Reinhard Global Tone Mapping

mediumPROTone MappingHDRCV: Computational Photography

Implement Reinhard's global tone mapping operator to map High Dynamic Range (HDR) images to a displayable range while preserving local contrast. This technique is crucial in Computational Photography as it enables the display of HDR images on standard devices.

The concept of tone mapping is based on the idea of compressing the dynamic range of an image, which is the ratio of the brightest and darkest areas, to fit within the limited range of a display device. Tone mapping operators like Reinhard's apply a non-linear transformation to the image's luminance values, preventing saturation and preserving details. The transformation involves calculating the log-average luminance of the image, which represents the average brightness, and then scaling the luminance values based on a key value that controls the overall brightness.

To achieve this, the following steps are involved:

Calculate the luminance of each pixel in the HDR image: $L = 0.299R + 0.587G + 0.114B$ .
Compute the log-average luminance of the image: $\bar{L} = \exp\left(\frac{1}{N}\sum_{x,y} \ln(L(x,y) + \delta)\right)$ with a tiny $\delta$ (e.g. $10^{-8}$ ) so black pixels don't break the logarithm.
Scale the luminance values based on the key value $a$ and log-average luminance: $L_s = \frac{a}{\bar{L}} L$ .
Apply the Reinhard compression function to the scaled luminance values.
Restore colour by multiplying each pixel's R, G and B by the same ratio $L_d / L$ , and clip the result to $[0, 1]$ .

L_d = \frac{L_s}{1 + L_s}

Return an array of the same shape as the input, $(H, W, 3)$ , with values in $[0, 1]$ .

This technique is widely used in image and video processing applications.

Example

Input

hdr = HDR image with values > 1.0
key = 0.18

Output

LDR image with values in [0, 1]

Reasoning

Take a $1 \times 2$ image with two grey pixels: $(4, 4, 4)$ (a highlight) and $(0.25, 0.25, 0.25)$ (a shadow), with key $a = 0.18$ . Grey pixels have $L$ equal to their channel value, since the weights sum to 1: $L = 4$ and $0.25$ .
Log-average luminance: $\bar{L} = e^{(\ln 4 + \ln 0.25)/2} = e^{0} = 1$ . As a geometric mean, it isn't dragged up by one huge highlight the way an arithmetic mean is.
Scale by $a / \bar{L} = 0.18$ : $L_s = 0.72$ and $0.045$ .
Compress: $L_d = 0.72 / 1.72 \approx 0.4186$ and $0.045 / 1.045 \approx 0.0431$ .
Each pixel's channels are multiplied by $L_d / L$ , giving $(0.4186, 0.4186, 0.4186)$ and $(0.0431, 0.0431, 0.0431)$ , shape $(1, 2, 3)$ as input.
Notice: the inputs differ by $16\times$ , the outputs by about $9.7\times$ , so highlights are squeezed hardest. $L_d$ can never reach 1, so grey pixels stay below 1, but a strongly coloured pixel can push one channel past 1, which is what the final clip to $[0, 1]$ handles.

Constraints

hdr: HDR image (H, W, 3)
key: Key value for exposure adjustment
Return: Tone-mapped LDR image

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Background Knowledge

In HDR imaging, scenes often have luminance values that span several orders of magnitude, while typical displays (and 8-bit images) can only show a limited dynamic range (e.g., 0–255). Tone mapping operators (TMOs) compress this large dynamic range into a smaller, displayable range while trying to preserve important visual properties like contrast and detail. Global TMOs apply the same mapping to every pixel based only on its luminance value; local TMOs consider spatial neighborhoods.

Reinhard’s global tone mapping operator is a classic, simple TMO. The idea is to first normalize scene (world) luminance by a “key” value that controls overall brightness, then pass it through a nonlinear curve:

Key scaling: $L = \dfrac{a}{\bar{L}} \cdot L_w$
Tone mapping: $L_d = \dfrac{L}{1 + L}$

Here $\bar{L}$ is the log-average luminance of the scene, which is more robust than a simple arithmetic mean for HDR data. The final $L_d$ is in $(0,1)$ and can be mapped to display intensities.

Algorithm / Approach

The general pattern for this problem:

Extract luminance from the HDR RGB image to get $L_w(x,y)$ .
Compute log-average luminance $\bar{L}$ over all pixels.
Apply key scaling with a given parameter $a$ (e.g., 0.18) to get scaled luminance $L$ .
Apply Reinhard curve $L_d = \frac{L}{1 + L}$ to compress dynamic range.
Re-apply color: scale original RGB channels so that their new luminance matches $L_d$ .
Clamp and convert to the desired output range (e.g., 0–255 uint8).

This is a straightforward per-pixel mapping with an initial global statistic ( $\bar{L}$ ).

Step-by-Step Strategy

Assume input is an HDR image img_hdr in floating point RGB.

Compute luminance channel Use standard Rec.709 coefficients:

L_w = 0.2126\,R + 0.7152\,G + 0.0722\,B

R = img_hdr[..., 0]
G = img_hdr[..., 1]
B = img_hdr[..., 2]
Lw = 0.2126 * R + 0.7152 * G + 0.0722 * B

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Constraints

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