Dithered quantization and noise shaping is well known in the audio community. The image processing community seems to be aware of this same theory only in bits and pieces, and frequently under conflicting terminology. This thesis attempts to show that dithered quantization of images is an extension of dithered quantization of audio signals to higher dimensions. Dithered quantization, or ``threshold modulation'', is investigated as a means of suppressing undesirable visual artifacts during the digital quantization, or requantization, of an image. Special attention is given to the statistical moments of the resulting error signal. Afterwards, noise shaping, or ``error diffusion'' methods are considered to try to improve on the dithered quantization technique. We also take time to develop the minimum-phase property for two-dimensional systems. This leads to a natural extension of Jensen's Inequality and the Hilbert transform relationship between the log-magnitude and phase of a two-dimensional system. We then describe how these developments are relevant to image processing.

Dithered quantization and noise shaping is well known in the audio community. The image processing community seems to be aware of this same theory only in bits and pieces, and frequently under conflicting terminology. This thesis attempts to show that dithered quantization of images is an extension of dithered quantization of audio signals to higher dimensions. Dithered quantization, or ``threshold modulation'', is investigated as a means of suppressing undesirable visual artifacts during the digital quantization, or requantization, of an image. Special attention is given to the statistical moments of the resulting error signal. Afterwards, noise shaping, or ``error diffusion'' methods are considered to try to improve on the dithered quantization technique. We also take time to develop the minimum-phase property for two-dimensional systems. This leads to a natural extension of Jensen's Inequality and the Hilbert transform relationship between the log-magnitude and phase of a two-dimensional system. We then describe how these developments are relevant to image processing.

This dissertation consists of three essays. The first essay studies the impact of indexing on stock price efficiency. Indexing has experienced substantial growth over the last two decades because it is an effective way of holding a diversified portfolio while minimizing trading costs and taxes. In this paper, we focus on one negative externality of indexing: the effect on efficiency of stock prices. Based on a sample of large and liquid U.S. stocks, we find that greater indexing leads to less efficient stock prices, as indicated by stronger post-earnings-announcement drift, greater deviations of stock prices from the random walk and greater return predictability from lagged order imbalances. We conjecture that reduced incentives for information acquisition and arbitrage induced by indexing are probably the main cause of the degradation in price efficiency, but we find no evidence supporting a direct impact from passive trading or any effect through liquidity. The second essay investigates the effect of price inefficiency on idiosyncratic risk and stock returns. I finds that price inefficiency in individual stocks contributes to expected idiosyncratic volatility. If idiosyncratic risk is priced, greater price inefficiency could be associated with higher expected returns. Consistent with this hypothesis, this paper then finds a positive relation between price inefficiency and future stock returns. This return premium of price inefficiency is not explained by traditional risk factors, illiquidity, or transactions costs. It is also evidently different from the return bias related to Jensen's inequality. This paper thus provides new insights about the determinants of expected stock returns, and new supporting evidence that idiosyncratic risk is priced. The third essay examines whether the upward return bias generated by Jensen's inequality could lead to better performance of equally-weighted (EW) indexes than value-weighted (VW) index when stock prices are not fully efficient. We find that, for a wide range of U.S. stock indexes, EW indexes deliver better four-factor adjusted returns than VW ones do even after deducting transaction costs. Consistent with our hypothesis that the outperformance of EW indexes comes from mispricing, we find that this outperformance concentrates in stocks with greater mispricing, as measured by deviation of stock prices from random walk. Findings in this essay not only imply a potentially winning investment strategy, but also provide new insight into a long-term debate on causes of the outperformance of the EW indexes. / Ph. D.