The relevant criminal statutes are ORS 163.700 and 163.702. Text is as follows: ORS 163.700. Invasion of personal privacy.
(1) Except as provided in ORS 163.702, a person commits the crime of invasion of personal privacy if:
(a)(A) The person knowingly makes or records a photograph, motion picture, videotape or other visual recording of another person in a state of nudity without the consent of the person being recorded; and
(B) At the time the visual recording is made or recorded the person being recorded is in a place and circumstances where the person has a reasonable expectation of personal privacy; or
(b)(A) For the purpose of arousing or gratifying the sexual desire of the person, the person is in a location to observe another person in a state of nudity without the consent of the other person; and
(B) The other person is in a place and circumstances where the person has a reasonable expectation of personal privacy.
(2) As used in this section:
(a) “Makes or records a photograph, motion picture, videotape or other visual recording” includes, but is not limited to, making or recording or employing, authorizing, permitting, compelling or inducing another person to make or record a photograph, motion picture, videotape or other visual recording.
(b) “Nudity” means uncovered, or less than opaquely covered, post-pubescent human genitals, pubic areas or a post-pubescent human female breast below a point immediately above the top of the areola. “Nudity” includes a partial state of nudity.
(c) “Places and circumstances where the person has a reasonable expectation of personal privacy” includes, but is not limited to, a bathroom, dressing room, locker room that includes an enclosed area for dressing or showering, tanning booth and any area where a person undresses in an enclosed space that is not open to public view.
(d) “Public view” means that an area can be readily seen and that a person within the area can be distinguished by normal unaided vision when viewed from a public place as defined in ORS 161.015.
(3) Invasion of personal privacy is a Class A misdemeanor. ORS 163.702. Exceptions to ORS 163.700.
The provisions of ORS 163.700 do not apply to:
(1) Any legitimate medical procedure performed by or under the direction of a person licensed to provide medical service for the purpose of medical diagnosis, treatment, education or research, including, but not limited to, the recording of medical procedures; and
(2) Any activity undertaken in the course of bona fide law enforcement or corrections activity or necessary to the proper functioning of the criminal justice system, including but not limited to the operation and management of jails, prisons and other youth and adult corrections facilities.
Also, I checked with the Portland City Code and could not find any ordinances that limit or restrict video taping in public places.
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