The following is a summary of revenue by category (in thousands):

Revenue recognized at a point in time for NRE services amounted to $3.0 million and $8.1 million for the years ended December 31, 2021 and 2020, respectively. For services performed on a customer’s owned asset, since the customer controls the asset being enhanced, revenue is recognized over time as services are rendered. Revenue recognized over time for NRE services using a cost-based input method amounted to $9.5 million and $8.0 million for the years ended December 31, 2021 and 2020, respectively. The Company is allowed to bill for services performed under the contract in the event the contract is terminated.

The opening and closing balances of our contract asset and liability balances from contracts with customers as of December 31, 2021 and 2020 were as follows (in thousands):

Remaining performance obligations represent the revenue that is expected to be recognized in future periods related to performance obligations included in a contract that are unsatisfied, or partially satisfied, as of the end of a period. As of December 31, 2021 and 2020, deferred revenue (both current and noncurrent) of $2.9 million and $7.5 million, respectively, represents the Company’s remaining performance obligations, of which $2.5 million and $6.1 million, respectively, is expected to be recognized within one year, with the remainder to be recognized thereafter.

Revenues for the years ended December 31, 2021 and 2020, include the following (in thousands):

Warranty Liabilities

Information regarding the changes in the Company’s product warranty liabilities for the years ended December 31, 2021 and 2020 is as follows (in thousands):