Context. Recent developments in time domain astronomy, such as Zwicky Transient Facility (ZTF), have made it possible to conduct daily scans of the entire visible sky, leading to the discovery of hundreds of new transients every night. Among these detections, 10 to 15 of these objects are supernovae (SNe), which have to be classified prior to cosmological use. The spectral energy distribution machine (SEDM) is a low-resolution (ℛ ~ 100) integral field spectrograph designed, built, and operated with the aim of spectroscopically observing and classifying targets detected by the ZTF main camera. Aims. As the current pysedm pipeline can only handle isolated point sources, it is limited by contamination when the transient is too close to its host galaxy core. This can lead to an incorrect typing and ultimately bias the cosmological analyses, affecting the homogeneity of the SN sample in terms of local environment properties. We present a new scene modeler to extract the transient spectrum from its structured background, with the aim of improving the typing efficiency of the SEDM. Methods. HyperGal is a fully chromatic scene modeler that uses archival pre-transient photometric images of the SN environment to generate a hyperspectral model of the host galaxy. It is based on the cigale SED fitter used as a physically-motivated spectral interpolator. The galaxy model, complemented by a point source for the transient and a diffuse background component, is projected onto the SEDM spectro-spatial observation space and adjusted to observations, and the SN spectrum is ultimately extracted from this multi-component model. The full procedure, from scene modeling to transient spectrum extraction and typing, is validated on 5000 simulated cubes built from actual SEDM observations of isolated host galaxies, covering a broad range of observing conditions and scene parameters. Results. We introduce the contrast, c, as the transient-to-total flux ratio at the SN location, integrated over the ZTF r-band. From estimated contrast distribution of real SEDm observations, we show that HyperGal correctly classifies ~95% of SNe Ia, and up to 99% for contrast c ≳ 0.2, representing more than 90% of the observations. Compared to the standard point-source extraction method (without the hyperspectral galaxy modeling step), HyperGal correctly classifies 20% more SNe Ia between 0.1 < c < 0.6 (50% of the observation conditions), with less than 5% of SN Ia misidentifications. The false-positive rate is less than 2% for c > 0.1 (> 99% of the observations), which represents half as much as the standard extraction method. Assuming a similar contrast distribution for core-collapse SNe, HyperGal classifies 14% additional SNe II and 11% additional SNe Ibc. Conclusions. HyperGal has proven to be extremely effective in extracting and classifying SNe in the presence of strong contamination by the host galaxy, providing a significant improvement with respect to the single point-source extraction.
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