CompPhoto: Chapter 10 Gradients

Feature Extraction

Process of identifying important structures (edges, corners, textures) in an image that describe meaningful information.

Image Feature

A part of an image that encodes it in a compact, distinctive form — such as edges, corners, or regions.

Edge

In computational photography, an edge is a location where image intensity changes rapidly.

Purpose of Edge Detection

To extract structural information and object boundaries from an image by identifying intensity discontinuities.

Types of Discontinuities

Edges can result from changes in surface color, depth, orientation (surface normal), or illumination.

Edge Detection Concept

Detects neighborhoods in the image with strong signs of change (large intensity gradients).

Neighborhood Size

In edge detection, the size of the neighborhood affects sensitivity and noise — small neighborhoods detect fine edges, large ones smooth over detail.

Change Metric

A quantitative measure (like a derivative) used to determine how intensity varies across pixels.

Image as a Function

An image can be modeled as F(x, y), where intensity varies continuously across spatial coordinates x and y.

Image Derivative

Describes how intensity changes with respect to position; large derivatives indicate edges.

Image Gradient

A vector representing both the magnitude and direction of intensity change at a pixel.

Gradient Formula

∇F = [∂F/∂x, ∂F/∂y], where ∂F/∂x and ∂F/∂y are partial derivatives in x and y directions.

Gradient Magnitude

‖∇F‖ = √((∂F/∂x)² + (∂F/∂y)²); represents the strength of the edge.

Gradient Direction

θ = arctan((∂F/∂y)/(∂F/∂x)); gives the direction of the most rapid intensity change.

Gradient Points In

Direction of maximum increase in intensity in the image.

Finite Difference Approximation

Method for estimating derivatives using discrete pixel differences.

Finite Difference Formulas

∂F/∂x ≈ F(x+1, y) − F(x, y); ∂F/∂y ≈ F(x, y+1) − F(x, y)

Kernel-Based Derivative

Uses convolution with small filters like [-1 1] or [-1 0 1] to approximate derivatives.

Differential Operator

A kernel (mask) used to compute derivatives in discrete images.

Common Gradient Operators

Sobel, Prewitt, and Roberts operators are common examples for approximating image gradients.

Sobel Operator

Combines smoothing and differentiation using two 3×3 kernels (one for x, one for y direction).

Edge Strength

Defined by the gradient magnitude; higher magnitude indicates stronger edge response.

Thresholding

Involves setting a limit to decide which gradient magnitudes correspond to edges.

Edge Map

A binary image where pixels above a gradient threshold are marked as edges.

Edge Direction

The orientation perpendicular to the gradient direction, indicating edge alignment.

Visualizing Gradients

Gradients can be represented as arrows showing direction and magnitude of intensity change.

Illumination Discontinuity

A change in lighting that creates a perceived edge (e.g., shadows).

Depth Discontinuity

Edges caused by a sudden change in depth between foreground and background.

Surface Normal Discontinuity

Occurs when the surface orientation changes abruptly, even if color and depth stay constant.

Reflectance Change

Edge caused by material or texture differences on a surface.

Applications of Edge Detection

Image segmentation, feature matching, object recognition, and motion analysis.