Extended Reinhard with White Point

mediumPROTone MappingCV: Computational Photography

Apply extended Reinhard tone mapping with white point control.

The basic Reinhard operator never reaches pure white (1.0) because it asymptotically approaches 1. The extended version allows control over which luminance value maps to white:

$L_d = \frac{L \cdot \left(1 + \frac{L}{L_{white}^2}\right)}{1 + L}$

where $L_{white}$ is the luminance value that should map to pure white (1.0).

Properties:

When $L = L_{white}$ : $L_d = 1.0$ (reaches pure white)
When $L \ll L_{white}$ : behaves like basic Reinhard
Allows intentional "burning out" of very bright highlights

This gives photographers control similar to choosing paper white in traditional printing.

Example

Input

reinhard_extended(1.0, 2.0)

Output

0.625

Reasoning

Inputs: $L = 1.0$ is the scene luminance of one HDR pixel, and $L_{white} = 2.0$ is the luminance you want to land exactly on pure white.
White-point term: the extra factor in the numerator is $1 + \frac{L}{L_{white}^2} = 1 + \frac{1.0}{4.0} = 1.25$ . It grows with $L$ , which is what lets the curve climb all the way to 1.
Plugging in: $L_d = \frac{1.0 \times 1.25}{1 + 1.0} = \frac{1.25}{2.0} = 0.625$ , rounded to 4 decimal places gives 0.625.
Compare with basic Reinhard: without the white-point term you would get $\frac{1.0}{2.0} = 0.5$ , so the extended curve brightens this midtone a little.
Notice that setting $L = L_{white} = 2.0$ gives $\frac{2.0 \times 1.5}{3.0} = 1.0$ exactly, and inputs brighter than $L_{white}$ go above 1 (for example $L = 3.0$ gives $1.3125$ ). That is the intentional burn-out of highlights the description mentions.

Constraints

L is HDR luminance value (non-negative)
L_white is the white point luminance (positive)
Return tone-mapped value rounded to 4 decimal places
When L = L_white, output should be 1.0

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Concept primerPREVIEW

You’re implementing a global tone-mapping operator that takes HDR luminance values and compresses them into displayable range $[0,1]$ using the extended Reinhard formula with a white point parameter.

1. Background Knowledge

HDR vs LDR and Tone Mapping HDR images can contain luminance values spanning many orders of magnitude (e.g., from dark shadows to bright sunlight). Standard displays (LDR) can only show a limited range, so we apply a tone-mapping operator (TMO) to compress the dynamic range while preserving important detail and contrast.
Reinhard Tone Mapping The original Reinhard global operator maps a luminance $L$ to display luminance $L_d$ using a simple curve that:
Is monotonic (brighter stays brighter),
Compresses large values,
Asymptotically approaches 1 but never reaches it. The extended version introduces a white point $L_{white}$ so that a specific luminance is forced to map to pure white (1.0), giving artistic control over how quickly highlights “burn out.”
White Point Control In the extended operator

L_d = \frac{L \cdot \left(1 + \frac{L}{L_{white}^2}\right)}{1 + L},

choosing a smaller $L_{white}$ causes highlights to clip earlier; choosing a larger $L_{white}$ preserves more highlight detail but may look flatter.

2. Algorithm / General Approach

The pattern for this type of global tone-mapping problem is:

Work in luminance space, not directly on RGB.
Apply a per-pixel mapping function $f(L)$ to compress luminance values into $[0,1]$ .
Reconstruct RGB using the ratio of new luminance to old luminance (so colors are preserved).
Clamp and gamma-correct as needed for the target display.

For this specific problem, the core is implementing the given scalar function for each luminance value using the provided formula and chosen $L_{white}$ .

3. Step-by-Step Strategy

Assume each pixel has RGB values $R,G,B$ in some linear HDR range.

Compute luminance per pixel Use a standard linear luminance formula, e.g.

float L = 0.2126f * R + 0.7152f * G + 0.0722f * B;

(Weights may differ depending on the assignment, but the idea is the same.)

Choose or receive $L_{white}$

Often given as a parameter in the problem.
If not, a typical strategy in practice is to pick something like a high percentile of luminance (e.g., 95th percentile), but in a coding exercise it’s usually provided.

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solution.py

Python 3.13

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