## 1 Introduction  
Time-domain ptychography refers to a specific class of iterative algorithms that reconstruct a complex valued waveform \(E(t)\) from a spectrogram,  
\[S(\tau,\omega)\propto\left|\int\limits_{-\infty}^{\infty}\mathrm{d}t\ E(t)\ P(t-\tau)\ \mathrm{e}^{-\mathrm{i}\omega t}\right|^{2}, \tag{1}\]  
with a known or unknown gate, \(P(t)\). The spectrogram itself is composed of spectra of the product field, \(E(t)P(t-\tau)\), recorded for a range of time delays, \(\tau\). Time-domain ptychography is derived from space-domain ptychography, which was originally proposed to solve the phase problem in crystallography [1]. The real space image of an unknown object is reconstructed iteratively from a series of far-field diffraction measurements. Each of those is recorded after shifting the object or the coherent illumination beam in the object plane. Far-field diffraction patterns must result from different, yet overlapping, areas of the object. Therefore, subsequent measurements must be transversely shifted by an amount smaller than the spatial extent, the \"support\" of the illumination beam. We have previously demonstrated that such concepts, including spatial reconstruction algorithms, can be transferred to the time domain [2; 3; 4; 5; 6] and thus yield numerous ramifications for ultra-fast optics, imaging and spectroscopy. In the time-domain, ptychography operates in the two conjugate one-dimensional spaces of time and frequency. That is, the two-dimensional real space vector, \(\overline{\tau}\), and the wave vector, \(\overline{k}\), are substituted by \(t\) and \(\omega\); the object, \(E(\overline{\tau})\), and the illumination beam, \(P(\overline{r})\), are substituted by the object waveform, \(E(t)\), and a gate, \(P(t)\), and instead of far-field diffraction images, \(S(\vec{k})\), spectra, \(S(\omega)\), are recorded. Combining the spectra recorded at different time delays, \(\tau\), results in a spectrogram, as described by Equation (1). We have previously demonstrated several applications of time-domain ptychography, for example, characterizing both typical and optically shaped ultrafast laser pulses [2; 3] as well as super continuum pulses [6]. We have also used time-domain ptychography to reconstruct single attosecond XUV pulses or trains of attosecond XUV pulses from streaked photo-electron spectra [4]. In photo-electron streaking, attosecond XUV pulses ionize an ensemble of atoms in the presence of an infrared field. The resulting energy spectrum of the photo-electrons released along this so-called streak field polarization direction reveals either excess or diminished kinetic energy, depending on the phase and strength of that streak field. Measured as a function of the time delay between XUV and streak pulse, a detailed theoretical analysis showed that this process can be made to reveal a spectrogram, given certain assumptions. Therefore, time-domain ptychography could be used to reconstruct the XUV pulses from the streaked photo-electron spectrogram. Photo-electron streaking has become the technique of choice for characterizing pulses emitted by X-ray free electron lasers (XFEL) [7]. Given the fluctuations that are typical of the electron acceleration as well as the stochastic nature of the self-amplification of spontaneous emission (SASE) nature of such sources, single shot measurement, not scanning, must be used for pulse characterization; no two pulses look alike at an FEL. In this manuscript, therefore, we propose a combination of so-called angular streaking [8; 9] with time-domain ptychography for the reconstruction of individual SASE X-ray pulses on a shot-to-shot basis. This manuscript is organized as follows. Section 2 summarizes the theoretical background of time-domain ptychography as applied to angular electron streaking for individual X-ray pulses, describes the X-ray pulse structures generated in a typical XFEL, describes photo-electron streaking with streak fields of different polarization states and discusses the extended ptychographic iterative engine (ePIE).