In many real world applications such as signal and image processing, there are many types of errors in the measurements during data formation, acquisition and transmission. Some errors will seriously damage the data; in such cases, there is no much information about the true signal in the damaged data. Moreover, using the already damaged parts of the data for signal reconstruction would be useless and may even worsen the performance of reconstruction methods. Developing ways to detect the damaged data accurately is very important. In this talk, I will describe a way to combine the reconstruction of signal and detection of damaged data together using binary matching pursuit in two applications: impulse noise removal and 1-bit compressive sensing. This method can be easily implemented and has high accuracy in detecting the damaged data.