Hyperspectral Reconstruction using Discrete LED-Structured Illumination

TL;DR

The paper demonstrates that continuous reflectance spectra can be reconstructed from a small set of randomly illuminated LEDs by using structured illumination and linear algebra-based reconstruction. By analyzing the information content with a singular value decomposition and recovering spectra via the Moore–Penrose pseudoinverse (with truncation to mitigate noise), the approach leverages sparsity in a suitable basis to achieve hyperspectral-like results with few measurements. Key findings include that ~25 LEDs suffice to accurately reconstruct sparse vegetation spectra and that reconstruction fidelity depends on LED bandwidth and singular-value decay. This work suggests a path toward low-cost, LED-based hyperspectral imaging tailored for focused applications such as precision agriculture.

Abstract

We consider the use of digital signal processing to reconstruct continuous reflectance spectra using a small finite set of randomly illuminated light emitting diodes (LEDs). We simulate the use of LEDs having identical spectral distance and Gaussian bandwidth whose illumination overlaps its nearest neighbors. An object, whose reflectance spectrum is to be determined, is illuminated by a series of random spectral patterns consisting of randomly chosen LEDs with random intensity. We quantify the information within the illumination patterns using the singular value decomposition (SVD) and reconstruct reflectance spectra, specifically hemoglobin and several green vegetation spectra using the pseudoinverse of the SVD for a given amount of noise. We show that for sparse plant spectra, it is possible to reconstruct the continuous green vegetation spectra with RMSE less than 1% with as few as 25 LEDs. Our study demonstrates that reconstructing sparse reflectance spectra based on random structured illumination can enable low-cost LED-based cameras to perform equally well as expensive cameras, especially for dedicated applications.

Figures (6)

• Figure 1: Schematic of the proposed experimental system. An array of LEDs is used to illuminate an object. For each illumination pattern, some of the LEDs are active with randomly chosen intensity. Many different configurations of LEDs and intensities are chosen.
• Figure 2: An example of the spectrum of one illumination pattern. In this example, 50 LEDs, each of 15 nm bandwidth, are in an array, but only 10 of them are active, each having a randomly chosen intensity. The normalization was performed by integrating over the 1D pattern and then dividing all terms by the sum.
• Figure 3: A graphical form of the singular value matrix $\mathbf{\Sigma}$ is shown. The element of the matrix is shown on the x-axis with its normalized value on the y-axis. In a) the singular values of $\mathbf{\Sigma}$ are shown for changing LED bandwidth. It clearly shows that the singular values drop precipitously with increasing bandwidth. The measurement matrix is normalized to the Frobenius norm and the example Gaussian white noise is scaled to $1\%$ (horizontal blue line). In b) the singular values for $\sigma=15$ nm are shown for 20, 30, 40, or 50 uniformly spectrally spaced LEDs. It is clear that the rank of the matrix is equal to the number of LEDs. The measurement matrix is normalized to the Frobenius norm and the example Gaussian white noise is scaled to $1\%$ (horizontal blue line).
• Figure 4: Oxyhemoglobin spectra vs example reconstruction. a) shows the first few elements of the discrete cosine transform of the spectra and the reconstruction of the DCT using the Moore-Penrose inverse. It can be seen that the DCT of the original spectrum and the reconstructed DCT remain close until the size of the singular values approaches the magnitude of the noise. b) shows the reconstruction vs the original oxyhemoglobin spectrum.
• Figure 5: The normalized RMSE values between the reconstructed and original oxyhemoglobin and pine spectra were calculated for a varying number of LEDs. Each reconstruction was performed 10 times with random illumination patterns. RMSE values were normalized to allow for direct comparison.