An Eternal Irradiance Camera

Jeremy Klotz; Shree K. Nayar

Figure 1. An irradiance camera and its applications. (a) We present a novel camera that measures the total illumination (irradiance) falling on every point of a sphere. Our prototype camera, called FluxCam, has a minimalist design: it uses a very small number of detectors to both measure irradiance and harvest energy. This allows FluxCam to be completely powered by the light falling on it, i.e., it can read out and wirelessly transmit its measurements at a framerate of 30 FPS or more without a battery, cable, or external power supply. FluxCam can be used for (b) estimating egomotion and (c) monitoring a workspace. In these cases, the camera is not only self-powered and untethered but also privacy preserving since it does not capture the visual details needed to identify individuals. FluxCam can also be used as (d) an untethered light probe for diffuse relighting and (e) an omnidirectional pyranometer for analyzing the sky’s illumination.

Figure 2. What is an irradiance camera? (a) A conventional omnidirectional camera uses millions of pixels to measure radiance from every direction. Each pixel can be modeled by a detector that views the scene through a lens. (b) If we remove the lenses, each detector measures irradiance received from a hemisphere. In effect, this camera measures the illumination falling on every point of a sphere. Since it measures the irradiance function, we call it an irradiance camera. (c) The irradiance function in any environment is extremely smooth, and hence an irradiance camera only needs a handful of detectors to measure it. (d) Since so few detectors are required, we can use large detectors that cover the entire surface of the sphere. Large detectors not only produce measurements with a wide dynamic range, but can also be used as solar cells that harvest energy from the incident illumination. This harvested energy can be used to fully power the camera using the light falling on it. As a result, the camera can operate forever, as an eternal camera, without a battery, cable, or external power supply.

Figure 3. Irradiance cameras in nature. The eyes of (a) a limpet and (b) a flatworm resemble an irradiance camera. These eyes do not have lenses. Instead, the photoreceptors (detectors) are arranged on a curved surface. Even though these eyes do not sense detailed images, they provide enough visual information for the organisms to function in their environments. Figures adapted from Land and Nilsson (2002) .

Figure 4. Spherical harmonic analysis of the kernel. (a) The irradiance function is the result of convolving the radiance function with the kernel

k(\vec{n})

, which is defined as the cosine of the zenith angle (

\theta

), but clamped to be non-negative. (b) The spherical harmonic coefficients of the kernel as a function of the spherical harmonic degree. The coefficients

k_{\ell}

tend toward 0 as the degree (frequency) increases, as a result of the kernel being a low-pass filter. We see from the plot that

k_{\ell}

is approximately zero for any degree

\ell\geq 7

(shaded gray region to the right of the vertical red line). Therefore, the kernel can be approximated by a weighted sum of the first seven spherical harmonics. (c) A 1D slice of the kernel through the gray dotted line in (a). The bandlimited approximation computed from the first seven spherical harmonics (red line) closely follows the kernel (black line).

	Kernel	Irradiance Function
Bandwidth ( $L$ )	Energy Captured (%)	D.O.F.	Approx. Error (%)
1	$37.50$	1	$25.23$
2	$87.50$	4	$9.08$
3	$99.22$	9	$1.77$
5	$99.81$	18	$0.77$
$\mathbf{7}$	$\mathbf{99.92}$	$\mathbf{31}$	$\mathbf{0.44}$

Table 1. Impact of the kernel’s approximate bandwidth. While the first three degrees of spherical harmonics capture

99\%

of the kernel’s energy (second column) ( Basri and Jacobs, 2003 ; Ramamoorthi and Hanrahan, 2001c ) , a higher bandwidth is needed to accurately approximate the irradiance function in real environments. The number of degrees of freedom in the approximated irradiance function (third column) is the number of non-zero spherical harmonic coefficients in the first

L

degrees. The error in the irradiance function (fourth column) is the error due to the bandwidth, as a percentage of the maximum irradiance in the scene, across 3,299 scenes. In this work, we approximate the kernel (and hence the irradiance function) using the first seven degrees of spherical harmonics (highlighted row).

Figure 5. A spherical harmonic sensor? (a) One approach to sensing the irradiance function would be to directly measure its spherical harmonic coefficients. Each coefficient could be measured using a detector and mask whose transmittance is given by the corresponding spherical harmonic. (b) In practice, this would require four detectors to measure a single spherical harmonic coefficient, since each detector can only receive light from a hemisphere, and a mask can only have positive transmittance values. This approach would require 124 detectors and hemispherical masks, while the irradiance function estimation problem only has 31 degrees of freedom.

Figure 6. Sampling the irradiance function in the spatial domain. Shown here is a roughly uniform arrangement of 49 samples (detectors) on the sphere, which is given by an approximate solution to the Tammes problem.

Figure 7. Simulations with real environment maps. (a) Example HDR radiance functions. Although the radiance function in each scene is a spherical image, we only show one hemisphere here. (b) Corresponding ground truth irradiance functions, visualized using a colormap. We simulate the irradiance camera by sampling the irradiance function at the 49 sample locations (blue dots). (c) The reconstructed irradiance functions computed from the 49 measurements. (d) A 1D slice of the reconstruction corresponding to the dashed gray line in (c). The 49 measurements are sufficient for accurately reconstructing the irradiance function across diverse real environments.

Figure 8. FluxCam: irradiance camera prototype. (a) Each sensor board consists of a photovoltaic attached to a custom circuit. The photovoltaic is used both for measuring irradiance and harvesting energy. The (b) front side and (c) back side of the processor board show

49\times

connectors for the sensor boards, readout circuitry, and an RF transmitter. (d) We have optimized the rotations of the detectors to minimize the camera’s size. Starting with a large sphere such that the bounding circles (blue) around each detector (dark brown) and its necessary padding (gold) do not overlap, we iteratively adjust the detector rotations and shrink the sphere until the detectors cannot be rotated without overlap. This yields a sphere with a smaller diameter. (e) We 3D printed the final design in four quadrants using stereolithography. (f) The inside of the camera. The supplemental video includes a time-lapse clip of the camera assembly.

Figure 9. Sensor performance. (a) Setup used to measure the radiometric characteristics of the sensor. (b) The radiometric response is linear over the entire dynamic range (up to

13{,}000\,

\mathrm{lx}

). (c) SNR of the sensor. At just $2.2\,$\mathrm{lx}

, the SNR is

2.4\,

\mathrm{dB}

, which is near the minimum detectable signal. This corresponds to a wide dynamic range of $75.4\,$\mathrm{dB}

. (d) Power harvested by the sensor. Due to the harvester’s overhead, the sensor board requires at least

71\,

\mathrm{lx}$$ to harvest power.

Figure 10. Reconstructed irradiance functions using FluxCam. (a) HDR images of two real scenes. (b) The corresponding ground truth irradiance functions. (c) Reconstructed irradiance functions using the 49 measurements produced by FluxCam in each scene. Across four different scenes, the measured irradiance functions attain an average RMSE of

3.93\%

with respect to the maximum irradiance in each scene. (d) Reconstruction accuracy for the 1D slice corresponding to the dashed gray line in (c).

Figure 11. Example applications of an irradiance camera. (a) FluxCam can be used as an optical gyroscope to determine the camera’s orientation. Here we show the computed orientation as the camera is rolled across a floor and tossed in the air. (b) The camera can serve as an untethered light probe for diffuse relighting. We placed the camera in a location that is partially occluded by another object (left image) and measured the irradiance function (inset image). This allows us to relight and insert a diffuse object (right image). (c) FluxCam can be used to monitor a workspace. Here we use the camera’s 49 measurements to determine which desks are occupied, detect when the door is open, and determine which lights are on (dashed black boxes in the left image). (d) FluxCam can be used as an omnidirectional pyranometer to estimate the best orientation of a solar panel (red dots). Additionally, the measurements can be used to reconstruct the sky, allowing us to quantify the current sky conditions. Please see the supplemental video.

$n$	$X$	$Y$	$Z$	$n$	$X$	$Y$	$Z$
1	$0.407$	$0.511$	$-0.757$	26	$0.956$	$-0.247$	$0.157$
2	$-0.569$	$-0.802$	$0.179$	27	$-0.131$	$0.967$	$0.217$
3	$0.307$	$0.865$	$-0.396$	28	$-0.840$	$0.042$	$0.540$
4	$0.654$	$-0.289$	$0.699$	29	$-0.263$	$-0.761$	$0.593$
5	$-0.710$	$-0.288$	$-0.643$	30	$0.403$	$-0.915$	$-0.012$
6	$-0.876$	$-0.398$	$0.272$	31	$0.046$	$0.213$	$-0.976$

Table S1. Sample locations. The coordinates of the 49 sample locations

\{\vec{n}_{1},\ldots,\vec{n}_{49}\}

on the unit sphere. These samples are an approximate solution to the Tammes problem ( Sloane et al., 1994 ) .

Figure S1. Conditioning of the reconstruction problem for different sample arrangements. The condition number of

A

using sample arrangements given by approximate solutions to the Tammes problem is shown in blue. The condition number of

A

after iteratively refining the sample locations is shown in black.

Component	Quantity	Description
Photovoltaic	49	Panasonic AM-5610CAR-DGK-T
Amplifier	49	TI OPA607
ADC	49	Analog Devices AD7683
Energy Harvester	49	TI BQ25570
Switch	49	Analog Devices ADG787
Readout Microcontroller	1	STM32L051C6T6

Table S2. Components in the prototype irradiance camera. The hardware design (circuit schematics and PCB layouts) is available online ( Klotz and Nayar, 2026 ) .

Figure S2. Settling time of transimpedance amplifier output. When the ENABLE signal goes high, the sensor board switches the photovoltaic from the energy harvester into the transimpedance amplifier. The output of the transimpedance amplifier, which is proportional to the photovoltaic’s short circuit current, takes roughly

20\,

\mathrm{ms}

to settle to its steady state value ( $550\,$\mathrm{mV}

in this case). The choice of when to sample the TIA output represents a tradeoff between producing high fidelity measurements and harvesting more energy.

Figure S3. Radiometric response function at low light levels.

Figure S4. Angular response of the detector. Across incident angles ranging from

-80

\mathrm{\SIUnitSymbolDegree}

to $+80$\mathrm{\SIUnitSymbolDegree}

, the measurements deviate from the ideal cosine response by

3.1\%

on average.

Figure S5. Performance of egomotion estimation. (a) Error in the predicted rotation as a function of camera height above the ground. (b) Error in the predicted rotation as a function of the translation distance. The solid blue line shows the median error, and the light blue region denotes the interquartile range.

Figure S6. Additional simulation results in scenes from the Laval Indoor and UrbanSky datasets. (a) HDR image of the scene (radiance function). (b) Ground truth irradiance function. (c) Reconstructed irradiance function from simulated measurements. (d) 1D slice of the reconstructed irradiance functions through the dashed gray lines in (c).

Figure S7. Irradiance functions measured by the prototype camera in four environments. (a) HDR image of the scene captured by a Ricoh Theta Z1. (b) Ground truth irradiance function computed using the HDR fisheye image. (c) Irradiance function measured by FluxCam. (d) A 1D slice through the dashed gray lines in (c).

An Eternal Irradiance Camera

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