When do absolute phrases lack a noun?

When do absolute phrases lack a noun?

absolute phrases are a noun + a modifier, although it is possible to use only a modifier.

When can a modifier, e.g. a participle, be used in an absolute phrase without being preceded by a noun?

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I'm trying to learn wikipedia's list of parenthetical phrases, but am really struggling with this part. What does the author mean by it?

Introductory phrase: Once upon a time, my father ate a muffin.[12]

Interjection: My father ate the muffin, gosh darn it!

Aside: My father, if you don't mind me telling you this, ate the muffin.

Appositive: My father, a jaded and bitter man, ate the muffin.

Absolute phrase: My father, his eyes flashing with rage, ate the muffin.

Free modifier: My father, chewing with unbridled fury, ate the muffin.

Resumptive modifier: My father ate the muffin, a muffin which no man had yet chewed.

Summative modifier: My father ate the muffin, a feat which no man had attempted.

So, to be specific, may I ask whether phrases such as that in bold below should or can be parenthesized with commas?

She is going to the store, burdened with shopping.