OutlineWhat are light fields
Acquisition of light fields from a 3D scene from a real world scene
Image rendering from light fields Changing viewing angle Changing the focal plane
Sampling and reconstruction Depth vs spectral support Optimal reconstruction Analysis of light transport
OutlineWhat are light fields
Acquisition of light fields from a 3D scene from a real world scene
Image rendering from light fields Changing viewing angle Changing the focal plane
Sampling and reconstruction Depth vs spectral support Optimal reconstruction Analysis of light transport
The Plenoptic Function Plenus – Complete, full.
Optic - appearance, look.
The set of things one can ever see
Light intensity as a function of◦ Viewpoint – orientation and position◦ Time◦ Wavelength
7D function!
𝑷 (𝒙 , 𝒚 , 𝒛 ,𝜽 ,𝝓 ,𝒕 ,𝝀)
The 5D Plenoptic Function Ignoring wavelength and time
We need a 5D function to describe light rays across occlusions◦ 2D orientation◦ 3D position
The Light Field (4D Assuming no occlusions
◦ Light is constant across rays◦ Need only 4D to represent the space of Rays
Is this assumption reasonable?
In free space, i.e outside the convex hull of the scene occluders
The Light Field Parameterizations
◦ Point on a Plane or curved Surface (2D) and Direction on a Hemisphere (2D)◦ Two Points on a Sphere◦ Two Points on two different Planes
Two Plane Parameterization Convenient parameterization for computational photography
Why?• Similar to camera geometry (i.e. film plane vs lens plane)• Linear parameterization - easy computations , no trigonometric functions, etc.
𝐿(𝑠 , 𝑡 ,: ,:)
𝐼 (𝑢 ,𝑣)=𝐿(: , : ,𝑢 ,𝑣)
The image a pinhole at
(u,v) captures
All views of a pixel (s,t)
Intuition
Light Field Rendering , Levoy Hanrahan '96.
OutlineWhat are light fields
Acquisition of light fields from a 3D scene from a real world scene
Image rendering from light fields Changing viewing angle Changing the focal plane
Sampling and reconstruction Depth vs spectral support Optimal reconstruction Analysis of light transport
Acquisition of Light Fields Synthetic 3D Scene
◦ Discretize s,t,u,v and capture all rays intersecting the objects using a standard Ray Tracer
OutlineWhat are light fields
Acquisition of light fields from a 3D scene from a real world scene
Image rendering from light fields Changing viewing angle Changing the focal plane
Sampling and reconstruction Depth vs spectral support Optimal reconstruction Analysis of light transport
Changing the View Point Problem: Computer Graphics
◦ Render a novel view point without expensive ray tracing
Solution:◦ Sample a Synthetic light field using Ray Tracing◦ Use the Light Field to generate any point of view, no need to Ray Trace
Light Field Rendering , Levoy Hanrahan '96.
Changing the View Point Conceptually: Use Ray Trace from all pixels in image plane
Actually: Use Homographic mapping from XY plane to the VU and TS, and lookup resulting ray radiance.
pinhole
Light Field Interpolation
Problem: Finite sampling of the Light Field – ◦ may not be sampled
Solution: Proper interpolation / reconstruction is needed◦ Nearest neighbor,◦ Linear,◦ Custom Filter
Detailed Analysis later on…
NN NN + Linear Linear
The camera operator Can define a camera as an operator on the Light Field.
◦ The conventional camera operator:
y
x [Stroebel et al. 1986]
Reminder - Thin lens formula
D D’1D’ D
1 const+ =
To focus closer - increase the sensor-to-lens distance.
Refocusing - Reparameterization
RefocusChange of distance
between planes
Reparameterization of the light field
Shearing of the Light field
Refocusing camera operator Shear and Integrate the original light field
*(cos term from conventional camera model is absorbed into L)
Computation of Refocusing Operator
•Naïve Approach
•For every X,Y go over all U,V and calculate the sum after reparameterization => O(n^4)
• Can we do better ????
y
x
′ ′′
Fourier Slice Theorem
• F – Fourier Transform Operator
• I – Integral Projection Operator
• S – Slicing Operator
𝐹 ∘ 𝐼=𝑆∘ 𝐹
Fourier Analysis of the Camera Operator
Recall that the Refocusing Camera Operator is:
And from the Last theorem we get The Fourier Slice Photography Theorem
Better Algorithm!
Fourier Slice Photography Thm – More corollaries
Two important results that are worth mentioning:
1. Filtered Light Field Photography Thm
2. The light field dimensionality gap
*K=?
The light field dimensionality gap
◦ The light field is 4D◦ In the frequency domain – The support of all the images with different focus depth is a 3D manifold
This observation was used in order to generate new views of the scene from a focal stack (Levin et al. 2010)
OutlineWhat are light fields
Acquisition of light fields from a 3D scene from a real world scene
Image rendering from light fields Changing viewing angle Changing the focal plane
Sampling and reconstruction Depth vs spectral support Optimal reconstruction Analysis of light transport
Light Field Sampling•Light Field Acquisition – Discretization
•Light Field Sampling is LimitedExample – Camera Array:
u,v
t,s
Sampling in frequency domain Aliasing in the frequency domain
Need to analyze Light Field Spectrum
* =
ALIASING!
No Aliasing!
Scene Depth and Light FieldLight Field Spectrum is related to Scene Depth
From Lambertian property each point in the scene corresponds to a line in the Light Field
Line slope is a function of the depth (z) of the point.
Plenoptic Sampling , Chai et al., 00.
Spectral Support of Light Field Constant Depth
Scene Light Field LF Spectrum
Plenoptic Sampling , Chai et al., 00.
Spectral Support of Light Field Varying Depth
Scene LF Spectrum
Plenoptic Sampling , Chai et al., 00.
Frequency Analysis of Light Transport
•Informally: Different features of lighting and scene causes different effects in the Frequency Content
•Blurry Reflections
•Shadow Boundries Low frequencyHigh frequency
A Frequency analysis of Light Transport , Durand et al. 05.
Frequency Analysis of Light Transport
Look at light transport as a signal processing system.◦ Light source is the input signal◦ Interaction are filters / transforms
Source Transport Occlusion Transport Reflection(BRDF)
Local Light Field We study the local 4D Light Field around a central Ray during transport
◦ In Spatial Domain◦ In Frequency Domain
* Local light field offers us the ability to talk about the Spectrum In a local setting
Local Light Field (2D) Parameterization
The analysis is in flatland, an extension to 4D light field is available
x-v parameterization x-Θ parameterization
A Frequency analysis of Light Transport , Durand et al. 05.
Light Transport – Spatial Domain Light Propagation Shear of the local Light Field
◦ No change in slope (v)◦ Linear change in displacement (X)
+¿
Occlusion Spatial domain:
Occlusion pointwise multiplication in the spatial domain
The incoming light field is multiplied by the binary occlusion function of the occluders.
Frequency domain
convolution in the frequency domain:
Reflection We consider planar surfaces * and rotation invariant BRDFs here
What happens when light hits a surface?
1. Multiplication by a cosine term
2. Mirror Reparameterization around the normal direction
3. convolution with the BRDF
* Similar analysis for curved surfaces is also presented in the paper
Reflection - cosine term Spatial domain - multiplication::
Frequency domain:
𝑙𝑅 ′ (𝑥 ,𝜃 )=𝑙𝑅 (𝑥 , 𝜃 ) cos+¿(𝜃 )¿
Reflection – Mirror reparameterization
Mirror Reparameterization around the normal direction ◦ Using the law of reflection
Frequency domain: mirror in the spatial domain => mirror in the frequency domain ◦ =
𝜃𝑖𝑛𝜃𝑜𝑢𝑡
Reflection - BRDF What is a BRDF?
◦ Bidirectional reflectance distribution function◦ A function of the incoming ant out going angles ◦ Tells us how much “light” comes out at a angle when illuminating the point from ◦ Different BRDFs model the reflectance properties of different materials
◦ A lot of BRDFs depend only on the difference between and the mirror reflection direction:
𝜋2
𝜋2
𝜋2
𝜋2
𝜌 (𝜃𝑖𝑛 , 𝜃𝑜𝑢𝑡 )=𝑐𝑜𝑛𝑠𝑡 𝜌 (𝜃𝑖𝑛 , 𝜃𝑜𝑢𝑡 )=𝛿 (𝜃 𝑟𝑒𝑓𝑙𝑒𝑐𝑡−𝜃𝑜𝑢𝑡 )
𝜋2
𝜋2
𝜌 (𝜃𝑖𝑛 , 𝜃𝑜𝑢𝑡 )= 𝑓 (𝜃𝑟𝑒𝑓𝑙𝑒𝑐𝑡 −𝜃𝑜𝑢𝑡 )
BRDF Intuition
𝜃
𝜋2
𝜋2
𝜃−𝜃
−𝑎 𝑎
𝜋2
𝜋2
𝜃 𝜃−𝜃* =
Assume a box BRDF, flat surface and a light source at infinity with angle .
Assume a Specular BRDF, flat surface and a light source at infinity with angle .
directiondirection
x (space)
x (space)
Reflection - BRDFSpatial domain:
The BRDF action on a light field is a convolution with the BRDF function
Frequency domain:
Convolution is changed into pointwise multiplication
=