Accurate estimation of heart rates from photoplethysmogram (PPG) signals during intense physical activity is a very challenging problem. This is because strenuous and high intensity exercise can result in severe motion artifacts in PPG signals, making accurate heart rate (HR) estimation difficult. In this study we investigated a novel technique to accurately reconstruct motion-corrupted PPG signals and HR based on time-varying spectral analysis. The algorithm is called Spectral filter algorithm for Motion Artifacts and heart rate reconstruction (SpaMA). The idea is to calculate the power spectral density of both PPG and accelerometer signals for each time shift of a windowed data segment. By comparing time-varying spectra of PPG and accelerometer data, those frequency peaks resulting from motion artifacts can be distinguished from the PPG spectrum. The SpaMA approach was applied to three different datasets and four types of activities: (1) training datasets from the 2015 IEEE Signal Process. Cup Database recorded from 12 subjects while performing treadmill exercise from 1 km/h to 15 km/h; (2) test datasets from the 2015 IEEE Signal Process. Cup Database recorded from 11 subjects while performing forearm and upper arm exercise. (3) Chon Lab dataset including 10 min recordings from 10 subjects during treadmill exercise. The ECG signals from all three datasets provided the reference HRs which were used to determine the accuracy of our SpaMA algorithm. The performance of the SpaMA approach was calculated by computing the mean absolute error between the estimated HR from the PPG and the reference HR from the ECG. The average estimation errors using our method on the first, second and third datasets are 0.89, 1.93 and 1.38 beats/min respectively, while the overall error on all 33 subjects is 1.86 beats/min and the performance on only treadmill experiment datasets (22 subjects) is 1.11 beats/min. Moreover, it was found that dynamics of heart rate variability can be accurately captured using the algorithm where the mean Pearson's correlation coefficient between the power spectral densities of the reference and the reconstructed heart rate time series was found to be 0.98. These results show that the SpaMA method has a potential for PPG-based HR monitoring in wearable devices for fitness tracking and health monitoring during intense physical activities.