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PROPERTIES AND APPLICATIONS

Outline What 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

Outline What 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 𝑷(𝒙, 𝒚, 𝒛, 𝜽, 𝝓, 𝒕, 𝝀) Light intensity as a function of ◦ Viewpoint – orientation and position ◦ Time ◦ Wavelength 7D function!

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.

2D light field Used for visualization. Assume the world is flat (2D)

Intuition The image a pinhole at (u,v) captures 𝐼(𝑢, 𝑣) = 𝐿(: , : , 𝑢, 𝑣) 𝐿(𝑠, 𝑡, : , : ) All views of a pixel (s,t) Light Field Rendering , Levoy Hanrahan '96.

Outline What 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

Acquisition of Light Fields Real world scenes Will be explained in more detail next week…

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 pinhole Actually: Use Homographic mapping from XY plane to the VU and TS, and lookup resulting ray radiance.

Light Field Interpolation 𝐼 𝑋, 𝑌 = 𝐿(𝐻 𝑈𝑉 𝑥, 𝑦 , 𝐻 𝑆𝑇 Problem: Finite sampling of the Light Field – ◦ 𝐻 𝑈𝑉 𝑥, 𝑦 , 𝐻 𝑆𝑇 𝑥, 𝑦 may not be sampled 𝑥, 𝑦 ) Solution: Proper interpolation / reconstruction is needed ◦ Nearest neighbor, ◦ Linear, ◦ Custom Filter

**Detailed Analysis later on…**

NN NN + Linear Linear

Changing the focal plane Fourier Slice Photography , Ng, 05

In-Camera Light Field Parameterization

The camera operator Can define a camera as an operator on the Light Field.

◦ The conventional camera operator: [Stroebel et al. 1986] x y

Reminder - Thin lens formula 1 D’ + 1 D = const

*D D’*

To focus closer - increase the sensor-to-lens distance .

Refocusing - Reparameterization Type equation here.

𝐿 𝐹 ′ 𝑢, 𝑥 = 𝐿 𝐹 (𝑢, 𝑢 + 𝑥 − 𝑢 𝛼 )

Reparametrization - 4D

Refocusing - Reparameterization Refocus Change of distance between planes 𝐹 ′ = 𝛼𝐹 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) y ′ ′ ′ x • Can we do better ????

Fourier Slice Theorem 𝐹 ∘ 𝐼 = 𝑆 ∘ 𝐹 • F – Fourier Transform Operator • I – Integral Projection Operator • S – Slicing Operator

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! **

𝑃𝑟𝑒𝑝𝑟𝑜𝑐𝑒𝑠𝑠: 𝑂 𝑛 4 log 𝑛 . 𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒 𝑛𝑒𝑤 𝐷𝑒𝑝𝑡ℎ 𝛼: 𝑂 𝑛 2 + 𝑂(𝑛 2 log 𝑛)

Fourier Slice Photography , Ng, 05

Fourier Slice Photography Thm – More corollaries Two important results that are worth mentioning: 1. **Filtered Light Field Photography Thm** 𝑳 𝑭 ′

***K=?**

**2. The light field dimensionality gap**

Filtered Light Field Photography Thm Theorem: **Filtered Light Field Photography**

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)

Light Field Sampling • Light Field Acquisition – Discretization • Light Field Sampling is Limited Example – Camera Array: t,s u,v

Sampling in frequency domain Aliasing in the frequency domain * Need to analyze Light Field Spectrum =

Scene Depth and Light Field Light 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.

Spectral Support of Light Field Plenoptic Sampling , Chai et al., 00.

Reconstruction Filters Optimal Slope for filter: Plenoptic Sampling , Chai et al., 00.

Limitations Assumptions ◦ Lambertian surfaces ◦ Free Space – No occlusions

Frequency Analysis of Light Transport • Informally: Different features of lighting and scene causes different effects in the Frequency Content • Blurry Reflections • Shadow Boundries High frequency Low 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.

Example Scenario Reflection 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) +

Light Transport – Frequency Domain Shear in spatial domain is also a shear in Frequency domain

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:

Occlusion – example

Reflection We consider planar surfaces * and rotation invariant BRDFs here What happens when light hits a surface?

1. Multiplication by a **cosine term ** 𝑙 𝑥, 𝜃 = 𝑙 𝑥, 𝜃 cos + (𝜃) 2. Mirror Reparameterization around the **normal direction ** 𝑙 𝑥, 𝜃 = 𝑙 𝑥, −𝜃 3. convolution with the **BRDF** 𝑙 𝑥, 𝜃 = 𝑙 𝑥, 𝜃 ∗ 𝑏𝑟𝑑𝑓(𝜃) * Similar analysis for curved surfaces is also presented in the paper

Reflection - cosine term Spatial domain - multiplication:: 𝑙 𝑅′ 𝑥, 𝜃 = 𝑙 𝑅 𝑥, 𝜃 cos + (𝜃) Frequency domain: 𝑙 𝑅′ Ω 𝑥 , Ω 𝜃 Ω 𝑥 , Ω 𝜃 ∗ 𝐹(cos + 𝜃 )

cosine term example Incoming Light field Light field After cosine term

Reflection – Mirror reparameterization Mirror Reparameterization around the normal direction ◦ Using the law of reflection 𝜃 𝑖𝑛 = 𝜃 𝑜𝑢𝑡 ◦ 𝑙 𝑅′ 𝑥, 𝜃 = 𝑙 𝑅 𝑥, −𝜃 𝜃 𝑖𝑛 𝜃 𝑜𝑢𝑡 ◦ Frequency domain: mirror in the spatial domain => mirror in the frequency domain 𝑙 𝑅′ Ω 𝑥 , Ω 𝜃 = 𝑙 𝑅 Ω 𝑥 , −Ω 𝜃

Reparameterization example Incoming Light field Light field After reparameterization

𝜋 2 Reflection - BRDF What is a BRDF?

◦ Bidirectional reflectance distribution function ◦ A function of the incoming ant out going angles 𝜌(𝜃 𝑖𝑛 , 𝜃 𝑜𝑢𝑡 ) ◦ Tells us how much “light” comes out at a angle 𝜃 𝑜𝑢𝑡 when illuminating the point from 𝜃 𝑖𝑛 .

◦ Different BRDFs model the reflectance properties of different materials ◦ A lot of BRDFs depend only on the difference between 𝜃 𝑜𝑢𝑡 and the mirror reflection direction: 𝜌 𝜃 𝑖𝑛 , 𝜃 𝑜𝑢𝑡 = 𝑐𝑜𝑛𝑠𝑡 𝜌 𝜃 𝑖𝑛 , 𝜃 𝑜𝑢𝑡 = 𝛿 𝜃 𝑟𝑒𝑓𝑙𝑒𝑐𝑡 − 𝜃 𝑜𝑢𝑡 𝜌 𝜃 𝑖𝑛 , 𝜃 𝑜𝑢𝑡 = 𝑓 𝜃 𝑟𝑒𝑓𝑙𝑒𝑐𝑡 − 𝜃 𝑜𝑢𝑡 𝜋 2 𝜋 2 𝜋 2 𝜋 2 𝜋 2

BRDF Intuition Assume a **Specular **BRDF, flat surface and a light source at infinity with angle 𝜃 .

𝜃 𝜃 −𝜃 𝜋 2 𝜋 2 Assume a **box **BRDF, flat surface and a light source at infinity with angle 𝜃 .

𝜃 * 𝜋 2 −𝑎 𝑎 𝜋 2 = 𝜃 −𝜃 x (space) x (space)

Reflection - BRDF Spatial domain: The BRDF action on a light field is a convolution with the BRDF function 𝑙 𝑅′ = 𝑙 𝑅 ∗ 𝜌(𝜃) Frequency domain: Convolution is changed into pointwise multiplication 𝑙 𝑅′ Ω 𝑥 , Ω 𝜃 = 𝑙 𝑅 Ω 𝑥 , Ω 𝜃 𝐹(𝜌(Ω 𝜃 ))

BRDF example Type equation here.

Incoming Light field ⊗ Light field After BRDF change ×

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