Hadamard transform

About: Hadamard transform is a research topic. Over the lifetime, 7262 publications have been published within this topic receiving 94328 citations.

On Finite Part Integrals and Hadamard-Type Fractional Derivatives

Abstract: This paper is devoted to investigating the relation between Hadamard-type fractional derivatives and finite part integrals in Hadamard sense; that is to say, the Hadamard-type fractional derivative of a given function can be expressed by the finite part integral of a strongly singular integral, which actually does not exist. Besides, our results also cover some fundamental properties on absolutely continuous functions, and the logarithmic series expansion formulas at the right end point of interval for functions in certain absolutely continuous spaces.

Uniform recovery in infinite-dimensional compressed sensing and applications to structured binary sampling

Abstract: Infinite-dimensional compressed sensing deals with the recovery of analog signals (functions) from linear measurements, often in the form of integral transforms such as the Fourier transform. This framework is well-suited to many real-world inverse problems, which are typically modelled in infinite-dimensional spaces, and where the application of finite-dimensional approaches can lead to noticeable artefacts. Another typical feature of such problems is that the signals are not only sparse in some dictionary, but possess a so-called local sparsity in levels structure. Consequently, the sampling scheme should be designed so as to exploit this additional structure. In this paper, we introduce a series of uniform recovery guarantees for infinite-dimensional compressed sensing based on sparsity in levels and so-called multilevel random subsampling. By using a weighted $\ell^1$-regularizer we derive measurement conditions that are sharp up to log factors, in the sense they agree with those of certain oracle estimators. These guarantees also apply in finite dimensions, and improve existing results for unweighted $\ell^1$-regularization. To illustrate our results, we consider the problem of binary sampling with the Walsh transform using orthogonal wavelets. Binary sampling is an important mechanism for certain imaging modalities. Through carefully estimating the local coherence between the Walsh and wavelet bases, we derive the first known recovery guarantees for this problem.

Optimal transformations for optical multiplex measurements in the presence of photon noise.

Abstract: Hadamard multiplexing is a measurement strategy that yields best sensitivity improvements over scanning measurements for signal-independent detector noise. The presence of photon noise degrades the performance of Hadamard multiplexing because of the increase of photon noise by the superposition of multiple signals. I derive the reduction of the sensitivity gain of a Hadamard measurement and an upper limit for the gain of any cyclic multiplexing strategy in the presence of photon noise. This upper limit clearly exceeds the reduced Hadamard gain and can be achieved by multiplexing sequences that differ from Hadamard S sequences but also share some similarities with respect to their autocorrelation. Examples of such sequences are given. As the analysis shows, the presence of photon noise limits the gain of multiplexing strategies to a finite value, which depends on the ratio between photon noise and detector noise and cannot be exceeded by increasing the number of multiplexed channels. In addition, only switching multiplex schemes, which superpose either all the light or no light of individual channels, can achieve the upper limit of the gain.

Scalable complete complementary sets of sequences

Abstract: A family of complete complementary sets of sequences with closed-form expression is presented in this paper. Firstly, by introducing the notion of Golay (1961)-paired matrix and providing its synthesis algorithms, an orthogonal Golay-paired matrix called Golay-paired Hadamard matrix is derived. Then, general procedures for constructing mutually orthogonal Golay-paired matrices are proposed. Finally, the complete complementary sets of sequences represented by Golay-paired Hadamard matrices are generated and their scalability is illustrated. The unique properties of this new family of scalable complete complementary sets of sequences make it an ideal candidate for applications in future advanced signal processing and communications systems.

S-sequence spatially-encoded synthetic aperture ultrasound imaging [Correspondence]

Abstract: Synthetic transmit aperture (STA) ultrasound imaging offers near-ideal reconstruction across an entire field of view. This performance comes at the cost of SNR compared with scanning using only dynamic receive focusing. SNR may be enhanced by using spatial encoding using a Hadamard sequence. An encoding based on a Hadamard sequence has two main drawbacks: the array must be capable of transmitting a pulse and an inverted pulse at the same time, and the inverted transmission must be symmetrical with respect to the non-inverted transmission. These are often not the case in practice, and thus Hadamard encoding may require twice as many transmission events and special consideration of the inverted waveform. As an alternative, we propose the use of Ssequences, which are similar to Hadamard sequences, but use half the elements and do not require an inverted pulse. This encoding is implemented on a commercial ultrasound system and compared with STA imaging using single-element emissions and Hadamard encoding in terms of SNR and resolution using a point target. We find that the two encodings perform very similarly despite the increased transmit power and doubling of transmit events in our implementation of Hadamard imaging. Both encodings give up to 19 dB signal improvement over single-element STA imaging, while maintaining resolution. Finally, we show sample in vivo human carotid images with all three methods which illustrate the suitability of Ssequence- encoded STA imaging for a clinical setting.