Lawrence “Larry” Shepp (1936–2013) was a polymathic mathematician whose work bridged the often-distant worlds of abstract probability theory and life-saving medical technology. Over a career spanning five decades—most notably during the "Golden Age" of Bell Labs—Shepp transformed how we look inside the human body and how we understand randomness in the physical world.
1. Biography: From Brooklyn to Bell Labs
Lawrence Alan Shepp was born on September 9, 1936, in Brooklyn, New York. A child of the New York public education system, he attended the legendary Stuyvesant High School before earning his undergraduate degree at Brooklyn College.
His mathematical trajectory was set at Princeton University, where he earned his Ph.D. in 1961. He studied under the titan of probability, William Feller, whose intuitive approach to mathematics deeply influenced Shepp’s own "problem-first" philosophy.
In 1962, Shepp joined AT&T Bell Laboratories in Murray Hill, New Jersey. At the time, Bell Labs was perhaps the most productive scientific environment in history. Shepp thrived there for 34 years, working alongside luminaries like John Tukey and Claude Shannon. After retiring from Bell Labs in 1996, he transitioned to academia, serving as the Board of Governors Professor of Statistics at Rutgers University and later as a professor at the Wharton School of the University of Pennsylvania. He remained active in research until his death on April 23, 2013.
2. Major Contributions: The Math of Seeing
Shepp was rare among mathematicians for his ability to solve "hard" pure problems while simultaneously inventing tools for industry.
Computerized Tomography (CT) and PET Scans
Shepp’s most tangible contribution to humanity is the development of the algorithms that make modern medical imaging possible. In the early 1970s, he co-developed the filtered back-projection algorithm, which allows a computer to take X-ray data from multiple angles and reconstruct a clear, cross-sectional image of the body. He later applied similar logic to Positron Emission Tomography (PET), using maximum likelihood estimators to improve image clarity.
The Shepp-Logan Phantom
To test his imaging algorithms, Shepp (along with Benjamin F. Logan) created a mathematical model of a human head composed of various ellipses. Known as the "Shepp-Logan Phantom," this image remains the industry standard for testing the accuracy of any new medical imaging reconstruction algorithm.
The XYZ Inequality
In 1982, Shepp proved a long-standing conjecture in combinatorics known as the XYZ inequality. This theorem deals with "partially ordered sets" (posets) and provides a rigorous proof for certain intuitive ideas about how different rankings or orderings correlate with one another.
Probability and Stochastic Processes
Shepp made fundamental discoveries regarding Brownian motion (the random motion of particles) and Gaussian processes. He solved the "Parking Problem"—a mathematical inquiry into how many random intervals (cars) can fit on a line (a street)—and developed the "Shepp’s Theorem" regarding the equivalence of Gaussian measures.
3. Notable Publications
Shepp was a prolific author with over 200 papers. His most influential works include:
- The Fourier reconstruction of a head section (1974, with B.F. Logan): Published in the IEEE Transactions on Nuclear Science, this paper introduced the Shepp-Logan filter and revolutionized the speed and quality of CT scans.
- Maximum likelihood reconstruction for emission tomography (1982, with Y. Vardi): This paper laid the statistical groundwork for modern PET scans, moving the field toward more accurate, iterative reconstruction methods.
- The XYZ inequality (1982): Published in the Annals of Probability, this settled a major question in the field of combinatorics and order theory.
- Connectedness of certain random graphs (1962): One of his early influential works in graph theory and probability.
4. Awards & Recognition
Shepp’s rare ability to impact both medicine and mathematics led to honors from multiple disciplines:
- National Academy of Sciences (1989): Elected for his contributions to mathematics and statistics.
- National Academy of Medicine (formerly Institute of Medicine): He was one of the very few mathematicians ever elected to this body, recognizing that his algorithms saved countless lives.
- Distinguished Scientist Award (SIAM): Awarded by the Society for Industrial and Applied Mathematics.
- Fellowships: He was a Fellow of the Institute of Mathematical Statistics (IMS) and the American Statistical Association (ASA).
5. Impact & Legacy
Larry Shepp’s legacy is twofold. In the medical realm, every time a patient undergoes a CT or PET scan, they are benefiting from Shepp’s mathematics. His work turned a blurry, theoretical possibility into a precise diagnostic tool.
In the mathematical realm, he is remembered for his "Feller-esque" intuition. He had an uncanny ability to look at a complex physical phenomenon—like a car parking or a signal traveling through a wire—and translate it into a solvable probability problem. He mentored dozens of Ph.D. students who went on to lead departments in statistics and financial mathematics.
6. Collaborations
Shepp was a social mathematician who thrived on collaboration.
- Benjamin Logan: His partner at Bell Labs for the seminal imaging work.
- Yehuda Vardi: A key collaborator on the statistical approach to PET scans.
- Persi Diaconis: The famed Stanford mathematician and magician, with whom Shepp discussed various problems in probability and card shuffling.
- The "Bell Labs Cohort": He frequently collaborated with Henry Landau and David Slepian on signal processing and information theory.
7. Lesser-Known Facts
- The "Phantom" is not a person: While it sounds like a ghost, the "Shepp-Logan Phantom" is actually a purely mathematical construct—a set of ten ellipses of varying densities. It is widely mistaken by students for a real scan of a human skull.
- A "Problem Solver" for Hire: At Bell Labs, engineers would often bring Shepp "impossible" problems. He was known for his "open-door" policy, where he would listen to a problem from a completely different field and sketch out a probabilistic solution on a napkin.
- Financial Math Pioneer: Though less famous for it, his work on "optimal stopping" and Brownian motion provided some of the foundational math later used in the Black-Scholes model and modern quantitative finance.
- A Character of New York: Despite his high academic standing, Shepp retained a sharp, Brooklyn-born wit and a no-nonsense attitude toward "overly fancy" math that didn't solve a real-world problem. He famously valued intuition and "the right picture" over dense, unreadable proofs.