Transforms & Texture Mapping

Computer Graphics and Imaging

UC Berkeley CS 184/284A

Discussion 03

Worksheet link!

https://tinyurl.com/2wmh4ewe

✨Icebreaker!✨

Week 3 Announcements

Disc 2 Review!

Review Discussion

1. Rasterization

• What is rasterization?
• How do you rasterize this triangle? →

2. Aliasing

• What is aliasing and when does it occur?
• What are some real-world examples?
• What anti-aliasing methods did we learn?

3. Nyquist Theorem

• What is the Nyquist theorem and why does it matter?

Disc 2 Leftovers!

3. Nyquist Theorem

Nyquist frequency

The Nyquist frequency is half of the sampling frequency of your device.

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Ex: If your camera has a frame rate of 24 (runs at 24 Hz), the Nyquist frequency is 12 Hz.

Nyquist theorem

There is no aliasing from frequencies in the signal that are less than the Nyquist frequency.

Ex (previous slide): Your camera can capture signals with frequency less than 12 Hz without aliasing.

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12

1/16

1/32

1/64

1/128

1/256

> 2 * 1/16 = ⅛

> 1/16

> 1/32

> 1/64

> 1/128

Hint:

In 1 rotation, a wheel with n spokes shows the same ”image” n times, so if that rotation took 1 second, it would have a frequency of n hz

Hint:

In 1 rotation, a wheel with n spokes shows the same ”image” n times, so if that rotation took 1 second, it would have a frequency of n hz

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Wheel A: 4 * 6 = 24 Hz → frame rate: 48

Wheel B: 6 * 5 = 30 Hz → frame rate: 60

4. Ashley’s Helicopter

What’s the Nyquist frequency of Ashley’s video camera?

The Nyquist frequency of Ashley’s video camera is 64 frames per second

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So long as the helicopter rotates at less than 64 cycles per second, Ashley will not see any aliasing. The helicopter has 8 blades, so rotating at 8 rotations/second achieves the Nyquist frequency of 64 cycles per second.

Moving onto this week's topic!

Transformation Matrices

1. Basic Transforms

Transformations

Isometric - transformation that preserves size, shape, and distance

Which transformation method holds this property?

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[Review] Matrix Vector Manipulation

Matrix-Vector Multiplication

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Matrix-Vector Multiplication

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Matrix-Vector Multiplication

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Matrix-Vector Multiplication

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Matrix-Vector Multiplication

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Matrix-Vector Multiplication

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θ

θ

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Why do we have points defined in 3D?

Translation

Goal: We want x + t_x, y + t_y

⇒ We need constant terms

Homogeneous Coordinates

Homogeneous Coordinates

• Represent a point in n-dimensional space using n + 1 coordinates.

Homogeneous Coordinates

• Represent a point in n-dimensional space using n + 1 coordinates.

→

Homogeneous Coordinates

• Homogeneous coordinates to represent translations as matrices!
• To shift right by t_x, shift up by t_y:

=

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What does this mean?

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NOTHING :)

Homogeneous Coordinates

• Represent a point in n-dimensional space using n + 1 coordinates.

Transformation Matrices

Transformation Matrices

Coordinate System�Change

Hint:

⚠️Order matters!⚠️

Slide link!

https://tinyurl.com/57ttm8kd

Attendance!

https://tinyurl.com/4p3xcpsx

Attendance Code:

In case you are VERY ambitious…

[Next week's overview & worksheet sols!]

Coordinate Spaces

How do we specify coordinate spaces?

• In original coordinate space, specify:
• Origin, o.
• Two unit vectors, u and v.

Coordinate System Transformation

• Transform from (u, v, o) frame to (x, y) frame using:

Coordinate System Transformation

• Transform from (u, v, o) frame to (x, y) frame using:

Coordinate System Transformation

• Transform from (u, v, o) frame to (x, y) frame using:

Affine Transformation (General Form)

A is a standard linear transform and b is the translation vector

Vector u represents y’s first basis vector inside of x space

Camera coordinates

Given an object’s camera coordinates, how do we find its world coordinates?

Camera has e, u, v in world coordinates

Given an object’s camera coordinates, how do we find its world coordinates?

Camera has e, u, v in world coordinates

Transformation matrix

Diagram

2. Rebecca Takes a Look

1. Realize that the view direction = -z axis of camera coordinates.

2. Draw the camera coordinate frame axes, and then we find that +z = -v

Barycentric Coordinates

Barycentric Coordinates

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• Represents the distance from a given point to the triangle’s vertices
• Alpha, beta, and gamma act as weights

https://www.geogebra.org/m/ZuvmPjmy

Barycentric Coordinates

• (x, y) as weighted sum of values at triangle vertices.
• If A, B, C are colors, αA + βB + γC is an in-between color.

Barycentric Coordinates

• (x, y) as weighted sum of values at triangle vertices.
• If A, B, C are colors, αA + βB + γC is an in-between color.

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Barycentric Interpolation

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3. Barycentric Coordinates

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Texture Mapping

Texture Coordinates

Texture Coordinates

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Texture Coordinates

Dealing with aliasing – Minecraft

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Dealing with aliasing – Minecraft

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Dealing with aliasing – Minecraft

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Antialiasing disabled (not using mipmaps): visible artifacts

Dealing with aliasing – Minecraft

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Antialiasing via mipmaps enabled

Antialiasing Textures

• Super-sampling, then down-sampling, is equivalent to low-pass filtering.
• Sample lower-resolution textures to avoid aliasing.
• Avoids expensive computations.

Antialiasing Textures

1. Pre-compute lower resolution versions of texture
2. Store textures in mipmap, using texture cache (if on GPUs).
3. Adaptively choose mipmap level, D, according to scene.

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Antialiasing Textures

1. Pre-compute lower resolution versions of texture
2. Store textures in mipmap, using texture cache (if on GPUs).
3. Adaptively choose mipmap level according to scene.

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Halve the dimensions each time →

Choosing Mipmap Levels

• Big jump in texture space → use a blurred texture → high D.
• Small jump in texture space → use a high resolution texture → low D.

Choosing Mipmap Levels

• Big jump in texture space → use a blurred texture → high D.
• Small jump in texture space → use a high resolution texture → low D.

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Mipmap Example

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512 x 512

256 x 256

128 x 128

64 x 64

Mipmaps

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How do we determine which Mipmap level to use?

Idea: Look at a pixel’s neighbors:

• If there’s a big jump in texture space, we should use a blurred texture
• If there’s a small jump in texture space, we should use a high res texture

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Level 0

Level 5

Mathematical Deep Dive - Mipmap Equation

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Mipmap Equation

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Bilinear Filtering

Only use values from�one mipmap level

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Trilinear Filtering

Linearly interpolate between�two mipmap levels

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Mathematical Deep Dive - Mipmap Equation

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Why max?

Consider this case:

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Using max(): →

Using min(): →

Using avg(): →

Using min would mean we lose 8-10 texture pixels worth of info per pixel in the y direction!

Intuition: We always want to use the “worst-case” mipmap to avoid aliasing

• Would rather we lose detail in one direction/have a blurry image than try to squeeze too much detail in the other direction

Mathematical Deep Dive - Mipmap Equation

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Consider are all 1: → D = 0

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are all 2: → D = 1

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are all 4: → D = 2

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Why log₂?

In Practice, we use one of two techniques:

• Bilinear Filtering - we only use values from one mipmap level
• Trilinear Filtering - we interpolate between two mipmap level

Bilinear: If we only want one level we use:

Exactly what we want!

Trilinear: log₂ keeps a consistent scaling as the size increases