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

Revenue recognized at a point in time for NRE services amounted   to $1.4 million and $3.1 million for the three months ended September 30, 2021 and 2020, respectively, and $4.9 million and $7.7 million for the nine months ended September 30, 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 $2.2 million and $1.9 million for the three months ended September 30, 2021 and 2020, respectively, and $5.6 million and $5.9 million for the nine months ended September 30, 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 September 30, 2021 and December 31, 2020 were as follows:

Revenues for the three and nine months ended September 30, 2021 and 2020, include the following:

Warranty Liabilities

Information regarding the changes in the Company’s product warranty liabilities for the three and nine months ended September 30, 2021 and 2020 is as follows (in thousands):