Picture of Brad Lucier

Bradley J. Lucier

Professor Emeritus of Mathematics and Computer Science
Purdue University
Math 809
150 North University Street
W. Lafayette, IN 47907-2067

lucier@math.purdue.edu

ORCID iD iconorcid.org/0000-0003-3808-939X

A professor can do as he pleases, but a professor emeritus can do as he damn well pleases. Nicolaas Bloembergen

You're an evil man. Linus Torvalds
Thank you.

Publications

A complete list of my papers can be found on my "bio" page . You can download various papers by me and my previous students on image processing and wavelets and numerical methods for partial differential equations and related topics.

New: A publication that doesn't quite fit in with those two lists is How I First Heard About Calculus which appears in the Journal of Humanistic Mathematics.

Conjecture

In Pointwise Besov Space Smoothing of Images we consider the following problem: Let an image $\epsilon$ be defined on $I=[0,1]^2$, the unit square, divided into $N\times N$ subsquares $I_i$ indexed by the multi-index $i=(i_1,i_2)$, $0\leq i_1,i_2<N$. The value of the image $\epsilon$ on $I_i$ is given by $\epsilon_i$, where the $\epsilon_i$ are i.i.d. $N(0,1)$ random variables. For various reasons it's interesting to estimate the expected value of the norm of $\epsilon$ in the dual space of $\text{BV}(I)$, the space of functions of bounded variation on $I$. Specifically, we want to estimate $$E\left(|\epsilon|_{\text{BV(I)}^*}\right)=E\left(\sup_{\|f\|_{\text{BV}(I)}\leq 1}\left|\int_I f\epsilon\right|\right).\tag{1}$$ We note in Section 8 of the previously mentioned paper that because of the strict norm inclusions $B^1_1(L_1(I))\subset\text{BV}(I)\subset B^1_\infty(L_1(I))$ between $\text{BV}(I)$ and the Besov spaces $B^1_1(L_1(I))$ and $ B^1_\infty(L_1(I))$, we can exploit wavelet atomic decompositions of the Besov spaces to show that there exist two constants $C_1$ and $C_2$ such that $$ C_1\frac{|\log N|^{1/2}}N\leq E\left(|\epsilon|_{\text{BV(I)}^*}\right)\leq C_2\frac{|\log N|^{3/2}}N. $$ I conjecture that the tighter bounds $$ C_1\frac{\log N}N\leq E\left(|\epsilon|_{\text{BV(I)}^*}\right)\leq C_2\frac{\log N}N. $$ hold. Update: We updated the supremeum in Equation (1) to replace $|f|_{\text{BV}(I)}$ by $\|f\|_{\text{BV}(I)}=|f|_{\text{BV}(I)}+\|f\|_{L^1(I)}$. Alternately, one could use $|f|_{\text{BV}(I)}$ and require $\int_I f=0$.

Courses

Test Images

High quality, grey-scale, test images for purposes of testing algorithms on natural images. Some results from my compression program are given.

The original Kodak Photo CD Photo Sampler color images from which the greyscale images above were derived, and the copyright terms under which the Kodak Photo CD Photo Sampler images were released. If anyone can tell me a standard way to convert these to YUV, YCrCb, or RGB, I'd be interested. Later note: it seems that various versions of Photoshop have good color conversion between Kodak Photo CD format and RGB, so that is what I used for the color versions. I'm still not happy with Photoshop's conversion to grey-scale. You can find a lot of useful information about working with Photo CD images at Ted's Unofficial Kodak Photo CD Homepage.

Non-blogs

Software

Books

Personal

My wife Maureen is an ATA-certified translator from French into English.

Math Department Home Page