Sigmoidal Tone Mapping

Why Sigmoid?

Sigmoidal curves mimic analog film response:

• Film was engineered over decades for good tone reproduction
• Sigmoid shape compresses both highlights and shadows smoothly
• Preserves midtone contrast

Sigmoidal Tone Curve

Shape

• Toe: Gentle rolloff in shadows
• Linear region: Steep slope for midtones
• Shoulder: Gentle rolloff in highlights

This S-curve naturally compresses extreme values while preserving detail in the midtones.

The Reinhard Formula

The standard two-parameter sigmoidal tone curve:

$R'(x, y) = \frac{R(x, y)^b}{(a \cdot L_m)^b + R(x, y)^b}$

where:

• $R(x, y)$ is the input linear pixel value
• $L_m$ is the geometric mean (log-average luminance) of the scene
• $a$ is the exposure parameter (controls brightness)
• $b$ is the contrast parameter (controls S-curve steepness)

Geometric Mean

The geometric mean normalises for scene brightness:

$L_m = \exp\left(\frac{1}{N} \sum_{(x,y)} \ln(L(x,y))\right)$

where $L(x,y)$ is the luminance at each pixel.

Why Geometric Mean?

• Less affected by extreme values than arithmetic mean
• Provides stable reference across different scenes
• Handles HDR values appropriately

Parameters

Exposure Parameter $a$

Controls overall brightness:

$a$ Value	Effect
Smaller $a$	Brighter image
Larger $a$	Darker image

Mathematical explanation: A smaller $a$ decreases $(a \cdot L_m)^b$ in the denominator, yielding larger output values.

Contrast Parameter $b$

Controls curve steepness:

$b$ Value	Effect
Higher $b$	More contrast (steeper midtone)
Lower $b$	Flatter response (less contrast)

Higher $b$ compresses highlights and shadows more aggressively.

Simplified Formula ($b = 1$)

When $b = 1$:

$R' = \frac{R}{a \cdot L_m + R}$

This simpler form is often used for quick approximations.

Advantages of Sigmoidal Curves

Advantage	Description
Film-like response	Familiar look for photographers
Smooth rolloff	No harsh clipping at extremes
Preserved midtones	Good detail where most visible
Parameter control	Adjustable exposure and contrast
Scene adaptation	Uses geometric mean for automatic scaling

Practical Implementation

Algorithm

1. Compute luminance $L(x,y)$ for each pixel
2. Calculate geometric mean $L_m$
3. Apply sigmoidal formula with chosen $a$ and $b$
4. Output is in [0, 1] range

Parameter Selection

Scene Type	Typical $a$	Typical $b$
Normal daylight	0.18	1.0
High contrast	0.12	1.5
Low contrast (fog)	0.25	0.8

Comparison with Other Methods

Method	Pros	Cons
Linear scaling	Simple	Harsh clipping
Reinhard sigmoid	Smooth, film-like	May need parameter tuning
Local operators	Detail preservation	Halo artefacts

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Summary

• Sigmoidal tone mapping mimics film response with S-curves
• Parameters $a$ (exposure) and $b$ (contrast) control the curve
• Geometric mean provides scene-adaptive normalisation
• Smooth rolloff in highlights and shadows preserves detail

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Past Paper Questions

2024 Paper 3 Question 3: Explain how sigmoidal tone mapping works. What do the parameters control?

2023 Paper 3 Question 4: What is the geometric mean and why is it used in tone mapping?

2022 Paper 3 Question 3: Compare sigmoidal tone mapping to simple exposure adjustment.