ = {𝑞𝑚𝑖𝑛: 𝑒: 𝑞𝑚𝑎𝑥}

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N

N

𝑓 𝑖, 𝑗 : 1: N × 1: N → Λ = {𝑞𝑚𝑖𝑛: 𝑒: 𝑞𝑚𝑎𝑥}

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N

N

𝑓 𝑖, 𝑗 : 1: N × 1: N → Λ = {𝑞𝑚𝑖𝑛: 𝑒: 𝑞𝑚𝑎𝑥}

𝑓 𝑖, 𝑗 = 𝑓 𝑥𝑖 = 𝑑𝑖, 𝑦𝑗 = 𝑑𝑗Λ

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𝑒𝑓 = 1

𝒇

𝒇N

N

𝑓 𝑖, 𝑗 : 1: N × 1: N → Λ = {𝑞𝑚𝑖𝑛: 𝑒: 𝑞𝑚𝑎𝑥}

𝑓 𝑖, 𝑗 = 𝑓 𝑥𝑖 = 𝑑𝑖, 𝑦𝑗 = 𝑑𝑗Λ

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N

N

𝑓 𝑖, 𝑗 = 𝛼 𝑓 𝑥𝑖 = 𝑑𝑖, 𝑦𝑗 = 𝑑𝑗 + 𝛽Λ

𝑒𝑓 = 0,2

𝒇

𝒇

𝑓 𝑖, 𝑗 : 1: N × 1: N → Λ = {𝑞𝑚𝑖𝑛: 𝑒: 𝑞𝑚𝑎𝑥}

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■N

N

𝑞𝑚𝑖𝑛

𝑞𝑚𝑎𝑥

𝑓 𝑖, 𝑗 : 1: N × 1: N → Λ = {𝑞𝑚𝑖𝑛: 𝑒: 𝑞𝑚𝑎𝑥}

𝑓 𝑖, 𝑗 = 𝛼 𝑓 𝑥𝑖 = 𝑑𝑖, 𝑦𝑗 = 𝑑𝑗 + 𝛽Λ

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𝑞𝑚𝑖𝑛

𝑞𝑚𝑎𝑥

N

N

𝑓 𝑖, 𝑗 : 1: N × 1: N → Λ = {𝑞𝑚𝑖𝑛: 𝑒: 𝑞𝑚𝑎𝑥}

𝑓 𝑖, 𝑗 = 𝛼 𝑓 𝑥𝑖 = 𝑑𝑖, 𝑦𝑗 = 𝑑𝑗 + 𝛽Λ

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𝑞𝑚𝑎𝑥

𝑞𝑚𝑖𝑛

𝒒𝒎𝒊𝒏 𝒒𝒎𝒂𝒙

𝑓 𝑖, 𝑗 = 𝛼 𝑓 𝑥𝑖 = 𝑑𝑖, 𝑦𝑗 = 𝑑𝑗 + 𝛽Λ

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𝑞𝑚𝑎𝑥

𝑞𝑚𝑖𝑛

𝒒𝒎𝒊𝒏 𝒒𝒎𝒂𝒙

𝑓 𝑖, 𝑗 = 𝛼 𝑓 𝑥𝑖 = 𝑑𝑖, 𝑦𝑗 = 𝑑𝑗 + 𝛽Λ

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𝑞𝑚𝑎𝑥

𝑞𝑚𝑖𝑛

𝒒𝒎𝒊𝒏 𝒒𝒎𝒂𝒙

𝑓 𝑖, 𝑗 = 𝛼 𝑓 𝑥𝑖 = 𝑑𝑖, 𝑦𝑗 = 𝑑𝑗 + 𝛽Λ

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𝑞𝑚𝑎𝑥

𝑞𝑚𝑖𝑛

𝒒𝒎𝒊𝒏 𝒒𝒎𝒂𝒙

𝑓 𝑖, 𝑗 = 𝛼 𝑓 𝑥𝑖 = 𝑑𝑖, 𝑦𝑗 = 𝑑𝑗 + 𝛽Λ

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𝑓 𝑖, 𝑗 = 𝛼 𝑓 𝑥𝑖 = 𝑑𝑖, 𝑦𝑗 = 𝑑𝑗 + 𝛽Λ

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data set data set data set image44 49 43 4D128 bytes

𝑓 𝑖, 𝑗 = 𝛼 𝑓 𝑥𝑖 = 𝑑𝑖, 𝑦𝑗 = 𝑑𝑗 + 𝛽Λ

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group element

TAG

VR length value

data set data set data set image44 49 43 4D128 bytes

𝑓 𝑖, 𝑗 = 𝛼 𝑓 𝑥𝑖 = 𝑑𝑖, 𝑦𝑗 = 𝑑𝑗 + 𝛽Λ

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group element

TAG

VR length

data set data set data set image44 49 43 4D128 bytes

EXPLICIT

group element length IMPLICIT

value

valueVR

𝑓 𝑖, 𝑗 = 𝛼 𝑓 𝑥𝑖 = 𝑑𝑖, 𝑦𝑗 = 𝑑𝑗 + 𝛽Λ

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data set

group element VR length value

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data set

group element VR length value

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data set

group element VR length value

Date

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data set

group element VR length value

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data set

group element VR length value

Date : 12/03/2014

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data set

group element VR length value

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𝒅

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𝒙

𝒚

𝒙

𝒚

𝒇 𝒔 = 𝒇 ∗ 𝒉

𝒛

𝒉 (𝑳𝑴𝑯)

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𝒙

𝒚

𝒙

𝒚

𝒇 𝒔 = 𝒇 ∗ 𝒉

𝒉

𝒅 × 𝒅

𝒛

(𝑳𝑴𝑯)

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𝒙

𝒔

𝒉𝟐𝒉𝟏

𝒔 = 𝒉𝟏 + 𝒉𝟐

𝒅𝒊𝒔𝒕 < 𝑳𝑴𝑯

𝒙

𝒇

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𝒙

𝒔

𝒉𝟐𝒉𝟏

𝒔 = 𝒉𝟏 + 𝒉𝟐

𝒅𝒊𝒔𝒕 < 𝑳𝑴𝑯

𝒅

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𝒙

𝒔

𝒉𝟐𝒉𝟏

𝒔 = 𝒉𝟏 + 𝒉𝟐

𝒅𝒊𝒔𝒕 < 𝑳𝑴𝑯

𝒅𝒊𝒔𝒕 ≥ 𝑳𝑴𝑯

𝒅

𝒙

𝒔

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𝒙

𝒔

𝒉𝟐𝒉𝟏

𝒔 = 𝒉𝟏 + 𝒉𝟐

𝒅𝒊𝒔𝒕 < 𝑳𝑴𝑯

𝒅𝒊𝒔𝒕 ≥ 𝑳𝑴𝑯

𝒅

𝒅 ≤𝒅𝒊𝒔𝒕

𝟐

𝒙

𝒔

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𝒙

𝒔

𝒉𝟐𝒉𝟏

𝒔 = 𝒉𝟏 + 𝒉𝟐

𝒅𝒊𝒔𝒕 < 𝑳𝑴𝑯

𝒅𝒊𝒔𝒕 = 𝑳𝑴𝑯

𝒅

𝒅 =𝑳𝑴𝑯

𝟐

𝒙

𝒔

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𝒅𝒊𝒔𝒕 = 𝑳𝑴𝑯

𝒅 =𝑳𝑴𝑯

𝟐

𝒙

𝒔

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𝒅𝒊𝒔𝒕 = 𝑳𝑴𝑯

𝒅 =𝑳𝑴𝑯

𝟐

𝒙

𝒔

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𝒙

𝒔

𝝂

𝒔

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𝒙

𝒔

𝝂

𝒔

𝒙

𝒆 ×𝒅

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𝒙

𝒔

𝝂

𝒔

𝒙

𝒆

𝒅

𝒙

𝒔𝒆 = 𝒔 × 𝒆

×

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𝒙

𝒔

𝝂

𝒔

𝒙

𝒆

𝒙

𝒔𝒆 = 𝒔 × 𝒆

𝝂

𝒆× ∗𝟏/𝒅𝒅

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𝒙

𝒔

𝝂

𝒔

𝒙

𝒆

𝒙

𝒔𝒆 = 𝒔 × 𝒆

𝝂

𝒆× ∗

𝝂

𝒔𝒆 = 𝒔 ∗ 𝒆

𝒅 𝟏/𝒅

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𝒙

𝒔

𝝂

𝒔

𝒙

𝒆

𝒙

𝒔𝒆 = 𝒔 × 𝒆

𝝂

𝒆× ∗

𝝂

𝒔𝒆 = 𝒔 ∗ 𝒆

𝒔𝒆 𝒙 = 𝒔 𝒙

𝒏=−∞

+∞

𝜹(𝒙 − 𝒏𝒅)

𝒅 𝟏/𝒅

𝒔𝒆 𝒙 =

𝒏=−∞

+∞

𝒔(𝝂 − 𝒏 𝝂𝒆)

𝝂𝒆 = 𝟏/𝒅

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𝝂

= 𝒔 ∗ 𝒆 =

𝒏=−∞

+∞

𝒔(𝝂 − 𝒏 𝝂𝒆)

𝟐 𝝂𝒆−𝟐 𝝂𝒆 −𝝂𝒆 𝟎

𝒔𝒆

𝝂𝒆

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𝝂

𝝂𝒆 𝟐 𝝂𝒆−𝟐 𝝂𝒆 −𝝂𝒆 𝟎 𝝂𝒎𝒂𝒙

𝝂 > 𝝂𝒎𝒂𝒙 ∶ 𝒔 𝝂 = 0

= 𝒔 ∗ 𝒆 =

𝒏=−∞

+∞

𝒔(𝝂 − 𝒏 𝝂𝒆) 𝒔𝒆

−𝝂𝒎𝒂𝒙

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𝝂

𝝂𝒆 𝟐 𝝂𝒆−𝟐 𝝂𝒆 −𝝂𝒆 𝟎 𝝂𝒎𝒂𝒙

𝝂

𝒘𝟏×

= 𝒔 ∗ 𝒆 =

𝒏=−∞

+∞

𝒔(𝝂 − 𝒏 𝝂𝒆) 𝒔𝒆

−𝝂𝒎𝒂𝒙

𝝂𝒆/𝟐−𝝂𝒆/𝟐

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𝝂

𝝂𝒆 𝟐 𝝂𝒆−𝟐 𝝂𝒆 −𝝂𝒆 𝟎 𝝂𝒎𝒂𝒙

𝝂

𝒘𝟏

= 𝒔 ∗ 𝒆 =

𝒏=−∞

+∞

𝒔(𝝂 − 𝒏 𝝂𝒆) 𝒔𝒆

? 𝒔𝒆 × 𝒘 = 𝒔

−𝝂𝒎𝒂𝒙

×

𝝂𝒆/𝟐−𝝂𝒆/𝟐

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𝝂

𝝂𝒆 𝟐 𝝂𝒆−𝟐 𝝂𝒆 −𝝂𝒆 𝟎 𝝂𝒎𝒂𝒙

𝝂

𝒘𝟏

= 𝒔 ∗ 𝒆 =

𝒏=−∞

+∞

𝒔(𝝂 − 𝒏 𝝂𝒆) 𝒔𝒆

𝝂𝒆 − 𝝂𝒎𝒂𝒙

? 𝒔𝒆 × 𝒘 = 𝒔

−𝝂𝒎𝒂𝒙

×

𝝂𝒆/𝟐−𝝂𝒆/𝟐

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𝝂

𝝂𝒆 𝟐 𝝂𝒆−𝟐 𝝂𝒆 −𝝂𝒆 𝟎 𝝂𝒎𝒂𝒙

𝝂

𝒘𝟏

= 𝒔 ∗ 𝒆 =

𝒏=−∞

+∞

𝒔(𝝂 − 𝒏 𝝂𝒆) 𝒔𝒆

?

𝝂𝒆 − 𝝂𝒎𝒂𝒙

𝝂𝒎𝒂𝒙 ≤ 𝝂𝒆/𝟐𝝂𝒆 − 𝝂𝒎𝒂𝒙 ≥ 𝝂𝒆/𝟐

𝒔𝒆 × 𝒘 = 𝒔

−𝝂𝒎𝒂𝒙

×

𝝂𝒆/𝟐−𝝂𝒆/𝟐

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𝝂

𝝂𝒆 𝟐 𝝂𝒆−𝟐 𝝂𝒆 −𝝂𝒆 𝟎 𝝂𝒎𝒂𝒙

𝝂

𝒘𝟏

𝒔𝒆 × 𝒘 = 𝒔𝒔𝒔𝒊

𝝂𝒆 ≥ 𝟐 𝝂𝒎𝒂𝒙

= 𝒔 ∗ 𝒆 =

𝒏=−∞

+∞

𝒔(𝝂 − 𝒏 𝝂𝒆) 𝒔𝒆

𝝂𝒆 − 𝝂𝒎𝒂𝒙

−𝝂𝒎𝒂𝒙

×

𝝂𝒆/𝟐−𝝂𝒆/𝟐

𝝂𝒎𝒂𝒙 ≤ 𝝂𝒆/𝟐𝝂𝒆 − 𝝂𝒎𝒂𝒙 ≥ 𝝂𝒆/𝟐

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𝝂

𝝂𝒆 𝟐 𝝂𝒆−𝟐 𝝂𝒆 −𝝂𝒆 𝟎 𝝂𝒎𝒂𝒙

𝝂

𝒘𝟏

𝒔𝒆 × 𝒘 = 𝒔 ⇔ 𝒔 = 𝒔𝒆 ∗ 𝒘𝒔𝒔𝒊

𝝂𝒆 ≥ 𝟐 𝝂𝒎𝒂𝒙

= 𝒔 ∗ 𝒆 =

𝒏=−∞

+∞

𝒔(𝝂 − 𝒏 𝝂𝒆) 𝒔𝒆

𝝂𝒆 − 𝝂𝒎𝒂𝒙

−𝝂𝒎𝒂𝒙

×

𝝂𝒆/𝟐−𝝂𝒆/𝟐

𝝂𝒎𝒂𝒙 ≤ 𝝂𝒆/𝟐𝝂𝒆 − 𝝂𝒎𝒂𝒙 ≥ 𝝂𝒆/𝟐

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𝝂

𝝂𝒆 𝟐 𝝂𝒆−𝟐 𝝂𝒆 −𝝂𝒆 𝟎 𝝂𝒎𝒂𝒙−𝝂𝒎𝒂𝒙

𝝂

𝒘𝟏

𝒔𝒆 × 𝒘 = 𝒔 ⇔ 𝒔 = 𝒔𝒆 ∗ 𝒔𝒊𝒏𝒄(𝒙/𝒅)𝒔𝒔𝒊

𝝂𝒆 ≥ 𝟐 𝝂𝒎𝒂𝒙

= 𝒔 ∗ 𝒆 =

𝒏=−∞

+∞

𝒔(𝝂 − 𝒏 𝝂𝒆) 𝒔𝒆

𝝂𝒆 − 𝝂𝒎𝒂𝒙

𝒔𝒊𝒏𝒄(𝒙)

×

𝝂𝒆/𝟐−𝝂𝒆/𝟐

𝝂𝒎𝒂𝒙 ≤ 𝝂𝒆/𝟐𝝂𝒆 − 𝝂𝒎𝒂𝒙 ≥ 𝝂𝒆/𝟐

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𝒙

𝒔

𝒙

𝒔𝒆

𝒔𝒆 𝒙 = 𝒔 𝒙

𝒏=−∞

+∞

𝜹(𝒙 − 𝒏𝒅)× 𝒆

𝝂𝒆 ≥ 𝟐 𝝂𝒎𝒂𝒙 ∶ 𝒅 ≤𝟏

𝟐 𝝂𝒎𝒂𝒙

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𝒙

𝒔

𝒙

𝒔𝒆

𝒔𝒆 𝒙 = 𝒔 𝒙

𝒏=−∞

+∞

𝜹(𝒙 − 𝒏𝒅)

𝒙

𝒔𝒔 𝒙 = 𝒔𝒆 𝒙 ∗ 𝒔𝒊𝒏𝒄(𝒙/𝒅)

∗ 𝒔𝒊𝒏𝒄

× 𝒆

𝝂𝒆 ≥ 𝟐 𝝂𝒎𝒂𝒙 ∶ 𝒅 ≤𝟏

𝟐 𝝂𝒎𝒂𝒙

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𝝂

𝝂𝒆 𝟐 𝝂𝒆−𝟐 𝝂𝒆 −𝝂𝒆 𝟎 𝝂𝒎𝒂𝒙−𝝂𝒎𝒂𝒙

= 𝒔 ∗ 𝒆 =

𝒏=−∞

+∞

𝒔(𝝂 − 𝒏 𝝂𝒆) 𝒔𝒆𝝂𝒆 < 𝟐 𝝂𝒎𝒂𝒙

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𝝂

𝝂𝒆 𝟐 𝝂𝒆−𝟐 𝝂𝒆 −𝝂𝒆 𝟎

= 𝒔 ∗ 𝒆 =

𝒏=−∞

+∞

𝒔(𝝂 − 𝒏 𝝂𝒆) 𝒔𝒆𝝂𝒆 < 𝟐 𝝂𝒎𝒂𝒙

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𝝂

𝝂𝒆 𝟐 𝝂𝒆−𝟐 𝝂𝒆 −𝝂𝒆

= 𝒔 ∗ 𝒆 =

𝒏=−∞

+∞

𝒔(𝝂 − 𝒏 𝝂𝒆) 𝒔𝒆𝝂𝒆 < 𝟐 𝝂𝒎𝒂𝒙

𝝂

𝟎

𝟎 𝝂𝒎𝒂𝒙−𝝂𝒎𝒂𝒙

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𝝂

𝝂𝒆 𝟐 𝝂𝒆−𝟐 𝝂𝒆 −𝝂𝒆

= 𝒔 ∗ 𝒆 =

𝒏=−∞

+∞

𝒔(𝝂 − 𝒏 𝝂𝒆) 𝒔𝒆𝝂𝒆 < 𝟐 𝝂𝒎𝒂𝒙

𝝂

𝟎

𝟎 𝝂𝒆−𝝂𝒆

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𝝂

𝝂𝒆 𝟐 𝝂𝒆−𝟐 𝝂𝒆 −𝝂𝒆

= 𝒔 ∗ 𝒆 =

𝒏=−∞

+∞

𝒔(𝝂 − 𝒏 𝝂𝒆) 𝒔𝒆𝝂𝒆 < 𝟐 𝝂𝒎𝒂𝒙

𝝂

𝟎

𝟎 𝝂𝒆−𝝂𝒆 𝝂𝒆/2−𝝂𝒆/2

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𝝂

𝝂𝒆 𝟐 𝝂𝒆−𝟐 𝝂𝒆 −𝝂𝒆 𝟎

= 𝒔 ∗ 𝒆 =

𝒏=−∞

+∞

𝒔(𝝂 − 𝒏 𝝂𝒆) 𝒔𝒆𝝂𝒆 < 𝟐 𝝂𝒎𝒂𝒙

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𝝂

𝝂𝒆 𝟐 𝝂𝒆−𝟐 𝝂𝒆 −𝝂𝒆 𝟎

= 𝒔 ∗ 𝒆 =

𝒏=−∞

+∞

𝒔(𝝂 − 𝒏 𝝂𝒆) 𝒔𝒆𝝂𝒆 < 𝟐 𝝂𝒎𝒂𝒙

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𝝂

𝝂𝒆 𝟐 𝝂𝒆−𝟐 𝝂𝒆 −𝝂𝒆 𝟎

= 𝒔 ∗ 𝒆 =

𝒏=−∞

+∞

𝒔(𝝂 − 𝒏 𝝂𝒆) 𝒔𝒆𝝂𝒆 < 𝟐 𝝂𝒎𝒂𝒙

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𝝂

𝝂𝒆 𝟐 𝝂𝒆−𝟐 𝝂𝒆 −𝝂𝒆 𝟎

= 𝒔 ∗ 𝒆 =

𝒏=−∞

+∞

𝒔(𝝂 − 𝒏 𝝂𝒆) 𝒔𝒆𝝂𝒆 < 𝟐 𝝂𝒎𝒂𝒙

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𝒔 𝒙 = 𝒔𝒆 𝒙 ∗ 𝒔𝒊𝒏𝒄(𝝂𝒆𝒙)

𝝂𝒆 = 𝟎, 𝟖 𝝂𝒎𝒂𝒙 𝝂𝒆 = 𝟏, 𝟐𝟓 𝝂𝒎𝒂𝒙

𝝂𝒆 = 𝟐 𝝂𝒎𝒂𝒙 𝝂𝒆 = 𝟓 𝝂𝒎𝒂𝒙

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𝒙

𝒚

𝒙

𝒚

𝒇

𝒔 = 𝒇 ∗ 𝒉

𝒛

𝒉 (𝑳𝑴𝑯)

𝑳𝑴𝑯 𝑳𝑴𝑯 𝒔 = 𝒇 𝒉

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𝒙

𝒚

𝒙

𝒚

𝒇

𝒛

𝒉 (𝑳𝑴𝑯)

𝑳𝑴𝑯 𝑳𝑴𝑯

𝝂𝒎𝒂𝒙 ≈𝟏

𝑳𝑴𝑯

𝝂

𝟏/𝑳𝑴𝑯

𝒉(𝝂)

𝒔 = 𝒇 ∗ 𝒉

𝒔 = 𝒇 𝒉

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𝒙

𝒚

𝒙

𝒚

𝒇

𝒛

𝒉 (𝑳𝑴𝑯)

𝑳𝑴𝑯

𝝂𝒎𝒂𝒙 ≈𝟏

𝑳𝑴𝑯 𝒉(𝝂)

𝝂𝒆 =𝟐

𝑳𝑴𝑯; 𝒅 =

𝑳𝑴𝑯

𝟐

𝒅 × 𝒅

𝒔 = 𝒇 ∗ 𝒉

𝒔 = 𝒇 𝒉

𝝂

𝟏/𝑳𝑴𝑯

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𝒙

𝒚

𝒛

𝑳𝑴𝑯 𝒉

𝟓 → 𝟏𝟓𝒎𝒎

𝐍 = 𝟐𝟓𝟔 ∶ 𝒅 =𝟓𝟎𝟎

𝟐𝟓𝟔≈ 𝟐𝒎𝒎 ≪

𝑳𝑴𝑯

𝟐

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𝒙

𝒚

𝒛

𝑳𝑴𝑯 𝒉

𝟓 → 𝟏𝟓𝒎𝒎

𝐍 = 𝟏𝟐𝟖 ∶ 𝒅 =𝟓𝟎𝟎

𝟏𝟐𝟖≈ 𝟒𝒎𝒎 ≈

𝑳𝑴𝑯

𝟐

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𝒙

𝒚

𝒛

𝒅 ≈ 𝟐 − 𝟑𝒎𝒎

Taille du détecteur

Démultiplexage

Non collinéarité

Portée moyenne

Astigmatisme radial

𝑳𝑴𝑯 𝒉

𝟒 → 𝟏𝟎𝒎𝒎

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= +

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= +

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𝝋 = 𝒇 ∗ 𝒉

𝝋 = 𝒇 𝒉

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𝝋 = 𝒇 ∗ 𝒉

𝝋 = 𝒇 𝒉

𝒉

𝝋𝒊,𝒋 =

𝒎=−𝟏

𝟏

𝒏=−𝟏

𝟏

𝒇𝒊−𝒎,𝒋−𝒏 𝒉𝒎,𝒏

𝒉 =1

4

0,25 0,5 0,250,5 1 0,50,25 0,5 0,25

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𝝋 = 𝒇 ∗ 𝒉

𝝋 = 𝒇 𝒉

𝒉

𝝋 𝜈𝑖 , 𝜈𝑗 = 𝒇 𝜈𝑖 , 𝜈𝑗 𝒉 𝜈𝑖 , 𝜈𝑗

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𝒉 𝜈 =1

1 +𝜈𝜈𝐶

2𝑛

𝟏/ 𝟐

𝝂𝑪

𝝂

𝒉 𝝂 𝒏

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𝒉 𝜈𝑖 , 𝜈𝑗 =1

1 +𝜈𝑖2 + 𝜈𝑗

2

𝜈𝐶

2𝑛

𝜈𝐶 = 60%𝑛 = 4

𝜈𝐶 = 60%𝑛 = 12

𝜈𝐶 = 30%𝑛 = 4

𝜈𝐶 = 30%𝑛 = 12

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𝒉 𝜈𝑖 , 𝜈𝑗 =1

1 +𝜈𝑖2 + 𝜈𝑗

2

𝜈𝐶

2𝑛

𝜈𝐶 = 50%𝑛 = 20

𝜈𝐶 = 25%𝑛 = 20

𝜈𝐶 = 25%𝑛 = 2

𝜈𝐶 = 50%𝑛 = 2

𝜈𝐶 = 75%𝑛 = 20

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𝛁𝑓(𝑥) = lim𝜀→0

𝑓 𝑥 + 𝜀 − 𝑓(𝑥 − 𝜀)

2𝜀𝑚𝑎𝑥(𝛁𝑓)

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𝑚𝑎𝑥(𝛁𝑓)𝛁𝑓(𝑥) = lim

𝜀→0

𝑓 𝑥 + 𝜀 − 𝑓(𝑥 − 𝜀)

2𝜀

𝒉 = 𝛁𝒙 =−0,5 0 0,5−1 0 1−0,5 0 0,5

𝒉 = 𝛁𝒚 =−0,5 −1 −0,50 0 00,5 1 0,5

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𝒉 = 𝛁𝒙 =−0,5 0 0,5−1 0 1−0,5 0 0,5

𝒉 = 𝛁𝒚 =−0,5 −1 −0,50 0 00,5 1 0,5

𝒉 = 𝛁𝒙2 + 𝛁𝒚

2

𝑚𝑎𝑥(𝛁𝑓)𝛁𝑓(𝑥) = lim

𝜀→0

𝑓 𝑥 + 𝜀 − 𝑓(𝑥 − 𝜀)

2𝜀

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𝚫𝑓(𝑥) = lim𝜀→0

𝑓 𝑥 + 𝜀 − 2𝑓 𝑥 + 𝑓(𝑥 − 𝜀)

𝜀²𝚫𝑓 = 0

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𝒉 =0 −0,25 0

−0,25 1 −0,250 −0,25 0

𝒉 =−0,125 −0,125 −0,125−0,125 1 −0,125−0,125 −0,125 −0,125

𝒉 =0,25 −0,5 0,25−0,5 1 −0,50,25 −0,5 0,25

𝚫𝑓 = 0 𝚫𝑓(𝑥) = lim𝜀→0

𝑓 𝑥 + 𝜀 − 2𝑓 𝑥 + 𝑓(𝑥 − 𝜀)

𝜀²

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𝒙

𝒚

𝒙

𝒚

𝒇

𝒛

𝒉

𝝋 = 𝒇 ∗ 𝒉

𝝋 = 𝒇 𝒉

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𝒙

𝒚

𝒙

𝒚

𝒇

𝒛

𝒉

𝝋 = 𝒇 ∗ 𝒉

𝒇 = 𝝋

𝒉𝒇 = 𝝋 ∗

𝟏

𝒉

𝝋 = 𝒇 𝒉

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𝒉𝒉

𝑥

𝐿𝑀𝐻

𝜈

𝜈𝑚𝑎𝑥

𝝋 = 𝒇 ∗ 𝒉 𝝋 = 𝒇 𝒉

𝑳𝑴𝑯

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𝟏

𝒉

𝟏

𝒉

𝒇 = 𝝋

𝒉

𝒇 = 𝝋 ∗ 𝟏/ 𝒉𝑥

𝐿𝑀𝐻

𝒉𝒉

𝑥

𝐿𝑀𝐻

𝜈

𝜈𝑚𝑎𝑥

𝝋 = 𝒇 ∗ 𝒉 𝝋 = 𝒇 𝒉

𝑳𝑴𝑯

𝜈

𝜈𝑚𝑎𝑥

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𝒇 = 𝝋 𝒎𝒏

𝒎𝒏 =1 − 1 − 𝒉2 𝑛

𝒉

𝒇 = 𝝋

𝒉

𝜈

𝜈𝑚𝑎𝑥

𝒎

1

𝒉

𝒏 = 𝟏

𝟐𝟒

𝟖

𝟏𝟔

𝟓𝟎

𝟏𝟎𝟎

Page 102: TRAITEMENT des IMAGES MEDICALES · group element TAG VR length 128 bytes 44 49 43 4D data set data set data set image EXPLICIT group element length IMPLICIT value VR value , = = ,

𝒇 = 𝝋 𝒎𝒏

𝒇 = 𝝋

𝒉

𝜈

𝜈𝑚𝑎𝑥

1

𝒉

𝒏 = 𝟏

𝟐𝟒

𝟖

𝟏𝟔

𝟓𝟎

𝟏𝟎𝟎

𝒏 = 𝟏

𝟒

𝟏𝟔

𝟏𝟎𝟎

𝒎𝒏 𝒎𝒏

𝒇 = 𝝋 ∗ 𝒎𝒏

𝒎𝒏 =1 − 1 − 𝒉2 𝑛

𝒉

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𝒇 = 𝝋

𝒉

𝜈

𝜈𝑚𝑎𝑥

1

𝒉

𝒏 = 𝟏

𝟐𝟒

𝟖

𝟏𝟔

𝟓𝟎

𝟏𝟎𝟎

𝒏 = 𝟏

𝟒

𝟏𝟔

𝟏𝟎𝟎

𝒇 = 𝝋 𝒎𝒏

𝒎𝒏 𝒎𝒏

𝒇 = 𝝋 ∗ 𝒎𝒏

𝑛 = 0,834 ln 𝐶 − 7,774

King et al. J Nucl Med 1988

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𝑥

𝑦

s

θ

s

Page 105: TRAITEMENT des IMAGES MEDICALES · group element TAG VR length 128 bytes 44 49 43 4D data set data set data set image EXPLICIT group element length IMPLICIT value VR value , = = ,

𝑥

𝑦

s

θ

s

𝑥

𝑦

s

θϕ

R

s

D

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𝑥

𝑦

s

θ

s

𝑥

𝑦

s

θϕ

D

s

ϕ

R

R

s θ = R cos θ − ϕ

s′ θ = R sin ϕ − θ = D

Page 107: TRAITEMENT des IMAGES MEDICALES · group element TAG VR length 128 bytes 44 49 43 4D data set data set data set image EXPLICIT group element length IMPLICIT value VR value , = = ,

𝑥

𝑦

s

θ

s

𝑥

𝑦

s

θϕ

D

s

R

ΔsΔθ

𝚫𝐬

𝚫𝛉= 𝐃

s θ = R cos θ − ϕ

s′ θ = R sin ϕ − θ = D

Page 108: TRAITEMENT des IMAGES MEDICALES · group element TAG VR length 128 bytes 44 49 43 4D data set data set data set image EXPLICIT group element length IMPLICIT value VR value , = = ,

𝑥

𝑦

s

θ

s

𝑥

𝑦

D

s

𝚫𝐬

𝚫𝛉= 𝐃

s θ = R cos θ − ϕ

s′ θ = R sin ϕ − θ = D

D

D

Page 109: TRAITEMENT des IMAGES MEDICALES · group element TAG VR length 128 bytes 44 49 43 4D data set data set data set image EXPLICIT group element length IMPLICIT value VR value , = = ,

𝑥

𝑦

s

θ

s

𝑥

𝑦

D

s

𝚫𝐬

𝚫𝛉= 𝐃

s θ = R cos θ − ϕ

s′ θ = R sin ϕ − θ = D

ω

k

TF

𝐤

𝛚= −𝐃

D

D

𝐤

𝛚

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𝑥

𝑦

s

θ

s

s ω

k

TF

𝐝𝐢𝐬𝐭 = 𝐁 + 𝐃

= 𝐁 −𝐤

𝛚𝑥

𝑦

s

θ

D

B

𝐤

𝛚

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𝑥

𝑦

s

θ

s

s ω

k

TF

𝐝𝐢𝐬𝐭 = 𝐁 + 𝐃

= 𝐁 −𝐤

𝛚𝑥

𝑦

s

θ

D

B

𝐤

𝛚

𝒇 = 𝝋 𝒎

𝑳𝑴𝑯