The recipient publishes their silent payment address, a single 32 byte public key: X = x*G
The sender chooses an input containing a public key: I = i*G
The sender tweaks the silent payment address with the private key that corresponds to their chosen input: X' = hash(i*X)*G + X
SinceiX == xI (Diffie-Hellman Key Exchange), the recipient can detect the payment by calculating hash(x*I)*G + X for each input key I in the blockchain and seeing if it matches an output in the corresponding transaction.