A method for spectral reflectance factor reconstruction based on wideband multi-illuminant imaging was proposed, using a programmable LED lighting system and modified Bare Bones Particle Swarm Optimization algorithms. From a set of 16 LEDs with different spectral power distributions, nine light sources with correlated color temperatures in the range of 1924 K - 15746 K, most of them daylight simulators, were generated. Samples from three color charts (X-Rite ColorChecker Digital SG, SCOCIE ScoColor paint chart, and SCOCIE ScoColor textile chart), were captured by a color industrial camera under the nine light sources, and used in sequence as training and/or testing colors. The spectral reconstruction models achieved under multi-illuminant imaging were trained and tested using the canonical Bare Bones Particle Swarm Optimization and its proposed modifications, along with six additional and commonly used algorithms. The impacts of different illuminants, illuminant combinations, algorithms, and training colors on reconstruction accuracy were studied comprehensively. The results indicated that training colors covering larger regions of color space give more accurate reconstructions of spectral reflectance factors, and combinations of two illuminants with a large difference of correlated color temperature achieve more than twice the accuracy of that under a single illuminant. Specifically, the average reconstruction error by the method proposed in this paper for patches from two color charts under A + D90 light sources was 0.94 and 1.08 CIEDE2000 color difference units. The results of the experiment also confirmed that some reconstruction algorithms are unsuitable for predicting spectral reflectance factors from multi-illuminant images due to the complexity of optimization problems and insufficient accuracy. The proposed reconstruction method has many advantages, such as being simple in operation, with no requirement of prior knowledge, and easy to implement in non-contact color measurement and color reproduction devices.