🎯 Course Objectives

By the end of this lecture, you should be able to:

• Explain what tracking means in Mixed Reality
• Understand how cameras form digital images
• Connect image processing to feature detection
• Understand how features enable 6DoF tracking
• See the link with the practical lab (feature detection for tracking)

PART A — Why Tracking in Mixed Reality?

What is Tracking in Mixed Reality?

Tracking = continuous estimation of pose:

• Position (x, y, z)
• Orientation (roll, pitch, yaw)

This applies to:

• The camera/headset
• The user’s body
• Objects in the environment
• Sometimes the entire scene (SLAM)

Why is Tracking Critical in MR?

Because MR requires spatial alignment between:

• The real world (captured by sensors)
• The virtual world (rendered by the engine)

If tracking is wrong:

• Virtual objects drift
• Registration breaks
• The experience feels “fake” or unstable

The Registration Problem

Core question:

Given an image from a moving camera, how do we estimate its 6DoF pose in the real world?

Conceptual pipeline:

Three Families of Tracking in XR

Tracking and Latency

Good tracking must be:

• Accurate
• Low-latency
• Robust to occlusion
• Stable over time

Why? Because MR rendering depends on it in real time.

PART B — From Camera to Digital Image

Cameras as Sensors in XR

A camera is a physical sensor that:

• Captures light from the environment
• Converts it into a digital image
• Introduces limitations that affect tracking

Key issues:

• Discretization
• Aliasing
• Noise
• Limited dynamic range

Discretization of Images

The real world is continuous.
A digital image is sampled on a grid of pixels.

This creates:

• Loss of detail
• Potential aliasing
• Dependence on resolution

Interpolation

Aliasing and Moiré

If sampling is too coarse:

• High-frequency patterns create artifacts (moiré)
• Edges look jagged (aliasing)

This matters for tracking:

• Artifacts can confuse feature detectors

Aliasing

Moiré

Demo: effect of downsampling on camera

Nyquist–Shannon Theorem

To properly sample a signal: Sampling frequency ≥ 2 × highest frequency in the signal

If violated → aliasing appears.

PART C — Projective Geometry (Core for Tracking)

Why Geometry Matters for Tracking

Tracking is fundamentally projective geometry.

We need to model:

• How 3D points project onto a 2D image
• How camera motion affects this projection

Extrinsic Parameters (R, t)

Describe the pose of the camera in the world:

• Rotation matrix R
• Translation vector t

👉 This is what tracking estimates.

The Projection Model (Simplified)

The core equation:

u = K [R | t] X

Where:

• X = 3D point in the world
• u = 2D pixel in the image
• K = intrinsics
• [R|t] = camera pose (tracking result)

Intuition of the Projection

1. Transform 3D point from world to camera space (R, t)
2. Project onto image plane (K)
3. Obtain pixel coordinates (u)

This is the mathematical backbone of visual tracking.

PART D — Image Processing as a Precursor to Tracking

Why Image Processing for Tracking?

Three main goals:

1. Reduce noise → more stable features
2. Enhance structures → clearer edges/corners
3. Normalize images → robustness across lighting changes

Grayscale Conversion

Most feature detectors work on grayscale images.

Classic formula:

Y = 0.299R + 0.587G + 0.114B

Reduces complexity while preserving structure.

Histogram Equalization

Improves contrast:

• Makes details more visible
• Helps feature detection in low-contrast images

Useful in challenging lighting conditions.

Contrast and Brightness Adjustment

Simple Blur

Gaussian Blur

Used to:

• Reduce high-frequency noise
• Smooth the image before detecting features

Important for robust tracking.

Sobel Filter

$f’(i)=\frac{f_{i+1}-f_{i-1}}{2}\ \text{is equivalent to}\ [\tfrac12,\ 0,\ -\tfrac12]$

Laplacian (Second-Order Derivative)

Detects regions of rapid change:

• Useful for feature detection
• Highlights corners and fine details

The standard discrete Laplacian is written:

$\nabla^2 f(i,j)=f_{i+1,j}+f_{i-1,j}+f_{i,j+1}+f_{i,j-1}-4f_{i,j}$

This corresponds to the kernel:

From Image Processing to Features

At this stage, we have a cleaned and enhanced image.

Next step: 👉 Detect meaningful points for tracking.

PART E — Feature Detection for Visual Tracking

What is a Feature?

A feature is a distinctive image pattern that can be reliably detected and matched across frames.

Examples:

• Corners
• Blobs
• Edges

What Makes a Good Feature?

A good feature should be:

• Local (robust to occlusion)
• Invariant (to translation, rotation, scale)
• Robust (to noise, lighting changes)
• Distinctive (easy to match)
• Repeatable (found again in next frame)

Why Corners are Ideal

Corners are:

• Stable across viewpoints
• Well-localized in both x and y
• Highly informative for motion estimation

Moravec Corner Detector (1977)

Idea:

• Measure how much intensity changes when shifting a small window in different directions.

Average intensity variation for a small displacement ((u,v))

$E(u,v)=\sum_{x,y} w(x,y),(I(x+u,y+v)-I(x,y))^2$

• (w) specifies the considered neighborhood (value 1 inside the window and 0 outside);
• (I(x,y)) is the intensity at pixel ((x,y)).

Results

Limitation:

• Not rotation invariant.

Harris Corner Detector (1988)

Improvement over Moravec:

• Uses image gradients
• More robust
• Rotation invariant

Widely used in vision-based tracking.

Average intensity variation for a small displacement $(u,v)$

$E(x,y)=\sum_{u,v} w(u,v),|I(x+u,y+v)-I(x,y)|^2$

• $w$ specifies the considered neighborhood (value 1 inside the window, 0 outside);
• $I(x,y)$ is the intensity at pixel $(x,y)$.

Taylor expansion

$I(x+u,y+v)=I(x,y)+u\frac{\partial I}{\partial x}(x,y)+v\frac{\partial I}{\partial y}(x,y)+o(u^2,v^2)$

Neglecting $o(u^2,v^2)$:

$E(x,y)=\sum_{u,v} w(u,v)\left|u\frac{\partial I}{\partial x}(x,y)+v\frac{\partial I}{\partial y}(x,y)\right|^2$

Quadratic form

Average intensity variation for a small displacement ((u,v)):

For small displacements $(u,v)$

$E(x,y)=[u\ v]M[u\ v]^T$

$M$: symmetric, positive definite $\Rightarrow$ eigenvalue decomposition possible.

Structure of the second-moment matrix

$M=\begin{pmatrix}A,C;C,B\end{pmatrix}$

with:

$A=\left(\frac{\partial I}{\partial x}\right)^2 \otimes w ; B=\left(\frac{\partial I}{\partial y}\right)^2 \otimes w ; C=\left(\frac{\partial I}{\partial x}\frac{\partial I}{\partial y}\right)\otimes w$

$w$: Gaussian window (isotropic).

A corner is characterized by a large variation of $E$ in all directions of $(x,y)$.

$\Rightarrow$ Compute the eigenvalues of $M$.

Corner response

Instead of computing the eigenvalues, we can compute:

$\det(M)=AB-C^2=\lambda_1\lambda_2$

$\text{trace}(M)=A+B=\lambda_1+\lambda_2$

and define the response:

$R=\det(M)-k,\text{trace}(M)^2$

Values of $R$:

• positive near corners,
• negative near edges,
• small in flat regions ($k=0.04$).

$\Rightarrow$ corners / interest points = local maxima of $R$.

Invariant by rotation: even after rotation, the matrix shape is unchanged, so the eigenvalues and the response $R$ are unchanged.

Not invariant by scale

Potential solution: compute Harris response at multiple scales (Harris–Laplace).

PART F — From Features to Tracking

Feature-Based Tracking Pipeline

Frame t:

1. Detect features
2. Describe features
3. Match with previous frame
4. Estimate motion (R, t)

Repeat continuously.

Feature Matching

Features are compared using descriptors.

Goal:

• Find correspondences between frames

This enables motion estimation.

Example: SIFT (Scale-Invariant Feature Transform)

SIFT provides:

• Scale invariance
• Rotation invariance
• Robust descriptors

Widely used in classical computer vision.

SIFT Pipeline

1. Build Gaussian pyramid (multi-scale representation)
2. Compute Difference of Gaussians (DoG)
3. Detect keypoints
4. Assign orientation
5. Compute descriptor from local gradients

Build Gaussian pyramid, compute DoG and detect keypoints

To obtain scale-independent descriptors, the image is resampled multiple times (building a “pyramid”).

The Gaussian difference is more efficient than a Laplacian calculation for each level.

Assign orientation and compute descriptor

The descriptor is a histogram of local gradients around the keypoint, weighted by a Gaussian window.

PART G — From 2D Features to 6DoF Tracking

Visual Odometry

If no 3D map is available:

• Track motion between consecutive frames
• Estimate relative movement of the camera

Used in SLAM systems.

SLAM in XR

Modern headsets use SLAM:

• Simultaneous Localization and Mapping
• Build a map of the environment
• Track the headset within that map

Examples:

• ARKit
• ARCore
• Meta Quest inside-out tracking

PART H — Tracking in Modern XR Systems

Optical Tracking (Vicon/OptiTrack)

Uses external cameras to track reflective markers.

Very accurate but requires a dedicated setup.

Hybrid Tracking Systems

Combine:

• Vision
• IMU
• Depth sensors

To improve robustness and reduce drift.