Who authorizes low visibility takeoff operations for airlines, and what is required?
The Part 121 regulation that governs low visibility takeoffs are specified in 14 CFR 121.651.  It references company OpSpecs and 91.175 and Part 97:



"Standard" takeoff minimums are specified in 14 CFR 91.175 and would be the highest of:


Non-standard takeoff minimums specified on the airport charts (FAR 97)
For aircraft, other than helicopters, having two engines or less — 1 statute mile visibility.
For aircraft having more than two engines — 1⁄2 statute mile visibility.
For helicopters — 1⁄2 statute mile visibility.


OpSpec C056 is issued to all 121 carriers, and basically just spells out the "standard" takeoff minimums that I listed above, and specifies that RVR must be used if it is available on the runway.



OpSpec C078 is normally issued and allows taking off with lower than standard visibility as long as the airport doesn't already have published higher than standard takeoff minimums.  This is a fairly involved OpSpec with several optional paragraphs, and operators may be approved for takeoff operations down to and including 300 RVR.  For those not familiar with the term, RVR (Runway Visual Range) is the distance that a pilot can see down the runway, measured by automatic equipment,  and is measured in feet in the US.

It offers approvals for the following visibilities:


1/4 SM / RVR 1600:

Touchdown zone RVR (or mid RVR if touchdown zone is inop) is controlling:
Requires "runway markings or runway lighting provide pilots with adequate visual reference to continuously identify the takeoff surface and maintain directional control throughout the takeoff run." (Basically as long as the pilot feels comfortable, but specifically mentions HIRL, CL, and RCLM as okay.)

RVR 1000 (optional)

Touchdown zone RVR and rollout RVR (A mid-RVR report may be substituted for either if they are inop, but still must have two systems) are both controlling
Requires Runway Centerline Lights (CL)

RVR 500 (optional)

Touchdown zone RVR, mid RVR, and rollout RVR are all controlling (one may be inop)
Requires Operative runway centerline lights (CL)
Requires Runway centerline markings (RCLM)

RVR 300 (optional)

Requires a takeoff guidance system
Requires operative high intensity runway lights (HIRL)
Requires operative runway centerline lights (CL)
Requires serviceable runway centerline markings (RCLM)
Front course guidance from the localizer (if applicable to the system used)
Crosswind less than 10 knots
Touchdown zone RVR, mid RVR, and rollout RVR are all controlling (one may be inop)
Requires special training requirements for both the PIC and SIC
Taxiways with operative taxiway centerline lighting must be available







Regulations/Sample OpSpecs


  §121.651 Takeoff and landing weather minimums: IFR: All certificate
  holders.
  
  (a) Notwithstanding any clearance from ATC, no pilot may begin a
  takeoff in an airplane under IFR when the weather conditions reported
  by the U.S. National Weather Service, a source approved by that
  Service, or a source approved by the Administrator, are less than
  those specified in—
  
  (1) The certificate holder's operations specifications; or
  
  (2) Parts 91 and 97 of this chapter, if the certificate holder's
  operations specifications do not specify takeoff minimums for the
  airport.
  
  §91.175 Takeoff and landing under IFR.
  
  ...
  
  (f) Civil airport takeoff minimums. This paragraph applies to persons
  operating an aircraft under part 121, 125, 129, or 135 of this
  chapter.
  
  (1) Unless otherwise authorized by the FAA, no pilot may takeoff from
  a civil airport under IFR unless the weather conditions at time of
  takeoff are at or above the weather minimums for IFR takeoff
  prescribed for that airport under part 97 of this chapter.
  
  (2) If takeoff weather minimums are not prescribed under part 97 of
  this chapter for a particular airport, the following weather minimums
  apply to takeoffs under IFR:
  
  (i) For aircraft, other than helicopters, having two engines or less—1
  statute mile visibility.
  
  (ii) For aircraft having more than two engines—1⁄2 statute mile
  visibility.
  
  (iii) For helicopters—1⁄2 statute mile visibility.
  
  C056, IFR Takeoff Minimums, Part 121 Airplane Operations - All
  Airports.
  
  a. Standard takeoff minimums are defined as 1 statute mile visibility
  or RVR 5000 for airplanes having 2 engines or less and ½ statute mile
  visibility or RVR 2400 for airplanes having more than 2 engines.
  
  b. RVR reports, when available for a particular runway, shall be used
  for all takeoff operations on that runway. All takeoff operations,
  based on RVR, must use RVR reports from the locations along the runway
  specified in this paragraph.
  
  c. When a takeoff minimum is not published, the certificate holder may
  use the applicable standard takeoff minimum and any lower than
  standard takeoff minimums authorized by these operations
  specifications. When standard takeoff minimums or greater are used,
  the Touchdown Zone RVR report, if available, is controlling.
  
  d. When a published takeoff minimum is greater than the applicable
  standard takeoff minimum and an alternate procedure (such as a minimum
  climb gradient compatible with aircraft capabilities) is not
  prescribed, the certificate holder shall not use a takeoff minimum
  lower than the published minimum. The Touchdown Zone RVR report, if
  available, is controlling.
  
  C078, IFR Lower Than Standard Takeoff Minimums, 14 CFR Part 121 Airplane Operations - All Airports.
  
  Standard takeoff minimums are authorized in operations specification
  paragraph C056. The certificate holder is authorized to use lower than
  standard takeoff minimums in accordance with the limitations and
  provisions of this operations specification as follows.
  
  a. Runway visual range (RVR) reports, when available for a particular
  runway, shall be used for all takeoff operations on that runway. All
  takeoff operations, based on RVR, must use RVR reports from the
  locations along the runway specified in this paragraph.
  
  b. When takeoff minimums are equal to or less than the applicable
  standard takeoff minimum, the certificate holder is authorized to use
  the lower than standard takeoff minimums described below:
  
  (1) Visibility or runway visual value (RVV) ¼ statute mile or
  touchdown zone RVR 1600, provided at least one of the following visual
  aids is available. The touchdown zone RVR report, if available, is
  controlling. The mid RVR report may be substituted for the touchdown
  zone RVR report if the touchdown zone RVR report is not available.
  
  (a) Operative high intensity runway lights (HIRL).
  
  (b) Operative runway centerline lights (CL).
  
  (c) Serviceable runway centerline marking (RCLM).
  
  (d) In circumstances when none of the above visual aids are available,
  visibility or RVV ¼ statute mile may still be used, provided other
  runway markings or runway lighting provide pilots with adequate visual
  reference to continuously identify the takeoff surface and maintain
  directional control throughout the takeoff run.
  
  [NOTE: If an operator is not authorized RVR 1000 the POI will not
  select RVR 1000 in the OPSS. If the OPSS is not available the POI
  should delete subparagraph b(2), b(3), &amp; b(4) from the word
  boilerplate.]
  
  (2) Touchdown zone RVR 1000 (beginning of takeoff run) and rollout RVR
  1000, provided all of the following visual aids and RVR equipment are
  available.
  
  (a) Operative runway centerline lights (CL).
  
  (b) Two operative RVR reporting systems serving the runway to be used,
  both of which are required and controlling. A mid-RVR report may be
  substituted for either a touchdown zone RVR report if a touchdown zone
  report is not available or a rollout RVR report if a rollout RVR
  report is not available.
  
  [NOTE: If an operator is not authorized RVR 500 the POI will not
  select RVR 500 in the OPSS. If the OPSS is not available the POI
  should delete subparagraph b(3), &amp; b(4) from the word boilerplate.]
  
  (3) Touchdown zone RVR 500 (beginning of takeoff run), mid RVR 500,
  and rollout RVR 500, provided all of the following visual aids and RVR
  equipment are available.
  
  (a) Operative runway centerline lights (CL).
  
  (b) Runway centerline markings (RCLM).
  
  (c) Operative touchdown zone and rollout RVR reporting systems serving
  the runway to be used, both of which are controlling, or three RVR
  reporting systems serving the runway to be used, all of which are
  controlling. However, if one of the three RVR reporting systems has
  failed, a takeoff is authorized, provided the remaining two RVR values
  are at or above the appropriate takeoff minimum as listed in this
  subparagraph.
  
  (4) At foreign airports which have runway lighting systems equivalent
  to U.S. standards, takeoff is authorized with a reported touchdown
  zone RVR of 150 meters, mid RVR of 150 meters, and rollout RVR of 150
  meters. At those airports where it has been determined that the runway
  lighting system is not equivalent to U.S. standards, the minimums in
  subparagraphs a(1) or (2), as appropriate, apply.
  
  c. Takeoff Guidance System, If Applicable. If the certificate holder
  is authorized to use takeoff minimums based upon the use of takeoff
  guidance systems, the minimums will be specified for the aircraft
  listed in the Table 1 below. The certificate holder shall conduct no
  other takeoffs using these takeoff minimums. If subparagraph c is not
  authorized, N/A will be annotated in each of the columns in the table.
  
  (1) Special provisions and limitations.
  
  (a) Operative high intensity runway lights (HIRL).
  
  (b) Operative runway centerline lights (CL).
  
  (c) Serviceable runway centerline markings (RCLM).
  
  (d) Front course guidance from the localizer must be available and
  used (if applicable to guidance systems used).
  
  (e) The reported crosswind component shall not exceed 10 knots.
  
  (f) Operative touchdown zone, and rollout RVR reporting systems
  serving the runway to be used, both of which are controlling, or three
  RVR reporting systems serving the runway to be used, all of which are
  controlling. However, if one of the three RVR reporting systems has
  failed, a takeoff is authorized, provided the remaining two RVR values
  are at or above the appropriate takeoff minimum as listed in this
  subparagraph.
  
  (g) The pilot-in-command and the second-in-command have completed the
  certificate holders approved training program for these operations.
  
  (h) All operations using these minimums shall be conducted to runways
  which provide direct access to taxi routings which are equipped with
  operative taxiway centerline lighting which meets U.S. or ICAO
  criteria for CAT III operations; or other taxiway guidance systems
  approved for these operations.
  
  (2) The certificate holder is authorized to use the following takeoff
  minimums for the airplanes listed below.
  
  Table 1 (N/A = Not Authorized) Airplane M/M/S Lowest Authorized RVR
  Required Takeoff Guidance System
  
  [NOTE: If an operator is not authorized pilot assessment the POI will
  not select this statement in the OPSS. If the OPSS is not available
  the POI should delete subparagraph d in its entirety from the word
  boilerplate.]
  
  d. Pilot Assessment of RVR for Takeoff (if applicable). In
  circumstances when the touchdown zone RVR reporting system has failed,
  is inaccurate, or is not available, the certificate holder is
  authorized to substitute pilot assessment of equivalent RVR for any
  touchdown zone RVR report required by this operations specification
  paragraph provided that:
  
  (1) The pilot has completed the FAA-approved training program for
  visibility assessment in lieu of RVR, and
  
  (2) Runway markings or runway lighting is available to provide
  adequate visual reference for the assessment.