Photorealism Steve Ash, Rhodes College 2006 T exture Mapping = - - PowerPoint PPT Presentation

Photorealism

Steve Ash, Rhodes College 2006

T exture Mapping

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T exture Mapping

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Given the intersection point P between the sphere and ray R, map the Cartesian coordinates (x, y, z) of P to (u, v) coordinates in the image to pick a color P(x, y, z) C(i, j)

T exture Mapping

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Given the intersection point P between the sphere and ray R, map the Cartesian coordinates (x, y, z) of P to (u, v) coordinates in the image to pick a color P(x, y, z) C(i, j)

Polar Coordinates

A point in the plane can be described by

(x, y) coordinates – Cartesian coordinates (r, θ) coordinates – Polar coordinates

image source: http://en.wikipedia.org/wiki/File:Polar_to_cartesian.svg

Polar => Cartesian

Given the (r, θ) polar coordinates what are the (x, y) Cartesian coordinates

x = r cosθ y = r sinθ

image source: http://en.wikipedia.org/wiki/File:Polar_to_cartesian.svg

Cartesian => Polar

Given the (x, y) Cartesian coordinates what are the (r, θ) Polar coordinates

r = sqrt( x2 + y2 ) θ = atan( y / x )

image source: http://en.wikipedia.org/wiki/File:Polar_to_cartesian.svg

Spherical Coordinates

A point in 3D space can be described by

(x, y, z) coordinates – Cartesian coordinates (r, θ, φ) coordinates – Spherical coordinates (θ, φ – latitude, longitude)

image source: http://en.wikipedia.org/wiki/File:Coord_system_SZ_0.svg

x z θ ϕ y r z x y N U V

Spherical => Cartesian

Given the (r, θ, φ) spherical coordinates what are the (x, y, z) Cartesian coordinates

x = r sin φ sinθ (based on the second diagram) y = r cos φ z = r sin φ cosθ

image source: http://en.wikipedia.org/wiki/File:Coord_system_SZ_0.svg

x z θ ϕ y r z x y N U V

Spherical => Cartesian

Given the (x, y, z) Cartesian coordinates what are the (r, θ, φ) Spherical coordinates

r = sqrt( x2 + y2 + z2 ) θ = atan ( x / z ) use C function atan2( a, b ) – range is [-π .. π] φ = acos ( y / r ) use C function acos( a ) – range is [0 .. π]

In the context of the ray-tracer the assumption is P(x, y, z) is on a sphere centered at the origin – for a sphere not centered at the origin correct the coordinates of P

image source: http://en.wikipedia.org/wiki/File:Coord_system_SZ_0.svg

x z θ ϕ y r z x y N U V

T exture Mapping

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Given the intersection point P between the sphere and ray R, map the Cartesian coordinates (x, y, z) of P to (u, v) coordinates in the image to pick a color

correct (x, y, z) as if sphere is centered at origin convert the Cartesian (x, y, z) to Spherical (r, θ, φ) keep only (θ, φ) range of θ is [-π .. π] and of φ is [0 .. π] compute u = f(θ) and v = g(φ) so that range of u and v is [0..1] compute i = u ∙ imageWidth j = v ∙ imageHeight set color of P to color at image[ i ][ j ]

P(x, y, z) C(i, j)

T exture Mapping (Interpolation)

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The computed coordinates (i, j) will not be integers, in

• general. To have smooth color change, interpolate the

colors of neighboring pixels.

Suppose i = 2 and j = 5.5 What should we use for C1 ?

P(x, y, z) C(i, j) C1 UL LL UR LR (2, 5) (3, 5) (3, 6) (2, 6)

T exture Mapping (Interpolation)

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The computed coordinates (i, j) will not be integers, in

• general. To have smooth color change, interpolate the

colors of neighboring pixels.

Suppose i = 2 and j = 5.5 What should we use for C1 ? C1 = 0.5 ∙ image[ UL ] + 0.5 ∙ image[ LL ]

P(x, y, z) C(i, j) C1 UL LL UR LR (2, 5) (3, 5) (3, 6) (2, 6)

T exture Mapping (Interpolation)

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The computed coordinates (i, j) will not be integers, in

• general. To have smooth color change, interpolate the

colors of neighboring pixels.

Suppose i = 2 and j = 5.7 What should we use for C1 ?

P(x, y, z) C(i, j) C1 UL LL UR LR (2, 5) (3, 5) (3, 6) (2, 6)

T exture Mapping (Interpolation)

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The computed coordinates (i, j) will not be integers, in

• general. To have smooth color change, interpolate the

colors of neighboring pixels.

Suppose i = 2 and j = 5.7 What should we use for C1 ? C1 = 0.7 ∙ image[ UL ] + 0.3 ∙ image[ LL ] C1 = frac ∙ image[ UL ] + (1-frac) ∙ image[ LL ]

P(x, y, z) C(i, j) C1 UL LL UR LR (2, 5) (3, 5) (3, 6) (2, 6)

T exture Mapping (Interpolation)

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The computed coordinates (i, j) will not be integers, in

• general. To have smooth color change, interpolate the

colors of neighboring pixels.

Let i0 = i – floor(i) and i1 = ceil(i) – i = 1 - i0

j0 = j – floor(j) and j1 = ceil(j) – j = 1 - j0

Compute C1 = j0 ∙ image[ UL ] + j1 ∙ image[ LL ] C2 = j0 ∙ image[ UR ] + j1 ∙ image[ LR ] C = i0 ∙ C2 + i1 ∙ C1

P(x, y, z) C(i, j) i0 j0 i1 j1 C1 C2 UL LL UR LR

T exture Mapping (Interpolation)

+ =

The computed coordinates (i, j) will not be integers, in

• general. To have smooth color change, interpolate the

colors of neighboring pixels.

Let i0 = i – floor(i) and i1 = ceil(i) – i = 1 - i0

j0 = j – floor(j) and j1 = ceil(j) – j = 1 - j0

Compute C1 = j0 ∙ image[ UL ] + j1 ∙ image[ LL ] C2 = j0 ∙ image[ UR ] + j1 ∙ image[ LR ] C = i0 ∙ C2 + i1 ∙ C1

P(x, y, z) C(i, j) i0 j0 i1 j1 C1 C2 UL LL UR LR use % for indices to wrap around the image when looking up neighbors

Bump Mapping

Bump Mapping

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Bump + T exture

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Bump Mapping

Perturb the normal slightly before returning it – this will cause the illumination equation to compute different rays, and therefore, different shadow patterns C P Norig ray Npert Lorig Lpert

Bump Mapping

Perturbing the normal in 3D

calculate vectors U, V that are perpendicular to N (i.e. tangential to sphere)

Calculating U, V, N vectors using Spherical coordinates:

N = (x, y, z) = (sin φ sin θ, cos φ, sin φ cos θ) U = ∂N/ ∂θ = ( cos θ, 0, - sin θ ) V = ∂N/ ∂φ = (cos φ sin θ, -sin φ , cos φ cos θ )

All vectors are already normalized N U V N U

Npert= N + u∙U + v∙V

V u∙U v∙V

Bump Mapping

Perturbing the normal in 3D

calculate vectors U, V that are perpendicular to N (i.e. tangential to sphere)

N U V N U

Npert= N + u∙U + v∙V

V u∙U v∙V The range of u, v is [-1..1] and they are calculated from the bump map using interpolation, i.e. a similar principle as in texture mapping For smoother approximation:

u = (image( i + 1, j ) – image( i - 1, j )) / 2 horizontal difference v = (image( i, j + 1 ) – image( i, j - 1 )) / 2 vertical difference

PPM Image Format

P6 PPM type identifier 475 238 width height 255 max value per color channel r, g, b ÙÙ×ÎÖÖÙÙ×ÙÙ×ÎÖÖ... triples of rgb values (one byte per channel), Rgbrgbrgbrgbrgb... range is [0..255] -- convert to [0..1]

Create a class image that stores the 2D array of colors stored in a PPM image

method that loads a PPM image (see Canvas class for saving a PPM image)

• perator () , so that image(j, j) returns an interpolated value

method that converts to grayscale – replace each pixel c(r, g, b) with c’(r’, g’, b’) where r’ = g’ = b’ = 0.299∙r + 0.587∙g + 0.114∙b

texture map bump map

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Shapes and Decorators

Shape Sphere T extureDecor BumpDecor getMaterial intersect(Ray r, Intersection i)

An intersection object stores: t value intersection point p normal at p color at p u, v values at p (in [0..1]) U, V vectors at p

data members: shape pointer image map

Shapes and Decorators

Shape Sphere T extureDecor BumpDecor getMaterial intersect(Ray r, Intersection i)

An intersection object stores: t value intersection point p normal at p color at p u, v values at p (in [0..1]) U, V vectors at p

Sphere::intersect(Ray r, Intersection& i) { same as Assignment 8 }

data members: shape pointer image map

Shapes and Decorators

Shape Sphere T extureDecor BumpDecor getMaterial intersect(Ray r, Intersection i)

An intersection object stores: t value intersection point p normal at p color at p u, v values at p (in [0..1]) U, V vectors at p

data members: shape pointer image map

Shapes and Decorators

Shape Sphere T extureDecor BumpDecor getMaterial intersect(Ray r, Intersection i)

An intersection object stores: t value intersection point p normal at p color at p u, v values at p (in [0..1]) U, V vectors at p

TextureDecor::intersect(Ray r, Intersection& i) {

• 1. call intersect on the shape
• 2. overwrite the color with the

interpolated color from the map }

data members: shape pointer image map

Shapes and Decorators

Shape Sphere T extureDecor BumpDecor getMaterial intersect(Ray r, Intersection i)

An intersection object stores: t value intersection point p normal at p color at p u, v values at p (in [0..1]) U, V vectors at p

BumpDecor::intersect(Ray r, Intersection& i) {

• 1. call intersect on the shape
• 2. overwrite the normal with the

perturbed normal from the map }

data members: shape pointer image map

Shapes and Decorators

Shape Sphere T extureDecor BumpDecor getMaterial intersect(Ray r, Intersection i)

void example() { Shape* s = new Sphere(…) Shape* textured = new TextDecor(s); Shape* bumped = new BumpDecor(s); Shape* both = new BumpDecor(textured); }

data members: shape pointer image map

void example() { Shape* p = new Plane(…) Shape* textured = new TextDecor(p); Shape* bumped = new BumpDecor(p); Shape* both = new BumpDecor(textured); }