It has been a deliberately slow, painstaking process, but 18 months after Mark Rothko's Black on Maroon was vandalised with quick drying and theoretically indelible graffiti ink Tate Modern has revealed the successful results of one of its most difficult restoration projects.

The work has gone back on public display alongside other paintings which make up Rothko's Seagram murals series. The Tate's director, Sir Nicholas Serota, said the restoration had been "far more successful than any of us dared hope".

Serota recalled the Sunday in October 2012 when his phone rang telling him that someone had stepped over a barrier and vandalised one of the gallery's most cherished possessions. "To hear that one has been damaged … it's an appalling feeling, a sickening feeling, partly because you have no idea whether it's going to be possible to restore it."

He said the Tate conservation team was one of the best in the world and had been given all the time and resources they needed. "Right from the outset I said, 'This takes as long as this takes. There is no rush on this.'"

Restoration team members spoke about the challenges of repairing the damage caused by Wlodzimierz Umaniec, also known as Vladimir Umanets, who was sentenced to two years in prison for writing his name and the words "a potential piece of yellowism" on the Rothko.

The vandalised and restored section of Black on Maroon 1958 Photograph: Sam Drake/Tate/PA

The ink he used was one of the most indelible there is. It is "designed specifically to be very black, very quick drying, very highly staining and permanent", said Patricia Smithen, Tate's head of conservation.

The job was further complicated by Rothko himself, as his paintings are not just oil on canvas - he painted with different kinds of paint in very deliberate and thin layers. "The surface is really delicate and it turned out that most of the solvent systems that could dissolve and remove the ink could potentially damage the painting as well," said Smithen.

The team, including Tate conservation scientist Bronwyn Ormsby, spent nine months figuring out what solvents and cleaning methods to use on the 1958 canvas.

Then it was down to conservator Rachel Barker who, using a zero-sized brush, finally applied a mix of benzyl alcohol and ethyl lactate on to a miniscule area of the black ink. She vividly recalls the day. "It is like a little bit of benzyl alcohol ethyl lactate etched in my brain, it was just the most amazing moment … the relief was palpable in the room."

Working on areas as small as 2mm at a time under a microscope at 20 times magnification, Barker slowly removed the ink speck by speck before embarking on the job of restoring Rothko's colour.

Serota said restorations of this kind had often involved extensive reworking of paintings. "We set out to do as little as possible" and it had been successful. "Ultimately now we have a painting which has been restored, the damage has been removed and what you see is what Rothko painted."

Umaniec initially said the act was part of a movement he called "yellowism." In a statement on Tuesday he apologised "to the British people" adding: "I suppose I wanted to change the art world ... but of course I did it in a very, very wrong way.

"I spent almost a year and a half in prison and the British people have paid huge restoration costs, so it definitely wasn't worth doing it. Probably the only good thing is that the art world has received a very strong message that something must be fundamentally changed about its process of restoration."

Serota said security had been reviewed but the last thing he wanted was to turn Tate Modern in to Fort Knox.

"We aim to make the works as available as possible to our visitors in the conditions in which artists originally made it. We will and do review our processes on security but essentially this is a gallery not a prison."

The black ink, described by conservators as a slick, did seep right through to the back of the canvas in places and the damage will always remain under the surface. But on the surface it is now invisible to the human eye.

The Rothko family donated a maroon primed canvas to help in the testing process. Christopher Rothko, the late artist's son, said: "The Rothko family has been repeatedly impressed by the thoroughness and dedication of the Tate conservation team. They have realised the only satisfactory resolution to a terrible situation: the work is once again on display for the public as our father intended."

Rothko donated his murals, originally commissioned for the Four Seasons restaurant in Manhattan, to Tate in 1969 and they arrived the following year. Tragically, it was also the day of his suicide.

Serota said the attack had, in effect, been on all nine murals.

During Umaniec's trial lawyers estimated the painting's value at between £5m and £9m but Serota said he knew neither the restoration cost nor how much the painting was worth. "We're never going to sell it," he said.