If corporate America needs a role model about reinvention, it should turn to Madonna.

Although the pop superstar is not as wealthy as Warren Buffett and might think QE2 is a luxurious cruise ship, she knows plenty about transforming her image to appeal to the ever-changing tastes of pop music fans. She even called her sixth world tour “The Re-Invention Tour.” Madonna’s strategy has enabled her to sell millions of albums and last for decades in a business that chews up and spits out acts at an alarming speed

Reinvention is difficult for pop stars and politicians alike, but it’s especially difficult for corporations, which spend tens of millions of dollars trying to convey a certain image to the public. If that message no longer resonates, it’s difficult to find a new strategy that works, particularly when economic times are tough. But companies eventually must innovate if they are to survive. This requires a willingness to take big risks and make big changes for a potential big payout.

Sometimes, companies make a change so massive that it alters the DNA of the companies involved. Here are three companies that have survived:

IBM

When Louis Gerstner became CEO of IBM (NYSE:IBM) in 1993, he was shocked by the poor state of the company. Big Blue had become complacent, suffocated by a stodgy, paternalistic culture. Businesses were run like personal fiefdoms, meaning that one end of the Armonk, N.Y.-based company didn’t know what the other end was doing, The results were chaos.

“An internal company study had revealed that Big Blue had failed to capture value from 29 separate technologies and businesses that the company had developed,” according to a study published last year by the Stanford Graduate School of Business.

One reason why so many promising technologies, including Internet routers (Cisco) and MS-DOS (Microsoft), passed the company by was that IBM executives were so focused on reaching short-term goals that they missed the bigger picture. Gerstner orchestrated a massive overhaul to change IBM’s culture by laying off thousands and unloading poorly performing assets.

After Gerstner retired in 2002, the changes continued. In 2004, IBM unload its PC unit to China’s Lenovo Group for $1.75 billion and took a minority stake in the joint venture running the business. This allowed IBM to focus on more profitable business customers. It turned out to be the correct move.

Wall Street has rewarded IBM’s shareholders handsomely. Its shares have more than doubled during the past five years.

Nokia

The name Jorma Ollila probably is not as familiar to most Americans as Gerstner, Apple (NASDAQ:AAPL) co-founder Steve Jobs, and co-founder and CEO of Google (NASDAQ:GOOG) Larry Page. That low profile suits the chairman of Nokia (NYSE:NOK) just fine.

Ollila, who was CEO from 1992 to 2006, transformed the sleepy Finnish conglomerate into the world’s biggest maker of cell phones. One of his first decisions was to focus on telecommunications and shed its rubber, cable and consumer electronics divisions.

The results were outstanding. Sales and profit soared thanks to Ollila’s emphasis on building quality phones with a long battery life. By 1998, Nokia held the top spot in the worldwide mobile phone market, and NOK more than doubled between January 1997 ($56.50) and January 1999 ($128.19).

In recent years, however, Nokia has suffered the other side of the innovation game, and the company has been eclipsed as cell phones evolved into smartphones. Ollila’s successor as CEO, Olli-Pekka Kallasvuo, was fired in 2010 and replaced by Microsoft executive Steven Elop. He described Nokia’s second-quarter 2011 results, where profit plunged 44%, as “clearly disappointing.”

Shares of Nokia are down nearly 73% over the past five years. Ollila, who also serves as non-executive chairman of Royal Dutch Shell (NYSE:RDS.A), will leave his job at Nokia in 2012.

Apple

In 2001, Apple CEO Steve Jobs was in the middle of rescuing the company he co-founded from oblivion. Mac sales were on the rise thanks to the candy-colored iMac, so Apple executives began to hunt for something that could boost sales even further. According to a 2006 Wired story, the answer was music.

“Music lovers were trading tunes like crazy on Napster,” the article says. “They were attaching speakers to their computers and ripping CDs. The rush to digital was especially marked in dorm rooms — a big source of iMac sales — but Apple had no jukebox software for managing digital music.”

The company quickly found that the MP3s on the market stunk. Under Jobs’ guidance, the Cupertino, Calif.-based company developed a blockbuster product along with a music downloading platform so simple that even the least technologically savvy people could work it. The impact was astonishing.

In 2001, digital music was a reminder to the record industry about how they were being ruined by piracy. Now, The International Federation of the Phonographic Industry estimates that digital music revenues rose 6% globally in 2010 to $4.6 billion, accounting for 29% of record companies’ trade revenue.

And the iPod’s impact on Apple can’t be understated. Shares of the company are up about 5,000% since the iPod was introduced.

Jonathan Berr does not own shares of any of any companies mentioned here. Follow him on Twitter at @jdberr.