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  • 1

    COMP 546

    Lecture 12

    Illumination and Reflectance

    Tues. Feb. 20, 2018

  • Illumination and Reflectance

    • Shading

    • Brightness versus Lightness

    • Color constancy

  • N(x)

    L 

    Shading on a sunny day

    𝑁

    𝐿

    Lambert’s (cosine) Law:

    𝐼 𝑋 = 𝑁(𝑋) ∙ 𝐿

  • Unit Surface Normal

    1

    𝜕𝑍 𝜕𝑋

    2

    + 𝜕𝑍 𝜕𝑌

    2

    + 1

    𝜕𝑍

    𝜕𝑋 , 𝜕𝑍

    𝜕𝑌 ,−1𝑁 ≡

    𝑁

    4

  • Shading on a sunny day

    5

  • Cast and Attached Shadows

    cast : 𝑁(𝑋) ∙ 𝐿 > 0 but light is occluded

    attached : 𝑁(𝑋) ∙ 𝐿 < 0

    𝐿

  • Shading models

    • sunny day (last lecture)

    • sunny day + low relief

    • cloudy day

  • Examples of low relief surfaces

    (un)crumpled paper

  • Low relief surface

    1

    𝜕𝑍 𝜕𝑋

    2

    + 𝜕𝑍 𝜕𝑌

    2

    + 1

    𝜕𝑍

    𝜕𝑋 , 𝜕𝑍

    𝜕𝑌 ,−1𝑁 ≡

    9

    𝜕𝑍

    𝜕𝑋 ≈ 0

    ≈0 ≈0

    𝜕𝑍

    𝜕𝑌 ≈ 0

    Thus,

    Suppose and are both small.

  • Linear shading model for low relief

    𝐼 𝑋, 𝑌 ≈ 𝜕𝑍

    𝜕𝑋 , 𝜕𝑍

    𝜕𝑌 ,−1 ∙ 𝐿𝑋 , 𝐿𝑌 , 𝐿𝑍

    10

    • shadows can still occur

    • one equation per point but two unknowns

  • Example: curtains

    𝑍 𝑋, 𝑌 = 𝑍0 + 𝑎 sin( 𝑘𝑋 𝑋 )

    𝐿 𝑁

    𝑋

    𝑍

  • Example: curtains

    𝑍 𝑋, 𝑌 = 𝑍0 + 𝑎 sin( 𝑘𝑋 𝑋 )

    𝜕𝑍

    𝜕𝑋 = 𝑎 𝑘𝑋 cos 𝑘𝑋 𝑋

    𝜕𝑍

    𝜕𝑌 = 0,

    𝑋

    𝑍

  • Example: curtains

    𝐼 𝑋, 𝑌 ≈ 𝜕𝑍

    𝜕𝑋 , 𝜕𝑍

    𝜕𝑌 , −1 ∙ 𝐿𝑋 , 𝐿𝑌 , 𝐿𝑍

    𝑍 𝑋, 𝑌 = 𝑍0 + 𝑎 sin( 𝑘𝑋 𝑋 )

    𝜕𝑍

    𝜕𝑋 = 𝑎 𝑘𝑋 cos 𝑘𝑋 𝑋

    𝐼 𝑋, 𝑌 = 𝑎 𝑘𝑋 cos 𝑘𝑋 𝑋 𝐿𝑋 − 𝐿𝑍

    𝜕𝑍

    𝜕𝑌 = 0,

  • Example: curtains

    𝑍 𝑋, 𝑌 = 𝑍0 + 𝑎 sin( 𝑘𝑋 𝑋 )

    𝐼 𝑋, 𝑌 = − 𝐿𝑍 + 𝑎 𝑘𝑋 𝐿𝑋 cos 𝑘𝑋 𝑋

    𝐿 = (𝐿𝑋, 𝐿𝑌, 𝐿𝑍)

    𝑋

    𝑍

    𝑁

    Q: where do the intensity maxima and minima occur?

  • Shading on a cloudy day (my Ph.D. thesis)

    15

  •  (x)

    Shading on a Cloudy Day

    Shadowing effects cannot be ignored.

  • Shading on a Cloudy Day

    17

    Shading is determined by shadowing and surface normal.

  • Shading on a Sunny Day

    Shading determined by surface normal only.

  • Shape from shading

    19

    Q: What is the task ? What problem is being solved?

    A: Estimate surface slant, tilt, curvature.

    How to account for (or estimate) the lighting ?

  • Illumination and Reflectance

    • Shading

    [Shading and shadowing models assume that intensity variations on a surface are entirely due to illumination. But surfaces have reflectance variations too. ]

    • Brightness versus Lightness

    • Color constancy

  • Which paper is lighter ?

    21

    Paper on the left is in shadow. It has lower physical intensity and it appears darker. But do both papers seem to be of same (white) material?

  • Which paper is lighter ?

    22

    Image is processed so that the right paper is given same image intensities as left paper. Now, right paper appears to be made of different material. Why?

  • 𝐼 𝑥, 𝑦 = 𝑖𝑙𝑙𝑢𝑚𝑖𝑛𝑎𝑡𝑖𝑜𝑛 𝑥, 𝑦 ∗ 𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑎𝑛𝑐𝑒 (𝑥, 𝑦)

    Abstract version

    “Real” example

  • 𝐼 𝑥, 𝑦 = 𝑖𝑙𝑙𝑢𝑚𝑖𝑛𝑎𝑡𝑖𝑜𝑛 𝑥, 𝑦 ∗ 𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑎𝑛𝑐𝑒 (𝑥, 𝑦)

    shading & shadows material

    luminance

    Physical quantities

  • low reflectance, high illumination

    low reflectance, low illumination

    high reflectance, low illumination

    𝐼 𝑥, 𝑦 = 𝑖𝑙𝑙𝑢𝑚𝑖𝑛𝑎𝑡𝑖𝑜𝑛 𝑥, 𝑦 ∗ 𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑎𝑛𝑐𝑒 (𝑥, 𝑦)

  • 𝐼 𝑥, 𝑦 = 𝑖𝑙𝑙𝑢𝑚𝑖𝑛𝑎𝑡𝑖𝑜𝑛 𝑥, 𝑦 ∗ 𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑎𝑛𝑐𝑒 (𝑥, 𝑦)

    “lightness”

    “brightness”

    Perceptual quantities

    (no standard term for perceived illumination)

  • All four indicated “squares” have same intensity.

    What is the key difference between the two configurations ?

    Adelson’s corrugated plaid illusion.

  • “Lightness” perception

    Q: What is the task ?

    What problem is being solved?

    A:

  • “Lightness” perception

    Q: What is the task ?

    What problem is being solved?

    A: Estimate the surface reflectance, by

    discounting the illumination.

    𝐼 𝑥, 𝑦 = 𝑖𝑙𝑙𝑢𝑚𝑖𝑛𝑎𝑡𝑖𝑜𝑛 𝑥, 𝑦 ∗ 𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑎𝑛𝑐𝑒 (𝑥, 𝑦)

    ?

  • “Lightness” perception: solution sketch

    Q: What is the task ?

    What problem is being solved?

    A: Estimate the surface reflectance, by

    discounting illumination effects.

    Compare points that have same illumination.

    𝐼 𝑥1, 𝑦1 = 𝑖𝑙𝑙𝑢𝑚𝑖𝑛𝑎𝑡𝑖𝑜𝑛 ∗ 𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑎𝑛𝑐𝑒 (𝑥1, 𝑦1)

    𝐼 𝑥2, 𝑦2 = 𝑖𝑙𝑙𝑢𝑚𝑖𝑛𝑎𝑡𝑖𝑜𝑛 ∗ 𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑎𝑛𝑐𝑒 (𝑥2, 𝑦2) =

  • Illumination and Reflectance

    • Shading

    • Brightness versus Lightness

    • Color constancy

  • Recall lecture 3 - color

    32

    Illumination

    Surface Reflectance (fraction)

    Absorption by photoreceptors (fraction)

    There are three different spectra here.

  • LMS cone responses

    𝐼 𝑥, 𝑦, 𝜆 = 𝑖𝑙𝑙𝑢𝑚𝑖𝑛𝑎𝑡𝑖𝑜𝑛 𝑥, 𝑦, 𝜆 ∗ 𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑎𝑛𝑐𝑒 𝑥, 𝑦, 𝜆

    Cone response

    𝐼 𝑥, 𝑦, 𝜆 𝐶𝐿𝑀𝑆(𝜆) 𝑑𝜆

  • Surface Color Perception

    34

    Q: What is the task? What is the problem to be solved?

    A:

    illumination

    Surface Reflectance

  • Surface Color Perception

    35

    Q: What is the task? What is the problem to be solved?

    A: Estimate the surface reflectance, by discounting the illumination.

    𝐼 𝑥, 𝑦, 𝜆 = 𝑖𝑙𝑙𝑢𝑚𝑖𝑛𝑎𝑡𝑖𝑜𝑛 𝑥, 𝑦, 𝜆 ∗ 𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑎𝑛𝑐𝑒 𝑥, 𝑦, 𝜆

    illumination

    Surface Reflectance

  • 𝐼𝑅𝐺𝐵 𝑥, 𝑦 = 𝑖𝑙𝑙𝑢𝑚𝑖𝑛𝑎𝑡𝑖𝑜𝑛𝑅𝐺𝐵 𝑥, 𝑦 ∗ 𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑎𝑛𝑐𝑒𝑅𝐺𝐵 (𝑥, 𝑦)

    For simplicity, let’s ignore the continuous wavelength 𝜆 and just consider 𝑅𝐺𝐵 (LMS).

    𝜆

    Cone response

    𝐼 𝑥, 𝑦, 𝜆 𝐶𝐿𝑀𝑆(𝜆) 𝑑𝜆

  • “Color Constancy”

    37

    Task: estimate the surface reflectance, by discounting the illumination

    𝐼𝑅𝐺𝐵 𝑥, 𝑦 = 𝑖𝑙𝑙𝑢𝑚𝑖𝑛𝑎𝑡𝑖𝑜𝑛𝑅𝐺𝐵 𝑥, 𝑦 ∗ 𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑎𝑛𝑐𝑒𝑅𝐺𝐵 (𝑥, 𝑦)

    illumination

    Surface Reflectance

  • Why we need color constancy

    • object recognition

    • skin evaluation (health, emotion, …)

    • food quality

    • …

  • Example 1: spatially uniform illumination

    = *

    𝐼𝑅𝐺𝐵 𝑥, 𝑦 = 𝑖𝑙𝑙𝑢𝑚𝑖𝑛𝑎𝑡𝑖𝑜𝑛𝑅𝐺𝐵 𝑥, 𝑦 ∗ 𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑎𝑛𝑐𝑒𝑅𝐺𝐵 (𝑥, 𝑦)

    Given this, how to estimate this?

  • Solution 1: ‘max-RGB’ Adaptation

    = *

    𝐼𝑅𝐺𝐵 𝑥, 𝑦 = 𝑖𝑙𝑙𝑢𝑚𝑖𝑛𝑎𝑡𝑖𝑜𝑛𝑅𝐺𝐵 𝑥, 𝑦 ∗ 𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑎𝑛𝑐𝑒𝑅𝐺𝐵 (𝑥, 𝑦)

    Divide each 𝐼𝑅𝐺𝐵 channel by the max value of 𝐼𝑅𝐺𝐵 in each channel.

    When does this give the correct answer?

  • Example 2: non-uniform illumination

    = *

    𝐼𝑅𝐺𝐵 𝑥, 𝑦 = 𝑖𝑙𝑙𝑢𝑚𝑖𝑛𝑎𝑡𝑖𝑜𝑛𝑅𝐺𝐵 𝑥, 𝑦 ∗ 𝑟𝑒