I consider myself a visual problem solver," says freelance artist Laura Ljungkvist. "I get faxes every week with problems I solve for clients. But making a book, I have to solve my own problem, and that's new for me."

For her debut picture book, Toni's Topsy-Turvy Telephone Day (Abrams), Ljungkvist wrestled with sequential imagery and storytelling. "I have never started over so many times on a piece of art, never lost so much sleep, never been so insecure in my process," she says, with mingled dismay and appreciation. She settled on a unified design in which a fluid black line twirls across every spread, resembling a phone cord and outlining characters' facial features as well. Although the line literally connects the speakers, they misunderstand one another's spoken messages.

In Toni, the pictures are done in Ljungkvist's signature style, a vibrant op-art array of lines and shapes. She communicates in abstract, ultramodern symbols (she calls them "advanced traffic signs") on a flat, opaque plane. Not surprisingly, one influence is J.otto Seibold: "Mr. Lunch is my idol," she says, referring to Seibold's dog hero. But where Seibold renders in Adobe software, Ljungkvist painstakingly mixes and applies gouache. "I could just click a mouse and make it happen," she says, "but there are imperfections that you might not see immediately. Subconsciously, I think you feel them."

Artistic success came quickly to Ljungkvist, who left her native Sweden for Manhattan in 1993. "New York welcomed me with open arms," she says happily. Her first assignment in the States was for the New Yorker, and gigs with the New York Times and Harper's Bazaar followed.

The book business was less receptive. Ljungkvist created a picture-book dummy (not Toni) in 1995. Because her agent is not in publishing, she pursued editors herself. "I didn't know the politics or the people, and being an impatient person, I found it frustrating," she says. There were no takers.

Ljungkvist might have abandoned this direction if not for a traumatic turn of events. In 1997, she suffered a brain injury in a horseback-riding accident, and spent three months in the hospital and six more months recuperating. During her rehabilitation, she refused to paint. "I was afraid I had lost my problem-solving skills," she recalls. When she found she could work again, her first goal was paying the rent; her second was making a picture book. "It took a hit on the head to make me decide, I'm going to do this again," she says.

On her second foray into publishing, Ljungkvist met Abrams editor Howard Reeves, who rejected her first pitch, but wanted to see more work. Cautiously optimistic, Ljungkvist consulted a friend, freelance editor David Colbert. "He sat down and taught me to conceptually understand what makes a children's book sing, and what makes you want to turn the page."

Ljungkvist found the germ of an idea in an old assignment for Mohawk Papers, in which she had painted two people connected by a phone line. As Toni took shape, she drew thumbnail sketches of a party host who calls six guests, unaware that the invitees cannot hear clearly. Then "I sat down with a dictionary, went through from A to Z," and wrote down like-sounding words; one guest mistakenly brings "cats and baboons" instead of "hats and balloons."

At Abrams, "Howard took two seconds to say, yes, we'll do it," Ljungkvist remembers, with evident relief. She hesitates to get specific about her next book project with Reeves ("It's full of patterns and fun things to look at"), but she anticipates more picture books and perhaps a line of "toys and other products." Whatever these objects might be, Ljungkvist wants them to be meaningful for young readers. She takes painting as a serious responsibility. "I have no complaints, no complaints!" she says, in the voice of one who takes nothing for granted. "I'm extremely fortunate to make a living at this—it's my 'hobby.' If I had to go to an office from 9 to 5 every day, I'd do this at night."