Both Canon SX1 IS and Canon G12 have 10.0 MP resolution sensors but Canon SX1 IS's sensor is 1/2.3-inch (6.17 x 4.55 mm ) and Canon G12's sensor is 1/1.7-inch (7.44 x 5.58 mm ).
Since Canon G12's has a larger sensor area with the same resolution, this means that it also has a larger pixel area hence better light collecting capacity for a given aperture compared to Canon SX1 IS.
Below you can see the SX1 IS and G12 sensor size comparison.
As seen above, Canon G12 has a 1.5x Larger sensor area than Canon SX1 IS. Larger sensors give photographer more control on the depth of field and blurry background compared to smaller sensor when shot in same focal length and aperture.