We demonstrate a general, computer automated procedure that inverts the q-space scattering data measured by spectroscopic imaging scanning tunneling microscopy (SI-STM) to determine the k-space scattering structure. This allows a detailed examination of the k-space origins of the quasiparticle interference (QPI) pattern in Bi2Sr2CaCu2O8+x. This new method allows the measurements of the differences between the positive and negative energy dispersions, the gap structure and it also measures energy dependent scattering length scale. Furthermore, the transitions between the dispersive QPI, the checkerboard and the pseudogap are mapped in detail allowing the exact nature of these transitions to be determined for both positive and negative energies. We are also able to measure the k-space scattering structure over a wide range of doping (p ~ 0.22 to 0.08), including regions where the octet model is not applicable. Our technique allows a complete picture of the k-space origins of the spatial excitations in Bi2Sr2CaCu2O8+x to be mapped out, providing for better comparisons between SI-STM and other experimental probes of the band structure and validating our new general approach for determining the k-space scattering origins from SI-STM data.
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