In SQL, how to split timestamps into several intervals but with the first record of the day as the beginning of the first interval?

Question

PostgreSQL 9.5

My dataset looks like this:

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I want to split the records of the same day into a 1-minute interval, but with the beginning of each interval as the min value of the interval.

With this dataset, my ideal output would look like this:

For example, in 2014-05-27, the min value is 15:09:40, so the first one-minute interval would be 15:09:40 - 15:10:40, and there're three records. For the next interval, 15:11:01 - 15:12:01, there's one record.

Hope I have made myself clear. Thanks in advance.