1. **State the problem:** Find the image of the point $(-10, -4)$ after a dilation centered at the origin with a scale factor of $\frac{1}{2}$.\n\n2. **Formula for dilation:** The image point $(x', y')$ after dilation with center at the origin and scale factor $k$ is given by:\n$$\left(x', y'\right) = \left(kx, ky\right)$$\nwhere $(x, y)$ is the original point.\n\n3. **Apply the formula:** Here, $k = \frac{1}{2}$, $x = -10$, and $y = -4$. Substitute these values:\n$$\left(x', y'\right) = \left(\frac{1}{2} \times (-10), \frac{1}{2} \times (-4)\right)$$\n\n4. **Calculate each coordinate:**\n$$x' = \frac{1}{2} \times (-10) = -5$$\n$$y' = \frac{1}{2} \times (-4) = -2$$\n\n5. **Final answer:** The image of the point $(-10, -4)$ after dilation is:\n$$\boxed{(-5, -2)}$$