At present, about 47 million people worldwide have Alzheimer's disease (AD), and because there is no treatment currently available to cure AD, people with AD (PWAD) are cared for. The estimated cost of care for PWAD in 2016 alone is about $236 billion, which puts a huge burden on relatives of PWAD. This work aims to reduce this burden by proposing an inexpensive indoor positioning system that can be used to monitor PWAD. For the positioning, freeform lenses are used to enable a novel optically pixeled LED luminaire (OPLL) that focuses beams from LEDs to various parts of a room, thereby creating uniquely identifiable regions which are used to improve positioning accuracy. Monte Carlo simulation with the designed OPLL in a room with dimensions 5m × 5m × 3m is used to compute the positioning error and theoretical analysis and experiments are used to validate the time for positioning. Results show that by appropriate LED beam design, OPLL has a positioning error and time for positioning of 0.735 m and 187 ms which is 55.1% lower and 1.2 times faster than existing multiple LED estimation model proximity systems.