Volume Rendering Lecture 8 February 13, 2020 Overview Scalar - - PowerPoint PPT Presentation
CS53000 - Spring 2020 Introduction to Scientific Visualization Volume Rendering Lecture 8 February 13, 2020 Overview Scalar Volumes Ray casting / Texture mapping Transfer functions CS530 / Spring 2020 : Introduction to Scientific
CS53000 - Spring 2020
Introduction to Scientific Visualization
February 13, 2020
CS530 / Spring 2020 : Introduction to Scientific Visualization.
February 13, 2020
Scalar Volumes Ray casting / Texture mapping Transfer functions
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Level sets
Images (2D) → curves Volumes (3D) → surfaces
Marching cubes (Lorensen & Cline 87)
Triangulated surface at a given isovalue
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Isosurfacing is "binary"
point inside isosurface? voxel contributes to image?
Is a hard, sharp boundary necessarily appropriate for the visualization task?
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Isosurface Volume Rendering Slice
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Marc Levoy, 1988 "Display of Surfaces from Volume Data"
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Pipelines: Isosurface vs. Volume Rendering
"no intermediate geometric structures"
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Volume Data Triangles Rendered Image Volume Data Rendered Image Isosurface extraction surface rendering volume rendering
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Pipelines: Isosurface vs. Volume Rendering
"no intermediate geometric structures"
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Volume Data Triangles Rendered Image Volume Data Rendered Image Isosurface extraction surface rendering volume rendering
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Any rendering process that maps from volume data to an image without introducing binary distinctions / intermediate geometry How to make the data visible?
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Any rendering process which maps from volume data to an image without introducing binary distinctions / intermediate geometry How to make the data visible? ➡ color and opacity
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Directly get a 3D representation of the volume data
light-emitting medium — Even gaseous phenomena can be simulated
emission, absorption, scattering)
interior structures)
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Isosurfacing poor for ...
measured, "real-world" (noisy) data amorphous, "soft" objects
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virtual angiography bovine combustion simulation
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Kajiya: exponential decay of light intensity
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e−τ
R t2
t1 σ(t)dt
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Kajiya: exponential decay of light intensity
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e−τ
R t2
t1 σ(t)dt
c(R) = D c(s(x(t)))µ(s(x(t)))e−
R t
0 µ(s(u))dudt
CS530 / Spring 2020 : Introduction to Scientific Visualization.
February 13, 2020
Kajiya: exponential decay of light intensity
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R t2
t1 σ(t)dt
c(R) = D c(s(x(t)))µ(s(x(t)))e−
R t
0 µ(s(u))dudt
CS530 / Spring 2020 : Introduction to Scientific Visualization.
February 13, 2020
Kajiya: exponential decay of light intensity
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e−τ
R t2
t1 σ(t)dt
c(R) = D c(s(x(t)))µ(s(x(t)))e−
R t
0 µ(s(u))dudt
CS530 / Spring 2020 : Introduction to Scientific Visualization.
February 13, 2020
Kajiya: exponential decay of light intensity
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R t2
t1 σ(t)dt
c(R) = D c(s(x(t)))µ(s(x(t)))e−
R t
0 µ(s(u))dudt
CS530 / Spring 2020 : Introduction to Scientific Visualization.
February 13, 2020
Kajiya: exponential decay of light intensity
11
e−τ
R t2
t1 σ(t)dt
c(R) = D c(s(x(t)))µ(s(x(t)))e−
R t
0 µ(s(u))dudt
CS530 / Spring 2020 : Introduction to Scientific Visualization.
February 13, 2020
Kajiya: exponential decay of light intensity
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R t2
t1 σ(t)dt
c(R) = D c(s(x(t)))µ(s(x(t)))e−
R t
0 µ(s(u))dudt
CS530 / Spring 2020 : Introduction to Scientific Visualization.
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Z D c(s(x(t)))µ(s(x(t)))e−
R t
0 µ(s(u(t))dudt
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Optical Mo
Markus Hadwiger, IEEE Visualization 2002 Tutorial Notes
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Optical Mo
Absorption active scattering scattering active Emission
Markus Hadwiger, IEEE Visualization 2002 Tutorial Notes
CS530 / Spring 2020 : Introduction to Scientific Visualization.
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Optical Mo
Absorption active scattering scattering active Emission
Markus Hadwiger, IEEE Visualization 2002 Tutorial Notes
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s
Ray
s0
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I(s) = I(s0)e−τ(s0,s) τ(s1, s2) = s2
s1
κ(s)ds
→
Integration
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s
Ray
s0
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I(s) = I(s0)e−τ(s0,s) τ(s1, s2) = s2
s1
κ(s)ds
→
Integration
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s
Ray s0
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Integration
s
s0
q(´ s)e−τ(s,´
s)d´
s
Markus Hadwiger, IEEE Visualization 2002 Tutorial Notes
Integration
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s
Ray s0 I(s0)
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Integration
S‘
~
q(s’)
s
s0
q(´ s)e−τ(s,´
s)d´
s
Markus Hadwiger, IEEE Visualization 2002 Tutorial Notes
Integration
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Discrete Approximation
I(si) I(si+1)
s0
si si+1 I(si+1) = αq(si+1) + (1 − α)I(si) I(s0) q(si), A(si) q(si+1), A(si+1) α = A(si+1)
Markus Hadwiger, IEEE Visualization 2002 Tutorial Notes
= q(si+1) OVER I(si)
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Discrete Approximation
Ray
I(si) I(si+1)
s0
si si+1 I(si+1) = αq(si+1) + (1 − α)I(si) I(s0) q(si), A(si) q(si+1), A(si+1) α = A(si+1)
Markus Hadwiger, IEEE Visualization 2002 Tutorial Notes
= q(si+1) OVER I(si)
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TF(si)
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Discrete Approximation
Ray
I(si) I(si+1)
s0
si si+1 I(si+1) = αq(si+1) + (1 − α)I(si) I(s0) q(si), A(si) q(si+1), A(si+1) α = A(si+1)
Markus Hadwiger, IEEE Visualization 2002 Tutorial Notes
= q(si+1) OVER I(si)
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TF(si+1) TF(si)
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Discrete Approximation
Ray
I(si) I(si+1)
s0
si si+1 I(si+1) = αq(si+1) + (1 − α)I(si) I(s0) q(si), A(si) q(si+1), A(si+1) α = A(si+1)
Markus Hadwiger, IEEE Visualization 2002 Tutorial Notes
= q(si+1) OVER I(si)
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TF(si+1) TF(si)
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Discrete Approximation
Ray Back-to-front Compositing with
I(si) I(si+1)
s0
si si+1 I(si+1) = αq(si+1) + (1 − α)I(si) I(s0) q(si), A(si) q(si+1), A(si+1) α = A(si+1)
Markus Hadwiger, IEEE Visualization 2002 Tutorial Notes
= q(si+1) OVER I(si)
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t2 t1 Light
Ray
P
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C(p), α(p) – p is a point in volume Functions of input data f(p)
C(f), α(f) – these are 1D functions
Can include lighting affects
C(f, N(p), L) where N(p) = grad(f)
Derivatives of f
C(f, grad(f) ), α(f, grad(f) )
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f
RGB
Map data value f to color and opacity
α
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Human Tooth CT
α(f)
RGB(f)
f
RGB
Map data value f to color and opacity
α
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Human Tooth CT
α(f)
RGB(f)
f
RGB
Shading, Compositing… Map data value f to color and opacity
α
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Human Tooth CT
α(f)
RGB(f)
f
RGB
Shading, Compositing… Map data value f to color and opacity
α
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Measured sources of volume data
CT (computed tomography) PET (positron emission tomography) MRI (magnetic resonance imaging) Ultrasound Confocal Microscopy
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Synthetic sources of volume data
CFD (computational fluid dynamics) Voxelization of discrete geometry
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Transfer function, with shading Skin/Air Bone/Soft tissue Bone/Air
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Voxels
basic unit of volume data
Interpolation
trilinear common, others possible
Gradient
direction of fastest change
Compositing
"over operator"
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f(0,0,0) f(1,0,0) f(0,1,0)
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Voxels
basic unit of volume data
Interpolation
trilinear common, others possible
Gradient
direction of fastest change
Compositing
"over operator"
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f(0,0,0) f(1,0,0) f(0,1,0)
CS530 / Spring 2020 : Introduction to Scientific Visualization.
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Voxels
basic unit of volume data
Interpolation
trilinear common, others possible
Gradient
direction of fastest change
Compositing
"over operator"
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f(0,0,0) f(1,0,0) f(0,1,0)
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∇f
∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z) ≈ ( (f(1,0,0) - f(-1,0,0))/2, (f(0,1,0) - f(0,-1,0))/2, (f(0,0,1) - f(0,0,-1))/2) Approximates "surface normal“ (of isosurface)
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cf = (0, 1, 0) αf = 0.4
c = αfcf + (1 − αf)αbcb α = αf + (1 − αf)αb
αb = 0.9
c = 0.4 1 + (1 − 0.4) × 0.9 1 = 0.54 0.4 α = 0.94
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cf = (0, 1, 1) αf = 0.4
c = αfcf + (1 − αf)αbcb α = αf + (1 − αf)αb
cb = (1, 0, 0) αb = 0.9
cm = (0, 1, 0) αm = 0.4
0.4 1 1 + (1 − 0.4) × 0.9 1 = 0.54 0.4 0.4 0.4 1 + (1 − 0.4) × 0.9 1 = 0.54 0.4
0.4 1 1 + (1 − 0.4) 0.54 0.4 = 0.324 0.64 0.4
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≠
c = αfcf + (1 − αf)αbcb α = αf + (1 − αf)αb
c = (0.324, 0.64, 0.4) α = 0.964
c = (0.324, 0.64, 0.24) α = 0.964
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Depth
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Depth First
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Depth Max intensity First
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Depth Max intensity Average First
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Depth Max intensity Average Accumulate First
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Depth Intensity First
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Depth Intensity Average
Synthetic Reprojection
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Depth Max Intensity
Maximum Intensity Projection Magnetic Resonance Angiogram
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Depth Intensity Accumulate
Make transparent layers visible; Uses a transfer function for color/opacity
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For each pixel ...
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straightforward use of over operator intuitively backwards
intuitively right not simple over operator facilitates early ray termination
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Ci ci ; ai Ci+1
Ci+1 = aici + (1-ai)Ci
(eye)
composite order
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Ci ; Ai ci ; ai Ci+1 ; Ai+1
Ci+1 = Ci + (1 - Ai)aici Ai+1 = Ai + (1 - Ai)ai
(eye)
composite order
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Ci+1 = Ci + (1 - Ai)ai ci Ai+1 = Ai + (1 - Ai)ai Ci+1 = ai ci + (1-ai)Ci
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Light
Ray
t1 t2
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Light
Ray
t1 t2
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Light
Ray
t1 t2
P
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Light
Ray
t1 t2
P
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Light
Ray
t1 t2
P
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Light
Ray
t1 t2
P
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for each voxel ...
contribution
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Ray casting Ray tracing Sweep plane algorithms (e.g., ZSWEEP) PT algorithm of Shirley and Tuchman
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Projected Tetetrahedra (Shirley & Tuchman)
Decompose each cell into tetrahedra Sort the tetrahedra back-to-front Project each tetrahedron and render its decomposition into 3 or 4 triangles
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Two different non-degenerate classes of the projected tetrahedra
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For the Volume Density Optical Model of Williams et
defined by the transfer functions κ(f(x,y,z)) and ρ(f(x,y,z)) with f(x,y,z) being the scalar function Usually the transfer functions are given as a linear or piecewise linear function, or as a lookup table
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For each rendered pixel the ray integral of the corresponding ray segment has to be computed Observation: The ray integral depends only on Sf, Sb, and l for the Volume Density Optical Model of Williams et al.
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Compute the three-dimensional ray integral by numerical integration and store the integrated chromaticity and opacity in a 3D texture Assign appropriate texture coords (Sf,Sb,l) to the projected vertices of each tetrahedron
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Object order method Hardware-accelerated approach Per-pixel exact rendering
Sort Required Slower than uniform volume rendering
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Slicing of the volume (proxy geometry) Composite the slices Back-to-Front
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color
simultaneous rays
1.0
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color
simultaneous rays
1.0
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Volume Data Eye
Image plane
Graphics Hardware
Slices
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Slices View direction
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Slices View direction
1 slice
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Slices View direction
1 slice 5 slices
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Slices View direction
1 slice 5 slices 20 slices
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Slices View direction
1 slice 5 slices 20 slices 45 slices
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Slices View direction
1 slice 5 slices 20 slices 45 slices 85 slices
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Slices View direction
1 slice 5 slices 20 slices 45 slices 85 slices 170 slices
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Volume Ray Casting
(Requires entire data set in memory) Can produce reflections, shadows, and complex illumination “relatively” easily Easily parallelizable (GPU friendly)
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Hardware Texture Mapping
Extremely Fast Correct illumination is hard Approximate accumulation Difficult to add detailed color and texture
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volume rendering:
Transfer functions make volume data visible by mapping data values to optical properties
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slices: volume data:
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f
RGB
Simple (usual) case: Map data value f to color and opacity
α
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Human Tooth CT
f
RGB
Simple (usual) case: Map data value f to color and opacity
α
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Human Tooth CT
α(f)
RGB(f)
f
RGB
Simple (usual) case: Map data value f to color and opacity
α
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Human Tooth CT
α(f)
RGB(f)
f
RGB
Shading, Compositing…
Simple (usual) case: Map data value f to color and opacity
α
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Human Tooth CT
α(f)
RGB(f)
f
RGB
Shading, Compositing…
Simple (usual) case: Map data value f to color and opacity
α
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Vol Range/ TF Domain Range
Data Value Color And Opacity
Space Vol
Domain
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with a standard graphics operator (“over”)
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v α v α v α v α
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Num voxels Data value Grey-Level Histogram
Material 1 Material 2 Material 3
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x v = f (x) “ here’s the edge ”
“Where’s the edge?”
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x v = f (x) “ here’s the edge ”
“Where’s the edge?” Result: edge pixels
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x v = f (x) “ here’s the edge ”
“Where’s the edge?” Result: edge pixels
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“ here’s the edge ” v α v v = f (x) v0 x
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“ here’s the edge ! ” v0 x v = f (x) x v = f (x)
“ here’s the edge ! ” Domain of the transfer function does not include position
Domain
Data Value
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imply data value isolation
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transfer function
experience
William Schroeder, Lisa Avila, and Ken Martin; Transfer Function Bake-off Vis ’00
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Reasoning:
➔ Make 3D histograms of value, 1st, 2nd deriv. By (1) inspecting and (2) algorithmically analyzing histogram volume, we can create transfer functions
Kindlmann & Durkin, VolVis ’98
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Edges at maximum
zero-crossing of 2nd
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Ideal Turbine Blade Engine Block
Project histogram volume to 2D scatterplots
guidance
boundary model at this stage
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x y z Volume Graphics Distance Map
3D position
data value
Signed distance to boundary
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x y z Volume Graphics Distance Map
(x,y,z)
3D position
data value
Signed distance to boundary
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x y z Volume Graphics Distance Map d
(x,y,z)
3D position
data value
Signed distance to boundary
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x y z Volume Graphics Distance Map d
(x,y,z)
New Distance Map
255
v
3D position
data value
Signed distance to boundary
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d(v) v v = f (x) x
v0 v0
d(v,g); g = gradient magnitude
via boundary model
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data value: v “distance”: x
distance function:
d(v)
boundary emphasis function: b(x)
α(v)
Automatically generated from histogram volume Created by user x α = b(x)
2 1
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Opacity function: α(v) = b(d(v))
data value: v “distance”: x
distance function:
d(v)
boundary emphasis function: b(x)
α(v)
Automatically generated from histogram volume Created by user x α = b(x)
2 1
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Opacity function: α(v) = b(d(v))
α(v,g) = b(d(v,g))
data value: v “distance”: x
distance function:
d(v)
boundary emphasis function: b(x)
α(v)
Automatically generated from histogram volume Created by user x α = b(x)
2 1
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f -- f ’ f -- f ’’ CT head slice
d(v)
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x
b(x)
2 1
Boundary emphasis function simple to set
a(v) = b(d(v))
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Basic Transfer Functions:
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Space Vol
Vol Range/ TF Domain Range
Value + Grad Mag Color And Opacity
Domain
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+
d(v,g)
Detected 4 distinct boundaries between 4 materials
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+
mapped 2D distance function plot are boundary centers Color transfer function
d(v,g)
Detected 4 distinct boundaries between 4 materials
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Mostly accurate isolation of all material boundaries
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“Interactive Volume Rendering Using Multi-Dimensional Transfer Functions and Direct Manipulation Widgets” Kniss, Kindlmann, Hansen: Vis ’01
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enamel dentin pulp background
RGB(f )
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Slice 1D TF output Rendering
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enamel dentin pulp background
RGB(f )
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Slice 1D TF output Rendering
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enamel dentin pulp background
RGB(f )
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1D transfer functions cannot accurately
Slice 1D TF output Rendering
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enamel dentin pulp background
RGB(f )
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1D transfer functions cannot accurately capture all material boundaries
Slice 1D TF output Rendering
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RGB( )
α
( )
Generalize…
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RGB( )
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( )
Generalize…
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RGB( )
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Generalize…
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RGB( )
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Generalize…
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RGB( )
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Generalize…
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RGB( )
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Generalize…
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|⇥f|
RGB( ) ( ) RGB( )
α
( )
Generalize…
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RGB( ) ( )
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RGB( ) ( )
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2D transfer functions give greater flexibility in boundary visualization
Display of Surfaces from Volume Data, Levoy 1988
Modify…
α
|⇥f|
RGB( ) ( )
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Trying to reintroduce dentin / background boundary … Modify…
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Second directional derivative
measured with Hessian
D2
ˆ ff
RGB
f, |rf| , D2
ˆ rff
⌘
α ⇣ f, |rf| , D2
ˆ rff
⌘
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Modify…
RGB
f, |rf| , D2
ˆ rff
⌘
α ⇣ f, |rf| , D2
ˆ rff
⌘
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Modify…
RGB
f, |rf| , D2
ˆ rff
⌘
α ⇣ f, |rf| , D2
ˆ rff
⌘
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Modify…
RGB
f, |rf| , D2
ˆ rff
⌘
α ⇣ f, |rf| , D2
ˆ rff
⌘
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Modify…
RGB
f, |rf| , D2
ˆ rff
⌘
α ⇣ f, |rf| , D2
ˆ rff
⌘
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Modify…
RGB
f, |rf| , D2
ˆ rff
⌘
α ⇣ f, |rf| , D2
ˆ rff
⌘
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Done
RGB
f, |rf| , D2
ˆ rff
⌘
α ⇣ f, |rf| , D2
ˆ rff
⌘
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enamel / background dentin / background dentin / enamel dentin / pulp
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enamel / background dentin / background dentin / enamel dentin / pulp
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enamel / background dentin / background dentin / enamel dentin / pulp
1D: not possible 2D: specificity not as good
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Higher dimensional transfer functions:
+ Better flexibility, specificity +Higher quality visualizations
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Higher dimensional transfer functions:
+ Better flexibility, specificity +Higher quality visualizations
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– Even harder to specify – Unintuitive relationship with boundaries – Greater demands on user interface