Mats Hummels accused referee Viktor Kassai of having "an extremely bad game" as 10-man Bayern Munich lost 6-3 on aggregate to Real Madrid in the Champions League quarter-finals, with the defender calling for video technology to be introduced to help match officials.

The tie was finely poised at 3-3 on aggregate when Arturo Vidal was harshly sent off after receiving a second yellow card when he looked to have won the ball in a tackle on Marco Asensio and two of the three goals scored by Cristiano Ronaldo appeared to be from offside positions.

Hummels said it hurt Bayern's players to see Ronaldo's controversial strikes in the aftermath of the match as the holders progressed to the semi-finals of the competition after a 4-2 victory on the night at the Santiago Bernabeu.

"Of course we are frustrated and disappointed, but we're also proud," Hummels told the German Football Association's website. "We gave all that we could out on the pitch and used up all our reserves. 

"We were so close and at 2-1 the whole stadium and Real were very nervous. We could have done it, we almost did. 

"When you go into the changing room and watch Real's second and third goals, you know that the referee has had an extremely bad game. That hurts."

Video assistant referees have been trialled with a view to being introduced across football and Hummels believes it is time for the match officials to be given more help via the aid of replays, with Vidal, Thomas Muller and Carlo Ancelotti having all criticised Kassai's performance.

"I've always been for it," Hummels said of video technology. "For many years I've been saying that because I think such decisions would be ruled correctly.

"It would have been very exciting to see how the game would have panned out if it was still 2-1 after 110 or 115 minutes."

Hummels played the full 120 minutes despite carrying a foot injury and the Germany international is hoping he will have no adverse effects from the game, with a DFB-Pokal semi-final against Borussia Dortmund looming for the Bundesliga leaders.

"We obviously spent many, many hours on the training ground receiving treatment and doing exercises, that worked somewhat," Hummels said. "I also spent some hours at home undergoing therapeutic procedures.

"I honestly didn't think it would work though. Especially after a minor setback on Saturday when my foot started swelling up again after a little pressure. But somehow it felt fine again on Sunday and it worked out fine.

"I only had to fight a little bit through the pain barrier and push myself beyond my limits after eight days of doing virtually nothing. Jerome Boateng and I just tried to shut out all our physical problems, which we managed well. I'm not expecting any consequences, but I don't know how my foot will have dealt with the 120 minutes."