科学界和鱼群

                                                                                                                                    邹谋炎

在80年代，法国的几个研究者发表了关于小波信号处理的奠基性工作。其中Stephane Mallat 提出的多分辨率分析和快速算法，推动了小波理论的应用发展。Mallat 在纽约和巴黎的大学中担任了多年的应用数学教授，他写的“信号处理的小波引导”为本领域的研究者所推崇。该书第3版 （2009）序言表述了作者的心路历程，可以供我们的研究者和数学教授们参考，现抄录如下：

“Preface to the Sparse Edition

I cannot help but find striking resemblances between scientific communities and schools of fish. We interact in conferences and through articles, and we move together while a global trajectory emerges from individual contributions. Some of us like to be at the center of the school, others prefer to wander around, and a few swim in multiple directions in front. To avoid dying by starvation in a progressively narrower and specialized domain, a scientific community needs also to move on. Computational harmonic analysis is still very much alive because it went beyond wavelets. Writing such a book is about decoding the trajectory of the school and gathering the pearls that have been uncovered on the way. Wavelets are no longer the central topic, despite the previous edition’s original title. It is just an important tool, as the Fourier transform is. Sparse representation and processing are now at the core.

In the 1980s, many researchers were focused on building time-frequency decompositions, trying to avoid the uncertainty barrier, and hoping to discover the ultimate representation. Along the way came the construction of wavelet orthogonal bases, which opened new perspectives through collaborations with physicists and mathematicians. Designing orthogonal bases with Xlets became a popular sport with compression and noise-reduction applications. Connections with approximations and sparsity also became more apparent. The search for sparsity has taken over, leading to new grounds where orthonormal bases are replaced by redundant dictionaries of waveforms.

During these last seven years, I also encountered the industrial world. With a lot of naiveness, some bandlets, and more mathematics, I cofounded a start-up with  Christophe Bernard, Jérome Kalifa, and Erwan Le Pennec. It took us some time to learn that in three months good engineering should produce robust algorithms that operate in real time, as opposed to the three years we were used to having for writing new ideas with promising perspectives. Yet, we survived because mathematics is a major source of industrial innovations for signal processing. Semiconductor technology offers amazing computational power and flexibility. However, ad hoc algorithms often do not scale easily and mathematics accelerates the trial-and-error development process. Sparsity decreases computations, memory, and data communications. Although it brings beauty, mathematical understanding is not a luxury. It is required by increasingly sophisticated information-processing devices.”

S. Mallat 书序言简译：                      

 “我不禁发现科学界和鱼群之间惊人的相似之处。我们在会议和通过文章相互接触，有人抛出一个贡献时就会出现一个全局性的轨迹，大家往一起凑。我们当中有人喜欢处于鱼群的中心，有人喜欢在周围游荡，也有人在前面朝多个方向游动。在一个越来越狭窄和专门的领域内为了不被饿死，科学界也需要往前凑。计算调和分析仍然非常活跃，因为它超出了小波的范畴。写本书的目的是为了解译群体的轨迹并把一路上发现的珍珠收集起来。小波不再是中心题目。它如同富氏变换那样只是一个重要工具。稀疏表示和处理当前处于核心位置。

 在 80 年代，许多研究人员集中关注建立时频分解，试图绕开不定性屏障，期望找出最终的表示方法。沿着构造小波正交基的路子，通过与物理学家和数学家的合作，开辟了新的前景。设计 X-let 相关的正交基变成了一种流行运动，连带着压缩和噪声抑制应用。近似和稀疏性的联系也变得更加明显。对稀疏性的研究已正当时，引导出新的基地：标准正交基被波形冗余词典所替代。

 在过去 7 年间我与工业界相遇。带着许多天真，和几个人共建了一个小公司。这让我们花了一点时间去学习到：在3个月内一个良好的工程应该生产出稳健算法可以实时运算；与此对照，在过去我们习惯于用3年时间来写那些有发展前景的新思想。是的，我们还活着，因为数学是信号处理工业创新的一个主要源泉。半导体技术提供了惊人的计算能力和灵活性。但是，特定算法常常不易估量，并且数学能够加速凑试发展过程。稀疏性使计算、存贮和数据搬运得以下降。虽然数学理解非常漂亮，但绝不奢侈。它是越来越精妙的信息处理元件所需要的。”

（译文摘自博文作者的中科院研究生院2011年暑期讲座材料“图像处理：某些发展动态和问题”）