Weak gravitational lensing has the potential to place tight constraints on the equation of the state of dark energy . However , this will only be possible if shear measurement methods can reach the required level of accuracy . We present a new method to measure the ellipticity of galaxies used in weak lensing surveys . The method makes use of direct deconvolution of the data by the total Point Spread Function ( PSF ) . We adopt a linear algebra formalism that represents the PSF as a Toeplitz matrix . This allows us to solve the convolution equation by applying the Hopfield Neural Network iterative scheme . The ellipticity of galaxies in the deconvolved images are then measured using second order moments of the autocorrelation function of the images . To our knowledge , it is the first time full image deconvolution is used to measure weak lensing shear . We apply our method to the simulated weak lensing data proposed in the GREAT10 challenge and obtain a quality factor of Q = 87 . This result is obtained after applying image denoising to the data , prior to the deconvolution . The additive and multiplicative biases on the shear power spectrum are then \sqrt { \mathcal { A } } = +0.09 \times 10 ^ { -4 } and \mathcal { M } / 2 = +0.0357 respectively .