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Mathematical Sciences (Kuroda) Lab
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Research on equations that clean up noisy images, approached from the viewpoint of variational problems.
AboutAbout the lab
The mathematics of equations that clean up images
Prof. Kuroda studies variational problems and partial differential equations with variational structure.
The main subject is the total variation flow equation, which arises when removing noise from images.
Because this equation is strongly singular, the lab uses subdifferentials of convex functions and the nonlinear semigroup theory they generate to study whether solutions exist and how they behave.
The lab also covers total variation flow equations that appear in mathematical models of crystal growth, and recently it has taken an interest in phenomena that occur when a domain becomes thin and degenerates.
Key terms
- Variational problem
- A problem of finding the shape or function that minimizes a quantity such as length, area, or energy
- Total variation
- The total amount by which a function's values move up and down. In an image, finer noise makes it larger
- Subdifferential
- A method that, even at points with a kink where the derivative does not exist, considers the set of slopes that serves in place of the derivative
ResearchResearch themes
The lab is interested in partial differential equations with variational structure, especially the relationship between subdifferential equations of convex functions and Mosco convergence. The subject is the total variation flow equation, which appears in image denoising and in mathematical models of crystal growth.
*Illustrative imageTotal Variation Flow for Cleaning Up Images
The lab studies a differential equation called the total variation flow equation, which appears in the process of removing noise from images. Besides image processing, the same equation also appears in mathematical models of crystal growth.
*Illustrative imageSubdifferentials and Nonlinear Semigroup Theory
The total variation flow equation is strongly singular. Using subdifferentials of convex functions and the nonlinear semigroup theory they generate, the lab investigates whether solutions exist and how they change over time.
*Illustrative imagePhenomena in Thinning, Degenerating Domains
Recently, the lab has also become interested in phenomena that occur when a domain becomes thin and degenerates.
MembersFaculty
- Associate ProfessorHirotoshi Kuroda
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Report an errorAbout the images
The top and research theme images are illustrative and were not taken at the lab. They are cropped from Wikimedia Commons images. Cropped CC BY-SA images are provided under the same CC BY-SA license as the original (version shown on each line).
- Top image: “Newly fallen snow crystals accumulating on a window sill” W.carter / CC0 (cropped)
- Total Variation Flow for Cleaning Up Images: “050.Pattern.050.Image.noise” Bautsch / CC0 (cropped)
- Subdifferentials and Nonlinear Semigroup Theory: “Blue colour drops” Sandya / CC BY-SA 3.0 (cropped)
- Phenomena in Thinning, Degenerating Domains: “Crystals growing from melted sulfur” AlexmarPhoto / CC BY 4.0 (cropped)
- Related lab (Mathematical Sciences (Masaki) Lab): “Baltic Sea Wave (Cien Water)” Marek Nowocień / CC BY 2.5 (cropped)
- Related lab (Analysis Lab (Hamamuki)): “Real snowflake macro photo” Alexey Kljatov / CC BY-SA 4.0 (cropped)
- Related lab (Geometry Lab (Inoguchi)): “Hopf Fibration” Niles Johnson / CC BY 4.0 (cropped)
- Related lab (Geometry Lab (Kobayashi)): “Chen-Gackstatter-Thayer surfaces” Anders Sandberg / CC BY-SA 3.0 (cropped)
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