We present new {it Hubble Space Telescope} images of high-velocity H-$alpha$ and Lyman-$alpha$ emission in the outer debris of SN~1987A. The H-$alpha$ images are dominated by emission from hydrogen atoms crossing the reverse shock. For the first time we observe emission from the reverse shock surface well above and below the equatorial ring, suggesting a bipolar or conical structure perpendicular to the ring plane. Using the H$alpha$ imaging, we measure the mass flux of hydrogen atoms crossing the reverse shock front, in the velocity intervals ($-$7,500~$<$~$V_{obs}$~$<$~$-$2,800 km s$^{-1}$) and (1,000~$<$~$V_{obs}$~$<$~7,500 km s$^{-1}$), $dot{M_{H}}$ = 1.2~$times$~10$^{-3}$ M$_{odot}$ yr$^{-1}$. We also present the first Lyman-$alpha$ imaging of the whole remnant and new $Chandra$ X-ray observations. Comparing the spatial distribution of the Lyman-$alpha$ and X-ray emission, we observe that the majority of the high-velocity Lyman-$alpha$ emission originates interior to the equatorial ring. The observed Lyman-$alpha$/H-$alpha$ photon ratio, $langle$$R(Lalpha / Halpha)$$rangle$ $approx$~17, is significantly higher than the theoretically predicted ratio of $approx$ 5 for neutral atoms crossing the reverse shock front. We attribute this excess to Lyman-$alpha$ emission produced by X-ray heating of the outer debris. The spatial orientation of the Lyman-$alpha$ and X-ray emission suggests that X-ray heating of the outer debris is the dominant Lyman-$alpha$ production mechanism in SN 1987A at this phase in its evolution.
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